\magnification=\magstep1
\font\s=cmr5
\font\a=cmbx6 scaled \magstep 4
\loadbold
\loadmsbm
\loadeufm
\redefine\l#1{\lambda_{#1}}
\define\ol{\overline \lambda}
\redefine\ll{\underline \lambda}
\define\lk{\underline \nu}
\define\ok{\overline \nu}
\define\oli{\overline \lambda_i}
\define\lli{\underline \lambda_i}
\define\os{\overline s}
\define\ls{\underline s}
\define\lm{\underline m}
\define\om{\overline m}
\define\lmu{\underline \mu}
\define\omu{\overline \mu}
\define\ra{\rightarrow}
\define\olij{\overline \lambda_{i_j}}
\define\llij{\underline \lambda_{i_j}}
\define\olin{\overline \lambda_{i_n}}
\define\llin{\underline \lambda_{i_n}}
\define\olijn{\overline \lambda_{{i_j},n}}
\define\llijn{\underline \lambda_{{i_j},n}}
\define\MN{{\s MIN}}
\define\MX{{\s MAX}}
\define\lam{\lambda}
\define\BbbR{\Bbb R}
\define\BbbZ{\Bbb Z}
\define\BbbC{\Bbb C}
\define\diam{\text{diam}}
\define\U{\text{U}}
\define\dU{|U|}
\define\g{\gamma}
\define\llijs{\olij^{\overline s}}
\define\din{\Delta_{i_1 \cdots i_n}}
\define\Lip{\text{Lip}}
\define\Holder{H\"older }
\redefine\dBF{\text{dim}_{B}F}
\redefine\dHF{\text{dim}_{H}F}
\redefine\dBFL{\underline {\text{dim}}_{B}F}
\redefine\dBFU{\overline {\text{dim}}_{B}F}
\define\h{\frac12}
\documentstyle{amsppt}
\hsize = 6.5 truein
\vsize = 9 truein
\NoRunningHeads
\topmatter
\title
ON THE DIMENSION OF DETERMINISTIC AND RANDOM CANTOR-LIKE SETS,
SYMBOLIC DYNAMICS, AND THE ECKMANN-RUELLE CONJECTURE
\endtitle
\author
YAKOV PESIN
and HOWARD WEISS
\endauthor
\leftheadtext{YAKOV PESIN and HOWARD WEISS}
\affil The Pennsylvania State University
\endaffil
\address{\aa{Yakov Pesin}{Howard Weiss}
\aa{Department of Mathematics}{Department of Mathematics}
\aa{The Pennsylvania State University}{The Pennsylvania State University}
\aa{University Park, PA 16802}{University Park, PA 16802}
\aa{U.S.A.}{U.S.A.}
\aa{Email: pesin\@math.psu.edu }{Email:weiss\@math.psu.edu}
\aa{}{}}
\endaddress
\thanks The work of the first author was partially supported by a
National Science Foundation grant \#DMS91-02887. The work of the
second author was partially supported by a National Science Foundation
Postdoctoral Research Fellowship. \endthanks
\keywords{Hausdorff dimension, box dimension, Cantor-like set,
geometric construction, random geometric construction, gauge function,
Eckmann-Ruelle Conjecture}
\endkeywords
\abstract In this paper we unify and extend many of the known results
on the dimensions of deterministic and random Cantor-like sets in
$\BbbR^n$ using their symbolic representation. We also construct
several
new examples of such
constructions that illustrate some new phenomena. These sets are defined
by geometric constructions with
arbitrary placement of subsets. We consider Markov constructions,
general symbolic constructions, nonstationary constructions, random
constructions (determined by a very general distribution), and
combinations of the above. One application to dynamical systems
is a counterexample to the $C^0$ version of the Ruelle-Eckmann conjecture.
\endabstract
\endtopmatter
\document
\CenteredTagsOnSplits
\TagsOnRight
\centerline{\bf Introduction}
\bigskip
In this paper we unify and extend many of the known results on the
dimension of deterministic and random Cantor-like sets in $\BbbR^n$.
These sets are defined by geometric constructions of different types.
Our most basic construction, which we call a {\it simple geometric
construction}, defines a Cantor-like set of the form
$$
F = \bigcap_{n=1}^{\infty} \bigcup_{(i_1 \cdots i_n)} \Delta_{i_1
\cdots i_n},
$$
where the basic sets on the $n$th step of the construction, $ \Delta_{i_1
\cdots i_n}, \, i_k = 1, \cdots, p$, are closed, disjoint, and
$\Delta_{i_1 \cdots i_n j} \subset \Delta_{i_1 \cdots i_n} $, for $j
= 1, \cdots, p$. We assume that the $\diam( \Delta_{i_1 \cdots i_n}) \ra 0$ as $n \to \infty$ and we emphasize that the placement of the sets $\Delta_{i_1
\cdots i_n} $ can be arbitrary as long as they satisfy the above
conditions. See Figure 1.
\vskip 0.25in
%\centerline{\epsfysize=2.75in \epsffile{pfig1.ps}}
\vskip 0.25in
\midinsert
\botcaption{Figure 1. \quad Simple Geometric Construction}
\endcaption
\endinsert
\medskip
We also consider a much broader class of geometric constructions
where the admissible basic sets $\Delta_{i_1 \cdots i_n} $ are determined by
a compact shift-invariant subset of the full shift on $p$ symbols. Our main results provide lower and upper estimates for the Hausdorff dimension and the box dimension of Cantor-like limit sets for large classes of geometric constructions. We also prove strict
positivity and boundedness of the Hausdorff measure of the limit set
for a broad class of constructions.
We will use two methods for obtaining a lower bound for the
Hausdorff dimension of a set: the uniform and non-uniform mass distribution principles.
See Appendix 2. The uniform mass distribution principle requires the
existence of a measure $m$
for which
$$ m(B(x, r)) \leq C r^s, \quad \tag 1
$$ where $B(x,r)$ is
the ball of radius $r$ centered at the point $x \in F$ and $C$ is a constant.
Then $s$
produces a lower bound for $\dHF$, the Hausdorff dimension of $F$. We
stress that (1) must hold for all $x \in F$ and the constant $C$
must be independent of $x$ and $r$. The non-uniform mass distribution principle requires that (1) hold for $m-$almost every $x \in F$ with $C = C(x)$ a measurable function.
Again, $s$ provides a lower bound for the Hausdorff dimension of the limit set.
If the uniform mass distribution principle holds, one can obtain
refined information about the $s-$Hausdorff measure of $F$. Although uniform mass distribution is stronger than the non-uniform version, we will establish
it for a broard class of geometric constructions.
One particular case is when a geometric construction
has sets at the $n$th step that are strictly
geometrically similar to the corresponding sets at the $(n-1)$th step.
This means that there is a collection of similarity maps (affine contractions) or more generally contraction maps $h_1, \cdots, h_p$ such that
$$
\Delta_{i_1 \cdots i_n } = h_{i_1} \circ \cdots \circ h_{i_n}(\Delta),
$$
where $\Delta$ denotes a ball in $\Bbb R^n$ (see Section 4).
This construction is called a {\it similarity
construction} since it exposes a self-similar character of the geometric
process. These special constructions
have been one of the main objects of study in dimension theory for many years.
About 50 years ago, Moran \cite{Mo} computed the Hausdorff dimension of
geometric constructions in $\Bbb R^n$ given by $p$ non-overlapping
balls $\Delta_{i_1 \dots i_n}$ satisfying $\diam( \Delta_{i_1 \dots i_n j})
= \lam_j \diam(\Delta_{i_1 \dots i_n})$ where $0 < \lam_j < 1$ for $j = 1, \dots, p$
are the {\it ratio coefficients}. We call this {\it a Moran construction}. It {\it need not} be a similarity construction. Moran discovered the formula $s = \dHF$,
where $s$ is the unique root of the equation
$$
\sum_{i=1}^p \lambda_i^t= 1. \quad \tag 2
$$
He also showed that
the $s-$dimensional Hausdorff measure of the limit set is finite and
strictly positive.
Moran's great insight was to realize that the similarity maps, or even the spacing of the balls
$ \{ \Delta_{i_1 \cdots i_n } \}$ are not important in the calculation of the Hausdorff dimension
of the limit set: the dimension depends only on the ratio coefficients. Moran proved this using
the (uniform) mass distribution principle applied
to the $s-$dimensional Hausdorff measure. We believe this is a seminal
paper that should be much better known. There are very few papers in the literature dealing with
geometric constructions that do not involve similarity maps.
In this paper we extend the Moran approach to much broader classes of geometric constructions.
We study constructions with general {\it geometry} of the basic sets and
general {\it symbolic representations}.
One can not expect to obtain any refined estimates for the Hausdorff and box dimensions of the
limit set $F$ of a construction with arbitrary { shape} and { spacing} of the basic sets. We control the geometry of the construction by either restricting the {shapes} or { sizes} of the basic sets, the { spacing} of the basics sets, or both. If one has strong control
over the sizes of the basic sets, then the spacing can be fairly arbitrary,
and vice-versa. In this paper we introduce a new approach to studying geometric
constructions having complicated geometry. Our approach is based on the notions of regularity and boundedness of the construction.
{\it Regular and bounded constructions} are those where the control over the geometry
is effected in the spirit of Moran processes by {\it generalized ratio coefficients} that encode
the information about {\it both} the shape and spacing of the basic sets. Example 3 in Section 5 illustrates the role of spacing.
For some constructions these coefficients are determined by the largest inscribed balls and smallest
circumscribed balls for the basic sets.
However, we construct an example where these numbers are completely independent of these balls (see Section 5, Example 8). We will compute the generalized ratio coefficients for all previously
studied classes of geometric constructions as well as several new ones.
One new class of geometric constructions that we introduce is {\it asymptotic geometric constructions} where the ratio coefficients depend on steps
of the construction and admit a {\it good} asymptotic behavior. We apply our techniques to study asymptotic constructions and show that the
non-uniform mass
distribution principle can be used to estimate the Hausdorff dimension of the limit set. For asymtotic constructions, one can not expect the uniform
mass distribution principle to hold. Hence the Hausdorff measure is
usually zero or infinite, even in simple examples (see Section 5, Example 4). In order to obtain more information about the
Hausdorff measure one can use a {\it gauge function} (see Appendix 4). In Section 5,
we present a family of simple asymptotic geometric constructions which admit infinitely
many gauge functions
depending on the rate of convergence of ratio coefficients. This
illustrates that the asymptotic category of constructions is quite rich
in that there exists a large variety of different limit sets exhibiting
many different structural properties.
We apply our study of asymptotic constructions to analyze
a new class of geometric constructions which are {\it random
Moran geometric constructions}. These constructions are essentially Moran
construction with ratio coefficients chosen randomly
from an arbitrary ergodic stationary process. We also consider random
one-dimensional constructions. In the literature, random constructions are
usually considered as a distinct class of geometric constructions. We
stress that our approach completely unifies the analysis of many types
of geometric constructions.
The Moran process has a natural symbolic description by the full shift on $p$ symbols,
where $p$ is the number of basic sets on the first step of the construction. We consider constructions modeled by a general symbolic
dynamical system.
For the Moran process, the crucial
observation is the existence of a measure $m$ on the set $F$ such that
the Hausdorff dimension of $F$ is equal to the Hausdorff dimension of
$m$. This
measure is the pullback of the equilibrium state
for the function $\phi (x)= \log \lambda_{i_1}$ where $x$ is
associated with a sequence $(i_1 i_2 \cdots)$ and $s$ is the unique
root of the equation
$$
P( s \phi) = 0. \quad \tag 3
$$
For the Moran constructions, $s$ is the Hausdorff dimension of
the limit set as well as its box dimension. We show that this is true for general symbolic Moran constructions (see Proposition 3 and Theorem 5.)
Equation (3) was discovered by Bowen \cite{Bo2} and seems to be
universal. We will show that all known equations previously used to
compute the Hausdorff dimension (for example equation (2)) coincide
with or are particular cases of (3).
For the general symbolic geometric constructions the measure $m$ is an equilibrium measure and admits the non-uniform mass distribution principle. There is a crucial
difference between Gibbs measures and equilibrium measures in
Statistical Physics (see \cite{R} and Appendix 3; see also Section 1, Remark 1). These notions coincide for subshifts of
finite type, but need not coincide for general symbolic systems.
We show that if $m$ is a Gibbs measure, then the
Hausdorff dimension of the limit set, $s$, can be studied using the uniform mass distribution
principle, and the $s-$dimensional Hausdorff measure is positive and finite (see Theorems 2 and 4). If $m$ is an equilibrium
state, then a non-uniform mass distribution principle can be applied (see Theorem 1).
Theorem 5 establishes the coincidence of the
Hausdorff dimension and the box dimension of a measure and is similar
in spirit to some results of Ledrappier and Young \cite{LY}. It
supports a general belief that the coincidence of the Hausdorff
dimension and box dimension of a {\it set} is a rare phenomenon and
requires {\it rigid} geometric constructions like some of those
considered in this paper. In Section 5, we provide an example of a simple
geometric construction in the plane with basic sets being rectangles for which the Hausdorff dimension $\dHF$,
the lower box dimension $ \dBFL$, and the upper box dimension $\dBFU$,
are distinct. The construction uses different ratio coefficients in
different directions and combines both vertical and horizontal
stacking of rectangles to effect the noncoincidence of the dimensions.
In \cite{PW}, the authors found that number theoretic properties of
the ratio coefficients, related to Pisot numbers, is another mechanism to
cause noncoincidence of dimensions. Another example which illustrates the non-coincidence of the Hausdorff and box dimensions was constructed in \cite{Mc}.
The coincidence of the Hausdorff dimension and box dimension of a {\it
measure} is more common. There is a general criterion proved by Young
\cite{Y} that guarantees the coincidence. Namely, let $m$ be a Borel
measure on $F$ such that for $m-$almost every $x \in F$ the limit
$$
\lim_{r \to 0} \frac{\log m(B(x,r))}{\log r} = d_m(x)
$$
exists. The
limit $d_m(x)$ is called the {\it pointwise dimension} at $x$. In
this case $m$ is called {\it exact dimensional}. If the limit does not exist,
one can consider the lower and upper limits $\underline d_m(x),
\overline d_m(x)$ to obtain the {\it lower and upper pointwise dimensions} at
$x$. If for $m$-almost every $x$
$$
d_m(x)= \text{const} \overset \text{def} \to \equiv s
$$
then the
Hausdorff dimension of $m$, $\text{dim}_Hm$, and the lower and upper
box dimensions of $m$, $\underline{\text{dim}}_Bm$ and $
\overline{\text{dim}}_Bm$, coincide and have the common value $s$.
This statement poses the problem of whether a given measure $m$ is
exact dimensional and moreover whether $d_m(x)=const$ almost everywhere. In \cite{C}, Cutler constructed an example of a continuous
map that preserves an ergodic exact dimensional measure $m$ with
$d_m(x)$ essentially non-constant. In Section 6.4, we present a much
simplier version of the construction of such a map that uses a special
simple geometric construction. If the map is smooth and $m$ is exact dimensional and ergodic, then $d_m(x) = const$ almost everywhere ( since $d_m(x)$ is invariant
under the map and measurable). Ledrappier and Misiurewicz \cite{LM} constructed a one-dimensional
smooth map preserving an ergodic measure that is not exact dimensional.
Eckmann and Ruelle conjectured that an ergodic measure
invariant under a $C^2$-diffeomorphism with non-zero Lyapunov exponents
is exact dimensional (and, hence, $d_m(x)=const$, see \cite{ER}). This
was proved in \cite{Y} for two-dimensional maps and in \cite{PY} for
some measures including Gibbs measures for Axiom A diffeomorphisms (see also \cite{L}). In Section 6, we construct a homeomorphism having
an ergodic invariant Gibbs measure with positive entropy that has
different upper and lower pointwise dimensions almost everywhere. In
other words, the {\it continuous} version of the Eckmann-Ruelle Conjecture fails.
Mauldin and Williams \cite{MW1} computed the Hausdorff dimension for a
Markov process given by similarity maps. They used symbolic dynamics,
large deviation theory, and graph theory. We give a considerable
generalization of their result as well as a considerable simplification
of their proof. The Moran construction in the Markov case was studied by
Stella \cite{St} with some additional strong assumptions. Afraimovich and
Shereshevsky \cite{AS} found a lower estimate for the Hausdorff
dimension of some simple geometric constructions. Similar types of
simple geometric constructions, given by two-dimensional self-affine
maps related to graphs of functions, were considered by Bedford and Urbanski in \cite{BU}. Shereshevsky \cite{S} also considered some one-dimensional Markov
geometric constructions.
Geometric constructions of random type have been considered by Falconer
\cite{F2}, Graf \cite{G}, Kahane \cite{K}, Graff, Mauldin and
Williams \cite{GMW}, and Mauldin and Williams \cite{MW2}. These
authors studied special types of branching processes that correspond to
the full shift on $p$ symbols with $p^n$ ratio coefficients at step
$n$ chosen randomly, essentially independently and with the same
distribution on $(0, 1)$. They also assume some independence
conditions over $n$. In this paper, we consider branching processes
that are associated with arbitrary compact shift-invariant subsets. We generate the ratio coefficients by chosing $p$ random
numbers on the interval $(a, b)$ \, where $ 0 < a \leq b < 1$. We do not
require that these numbers be independent nor be indentically
distributed.
A good general reference for dimension theory is \cite{F1}.
\medskip \centerline{\bf Acknowledgement}
\medskip
The authors would like to
thank H. Furstenberg and D. Mauldin for helpful discussions and would like to especially thank F. Ledrappier, Y. Peres, L. Barreira, and A. Manning for their fruitful comments and examples. The first author wishes to thank IHES
for their support and hospitality during his visit in the Fall 1992 which
enabled him to think about some problems in this paper.
\bigskip
\head{\bf Section 1: Geometric Constructions} \endhead
\bigskip
We define geometric constructions of Cantor-like subsets of $\Bbb R^d$ by using a symbolic description in the space of all one-sided infinite sequences $(i_1 i_2 \cdots)$ on $p$ symbols. We denote this space by $\Sigma^+_p$ and endow it with its usual topology (see Appendix 3).
\bigskip
A {\bf symbolic construction} is defined by
\roster
\item"{\bf a)}" a compact set $Q \subset
\Sigma_p^+$ invariant under the shift $\sigma$ (i.e.,
$\sigma(Q)=Q$)
\item"{\bf b)}" a family of closed sets called {\bf basic sets} $ \{
\Delta_{i_1 \cdots i_n} \} \subset \Bbb R^d$ \, for \, $ i_j = 1, 2,
\dots, p $ and $n \in \Bbb N$ where the $n$-tuples $(i_1 \cdots i_n)$
are admissible with respect to $Q$ (i.e., there exists $\omega = (i_1', i_2', \cdots) \in Q$
such that $i_1' = i_1, i_2' = i_2, \cdots, i_n' =i_n)$ and these sets satisfy
$$
\lim_{n \to \infty} \max_{(i_1 \cdots i_n)} \diam( \Delta_{i_1 \cdots i_n}) = 0. \quad \tag 4
$$
\endroster
For any admissible sequence $(i_1 \cdots i_{n+1}) \in \{1, \cdots, p \}^{n}$, we require that
$$
\Delta_{i_1 \cdots i_n i_{n+1}} \subset \Delta_{i_1 \cdots i_n}
$$
and
$$
\Delta_{i_1 \cdots i_n} \cap \Delta_{i'_1 \cdots i'_n} = \emptyset
\quad \text{ if } \, (i_1 \cdots i_n) \neq (i'_1,\cdots,i'_n). \quad
\tag 5
$$
The {\bf limit set} $F$ for this construction is defined by
$$
F = \bigcap_{n=1}^{\infty} \bigcup_{\underset \text{admissible} \to
{(i_1 \cdots i_n)}} \Delta_{i_1 \cdots i_n}.
$$
The set $F$ is a generalized Cantor set,
i.e., it is a perfect, nowhere dense, and totally disconnected set.
We stress that the placement of the sets $\{ \Delta_{i_1 \cdots i_n}
\}$ is completely arbitrary, and we make no assumptions on the
regularity of the boundaries of the sets $ \{ \Delta_{i_1 \cdots i_n}
\}$ which can be fractal. The basic sets need not even be connected.
\medskip
Given $x \in F$ and $n > 0$, there
exists a unique set $\Delta_{i_1 \cdots i_k}(x)$ that contains $x$ and
hence $x = \bigcap_{k=1}^{\infty} \Delta_{i_1 \cdots i_k}(x)$. This
gives a unique one-sided infinite sequence $(i_1 i_2 \cdots)$ such that
the mapping $\chi: F \ra Q$ defined by $x \mapsto (i_1 i_2
\cdots)$ is a homeomorphism from $F$ onto $Q$. The associated
symbolic dynamics is one of our main tools to compute the Hausdorff dimension
of the limit set $F$.
Consider the symbolic dynamical system $(Q, \sigma)$, where $Q \subset \Sigma^+_p$. Given a
$p-$tuple $\alpha = (\alpha_1, \cdots, \alpha_p)$ such that $0 < \alpha_i < 1$,
there exists a uniquely defined number $ s_{\alpha}$ such that
$P( s_{\alpha} \log \alpha_{i_1}) = 0$, where $P$ denotes the topological
pressure (see Appendix 3). Let $\mu_{\alpha} $ denote an
equilibrium measure for the function $(i_1, i_2, \cdots) \mapsto s_{\alpha} \log \alpha_{i_1}$ on $Q$,
and let $m_{\alpha} $ be the pull back measure on $F$ under the coding map $\chi$.
\medskip
\subhead{\bf 1. Lower estimates for the Hausdorff dimension} \endsubhead
\medskip
Since we allow arbitrary { spacing and { shape} of the basic sets,
one can not expect, in general, to obtain any refined estimates for the Hausdorff and box dimensions of the limit set $F$ (see Section 5, Example 8.) To obtain refined estimates, one needs to
control the {\it geometry} of the construction by either controlling the spacing of the basic sets, sizes or shapes of the basic sets, or both. If one has strong control
over the sizes of the basic sets, then the spacing can be fairly arbitrary,
and vice-versa. One well known method to
control the geometry is to effect the construction using similarity maps (see Section 4). This class of construction is very restrictive since one has strong control over both the spacing of the basic sets and their sizes in all directions. In \cite{Mo}, Moran presented a more general construction where
one has complete control over the sizes of the basic sets but where the spacing
is arbitrary. Moran computed the Hausdorff dimension of the limit set for
such a construction using the uniform mass distribution principle. We will present a more general method to control the geometry of the construction and extend the Moran approach from the full shift to a general symbolic system. This requires the non-uniform mass distribution principle (instead of the uniform one). We will estimate the dimensions for a large class of constructions called regular constructions, which include many constructions where one has only moderate control over the spacing and/or the sizes of the basic sets.
In a regular construction, control over the geometry is given by numbers
$\g_1, \cdots, \g_p$ such that one can approximate the basic sets $\Delta_{i_1 \cdots i_n}$ by balls of radius $\prod_{j=1}^n \g_{i_j}$.
We believe that one can compute the numbers $\g_1, \cdots, \g_p$ and construct the balls using detailed information about shape and spacing of the basic sets. In some cases these balls coincide with the largest balls that can be inscribed in the basic sets. However, we construct an example where the optimal numbers $\g_1, \cdots, \g_p$ are completely independent of the radius of the largest inscribed balls (see Section 5, Example 8). We will compute the numbers $\g_1, \cdots, \g_p$ for all previously studied classes of geometric constructions as well as several new ones, and we will produce refined estimates for the Hausdorff dimension from below.
Given $0 < r < 1$ and a vector of numbers $\g = (\g_1, \cdots, \g_p), \, 0 < \gamma_i < 1, i=1, \dots, p$, we first define a special cover $\frak U_r = \frak U_r (\g)$ of the limit set $F$. For any $x \in F$, let $n(x)$ denote the unique positive integer such that $\gamma_{i_1}
\gamma_{i_2} \cdots \gamma_{i_{n(x)}} > r$ and
$\gamma_{i_1} \gamma_{i_2} \cdots
\gamma_{i_{n(x)+1}} \leq r $ where $\chi(x)=(i_1 i_2 \cdots)$. It is
easy to see that $n(x) \to \infty$ as $r \to 0$ uniformly in $x$. Fix
$x \in F$. Consider the set $\Delta_{i_1 \cdots i_{n(x)}}$. We have $x
\in \Delta_{i_1 \cdots i_{n(x)}}$, and if $y \in \Delta_{i_1 \cdots
i_{n(x)}}$ and $n(y) \geq n(x)$, then
$$
\Delta_{i_1 \cdots i_{n(y)}}
\subset \Delta_{i_1 \cdots i_{n(x)}}.
$$
Let $\Delta(x)$ be the
largest set containing $x$ with the property
that $\Delta(x) = \Delta_{i_1 \cdots i_{n(z)}}$ for some $z \in \Delta(x)$ and
$ \Delta_{i_1 \cdots i_{n(y)}}
\subset \Delta(x)$ for any $y \in \Delta(x)$. The sets $\Delta(x)$
corresponding to different $x \in F$ either coincide or are disjoint.
We denote these sets by $\Delta_r^{(j)}, \, j=1, \cdots, N_r$. There exist
points $x_j$ such that $\Delta_r^{(j)} = \Delta_{i_1 \cdots i_{n(x_j)}}
$. These sets form a disjoint cover of $F$.
Consider the open Euclidean ball $B(x, r)$ of radius $r$ centered at a point $x$.
Let $N(x, r) $ denote the number of sets $\Delta_r^{(j)}$ that have non-empty intersection
with $B(x, r)$. We call a vector $\g$ {\bf l-estimating}
if $ N(x, r) \leq $ constant, uniformly in $x$ and $ r$. We call a symbolic construction
{\bf regular} if it admits an l-estimating vector. If $\g=(\g_1, \dots, \g_p)$ is an l-estimating vector for a regular symbolic construction,
then any vector $\tilde \g=(\tilde \g_1, \dots, \tilde \g_p)$ for which $\g_i \geq \tilde \g_i,
i=1, \dots, p$ is also l-estimating. It is
easy to see that $s_{\g} \leq s_{\tilde \g}$. On the
other hand a vector $\tilde \g=(\tilde \g_1, \dots, \tilde \g_p)$ for which $\tilde \g_i$
is sufficiently close to $1$ for some $i$ is not l-estimating.
Below we will give several examples of regular constructions although there are many constructions which are not regular. For example, if the Hasudorff dimension of the limit set is zero, then the corresponding construction is not regular. Let $F_1, F_2$ be two
limit sets for two symbolic constructions on the line: the first construction
is defined on the interval $[0, 1]$ and the second one on the interval $[2, 3]$ . It is easy
to see that if $\dim_H F_1 = 0$ and $\dim_H F_2 > 0$ then the set $F_1 \cup F_2$ is the limit set for a construction
which is not regular but whose limit set has positive Hausdorff dimension.
In Section 2 we present conditions which guarantee regularity of a large class
of one-dimensional geometric constructions. We show that, in particular, if the geometric construction satisfies
$$
\inf_n \inf_{(i_1 i_2 \dots i_n)} \frac{\log \diam(\Delta_{i_1 i_2 \dots i_n} )}{n} \geq constant > 0,
$$
then the construction is regular. This condition provides strong control over the ratio coefficients and requires them to be uniformly bounded away from zero. It is easy to show that if the ratio coefficients go to zero
uniformly in $n$, then the Hausdorff dimension of the limit set is zero.
Barreira \cite{Ba} has shown that the above assumption is almost
optimal. He exhibited a one dimensional simple geometric construction with ratio coefficients not uniformly bounded away from zero such that: (1) the
construction is not regular, (2) the limit set $F$ satisfies the following strong homogeneity property
$$
\dim_H F \cap B(x, r) = constant > 0
$$
for all $x \in F$ and $r > 0$.
If one considers a geometric construction in the plane with limit set $F = F_1 \times F_2$ where $F_1$ and $F_2$ are limit sets for one-dimensional geometric
constructions and $\dim_H F_1 = 0$, then the construction is not regular.
We believe that a geometric construction is regular if the projection of the basic sets at step $n$ onto every line
satisfies the above uniformity condition uniformly over the direction of the
projection.
\medskip
\proclaim{\bf Theorem 1} Let $F$ be the limit set for a regular symbolic construction. Then
$ s_{\g} \leq \dHF $ for any l-estimating vector $\g$. Hence $ \sup s_{\g} \leq \dHF $
where the supremum is taken over all l-estimating vectors $\g$.
\endproclaim
\medskip
The lower bound for the Hausdorff dimension of the limit set follows from
the lower bound for the lower pointwise dimension of the measure $m_{\g}$ restricted to a
sequence of sets that exhaust $F$. A more delicate question in dimension theory is whether
the Hausdorff measure of the limit set is positive. The answer is presumably negative for a general symbolic geometric construction. For a regular constrution, the positivity of the Hausdorff measure may depend on whether the equilibrium measure $m_{\g}$ is a Gibbs measure. If this measure is Gibbs, then it satisfies the uniform mass distribution principle and we have our most refined estimates. If the construction is Markov, then the measure $m_{\g}$ is Gibbs. Little is known about the existence of Gibbs measures for general symbolic dynamical systems (see Remark 1). We denote by $m_H(t, F)$ the $t-$dimensional Hausdorff measure of $F$ (see Appendix 1).
\medskip
\proclaim{\bf Theorem 2} Let $F$ be the limit set a regular symbolic construction. Assume that there exists an l-estimating vector $\g$ such that the measure $m_{\g}$ is a Gibbs measure. Then
\itemitem{1)} the measure $m_{\g}$ satisfies the uniform mass distribution principle
\itemitem{2)} $0 < m_{\text{H}}(s_{\g}, F)$, moreover, $m_{\g}(Z) \leq C
\, m_{\text{H}}(s_{\g}, Z)$ for any $Z \subset F$ where $C> 0$ is a constant
\itemitem{3)} $s_{\g} \leq {\underline d}_{m_{\g}}(x)$ for every $x \in F$.
\endproclaim
\medskip
The second statement in Theorem 2 is nontrivial only when $s_{\g} = \dHF$. Otherwise, $m_H(s_{\g},
F) = \infty$. If $s_{\g}< s = \dHF$, then the $s-$Hausdorff measure may be zero or infinite.
\medskip
\subhead {\bf 2. Upper estimates for the upper box dimension} \endsubhead
In order to obtain upper estimates for the upper box dimension, we require that the diameters of
the basic sets decrease exponentially.
More precisely, we say that a vector $\lam = (\lam_i, \dots, \lam_p), \, 0 < \lam_i < 1$ is a
{\bf u-estimating} vector for a construction if
$$
\diam(\Delta_{i_1 \cdots i_n}) \leq C \prod_{j=1}^n \lambda_{i_j}
$$
where $C > 0$ is a constant. The symbolic construction is called {\bf bounded} if there exists a u-estimating vector $\lam$. It is easy to see that if $\lam = (\lam_1, \dots, \lam_p)$ is a u-estimating vector for a bounded construction, then any vector $\tilde \lam = (\tilde \lam_1, \dots, \tilde \lam_p)$ for which $\tilde \lambda_i \geq \lambda_i, \, i=1, \dots, p$ is a u-estimating vector
and $s_{\lam} \leq s_{\tilde \lam}$.
We now wish to estimate the upper box dimension of the limit set $F$ for
a symbolic geometric construction. We need not assume that the construction is regular, but only bounded.
\proclaim{\bf Theorem 3} Let $F$ be the limit set for a bounded symbolic construction. Then
\roster
\item $ \dBFU \leq s_{\lam} $ for any u-estimating vector $\lam$. Hence
$ \dBFU \leq \inf s_{\lam} $, where the infinum is taken over all u-estimating vectors $\lam$
\item ${\overline d}_{m_{\lam}}(x) \leq s_{\lam}$ for any u-estimating vector $\lam$ and $ m_{\lam}-$almost every $x \in F$.
\endroster
\endproclaim
The arguments in the proof of the upper bound for the upper box dimension are essentially due to Bowen and were pointed out to us by Ledrappier. In the case when the measure $m_{\lambda}$ is a Gibbs measure, we can establish finiteness of the $s_{\lam}-$Hausdorff measure and obtain a stronger statement about the upper pointwise dimensions.
\medskip
\proclaim{\bf Theorem 4} Let $F$ be the limit set for a bounded symbolic construction. Assume that there exists a u-estimating vector $\lam$ such that
the measure $ m_{\lambda}$ is a Gibbs measure. Then
\roster
\item $m_{\text{H}}(s_{\lam}, F) < \infty$, moreover \, $m_{\text{H}}(s_{\lam}, Z) \leq C m_{\lam}(Z)$ for any $Z \subset F$ where $C> 0$ is a constant
\item $ {\overline d}_{m_{\lam}}(x) \leq s_{\lam}$ for every $x \in F$.
\endroster
\endproclaim
\medskip
The next statement provides an upper estimate for the number $s_{\lam}$.
\proclaim{\bf Proposition 1} Let $F$ be the limit set for a bounded symbolic construction and $\lam$ a u-estimating vector.
\roster
\item We have
$$
s_{\lam} \leq \frac{h(\sigma \, | \, Q)}{ - \log \lambda_{\max}}
$$
where $\lambda_{\max} = \max_{k=1}^p \{\lambda_k \}$, and $h(\sigma \, | \, Q) $ denotes the topological entropy. Equality occurs if $\lambda_i = \lambda$ for $i=1, \cdots, p$.
\item See \cite{Fu}. If $\lam_i = \lambda$ for $i =1, \cdots, p$, then
$$
s_{\lam} = \dHF = \dBFL = \dBFU = \frac{h(\sigma \, | \, Q)}{-\log \lambda}. $$
\item If ${h(\sigma \, | \, Q)}
= 0$, then $s_{\lam} = \dHF = \dBFL = \dBFU = 0$.
\endroster
\endproclaim
\bigskip
The following Theorem is an immediate consequence of Theorems 1--4.
\proclaim{\bf Theorem 5 }
Let $F$ be the limit set for a regular and bounded symbolic construction. Then for any l-estimating vector $\g$ and any u-estimating vector $\lam$ we have
\roster
\item $s_{\g} \leq \dHF \leq \dBFL \leq \dBFU \leq s_{\lam}$ where
$s_{\g}$ and $s_{\lam}$ are the unique roots of the
equations $P(s_{\g} \log \g_{i_1}) = 0$ and $P(s_{\lam} \log \lam_{i_1}) = 0$,
respectively
\item if $\g_i = \lam_i$ for $i = 1, \cdots , p$ then
$$
s=s_{\g} = s_{\lam} = \dHF = \dBFL = \dBFU.
$$
Moreover, $m =m_{\g} = m_{\lam} $, and if the measure $m $ is Gibbs, then $ m_H(s, *)$ is equivalent to $m$ and $\underline
d_m(x) = \overline d_m(x) = s$ for every $x \in F$.
\endroster
\endproclaim
\medskip
The coincidence of the Hausdorff dimension and the box dimension was established by Falconer \cite{F3}, however he did not
compute the actual value of the dimension.
In general, the lower and upper pointwise dimensions
${\underline d}_{m_{\g}}(x)$ and $ {\overline d}_{m_{\g}}(x)$ (as well as
${\underline d}_{m_{\lam}}(x)$ and $ {\overline d}_{m_{\lam}}(x)$) do not coincide almost everywhere (see Sections 6.2 and 6.3).
\subhead {\bf 3. Markov construction} \endsubhead
We will consider two important special cases of symbolic constructions specified by a subshift of finite type or the full shift. Let $A$ denote a
$p \times p$ transfer matrix with entries $ A(i,j) = 0$ or $ 1$ \,
and let $\Sigma_A^+$ consists of admissible sequences $(i_1 i_2 \cdots )$
with respect to $A$ (i.e., $A(i_j, i_{j+1})=1$ for $j=1, 2 \cdots
$). The
construction is called {\bf Markov} if $Q = \Sigma_A^+$. In the case when
the set $Q = \Sigma_p^+$ the
construction is called {\bf simple}.
Consider a subshift of finite type $(\Sigma_A^+, \sigma)$. Given $p$ numbers $0 < \alpha_1, \cdots, \alpha_p < 1$, we define a $(p \times p)$ diagonal matrix
$M_t(\alpha) = \text{diag}(\alpha_1^t, \cdots, \alpha_p^t)$. Let $\rho(B)$ denote the
spectral radius of the matrix $B$.
\medskip
\proclaim{\bf Proposition 2} The equation $P(s \log \alpha_{i_1}) = 0$ is equivalent
to the equation $\rho(A^*M_s(\alpha)) =1$,
where $A^*$ denotes the transpose of the matrix $A$.
\endproclaim
\medskip
We will assume that the transfer matrix $A$ is
transitive, i.e., there exists $N \in \Bbb N$ such that $(A^N)_{i,j} >
0$ for all $i$ and $j$ (see Remark (4) below).
The following corollaries are immediate consequences of Theorem 5 and Proposition 2.
\medskip
\proclaim{\bf Corollary 1}
Let $F$ be the limit set for a {\bf Markov} regular and bounded construction. Then for any l-estimating vector $\g$ and any u-estimating vector $\lam$ we have
$$
s_{\g} \leq \dHF \leq \dBFL \leq \dBFU \leq s_{\lam}
$$
where
$s_{\g}$ and $s_{\lam}$ are the unique roots of the equations
$$
\rho(A^*M_s(\g)) =1, \quad \rho(A^*M_s(\lam)) =1
$$
respectively.
\endproclaim
\medskip
\proclaim{\bf Corollary 2 }
Let $F$ be the limit set for a {\bf simple} regular and bounded construction. Then for any l-estimating vector $\g$ and any u-estimating vector $\lam$ we have
\roster
\item $s_{\g} \leq \dHF \leq \dBFL \leq \dBFU \leq s_{\lam}$ where
$s_{\g}$ and $s_{\lam}$ are the unique roots of the
equations
$$
\sum_{i=1}^{p} \g_i^{t} = 1 \quad
\text{ and } \quad \sum_{i=1}^{p} \lam_i^{t} = 1
$$
respectively
\item
the Gibbs measures $\mu_{\g}$ and $ \mu_{\lam}$ on $\Sigma^+_p$ satisfy
$$
\mu_{\g}(C_{i_1 \cdots i_n}) = \prod_{j=1}^n \g_{i_j}^{s_{\g}} \, \text{ and } \,
\mu_{\lam}(C_{i_1 \cdots i_n}) = \prod_{j=1}^n \lam_{i_j}^{s_{\lam}}
$$
where $C_{i_1 \cdots i_n}$ is a cylinder set.
\endroster
\endproclaim
\bigskip
\proclaim{\bf Remarks} \endproclaim
{\bf (1)} It follows from the expansiveness of the shift map that for any general symbolic system
and any continuous function, there exists an equilibrium measure corresponding to this function.
However, in general, this measure need not be a Gibbs measure. It is well known that if a
symbolic system has the specification property, then the equilibrium measure is Gibbs and is
unique provided the function is H\"older continuous \cite{R}. Thus,
Theorem 5 provides refined estimates for the Hausdorff and box dimensions as well as the
Hausdorff measure for a symbolic geometric construction modeled by a symbolic system which
satisfies specification.
Subshifts of finite type and their finite factors (sophic systems) are known to satisfy specification \cite{R, We}. The only other example we know is the beta-shift \cite{BM} for special values of $\beta$.
If the topological entropy of $\sigma \, | \, Q$ is zero, then by
Proposition 1, the Hausdorff dimension and box dimension coincide and
are zero. In this case the measure $\mu_{\lam}$ is a measure of maximal entropy and is
not a Gibbs measure.
We believe that one can find a symbolic system with
positive topological entropy and can build a
$\g-$regular and $\lam-$bounded symbolic construction such that the two measures $m_{\g}$ and
$m_{\lam}$ are not Gibbs. Moreover, $m_{\lam}$ will not be equivalent to the $s_{\lam}-$Hausdorff
measure. There is an interesting question of whether one can build a regular (or bounded) symbolic
construction admitting two l-estimating vectors $\g_1, \g_2$ ( two u-estimating vectors
$\lam_1, \lam_2$) such that $m_{\g_1}$ is Gibbs and $m_{\g_2}$ is not ( $m_{\lam_1}$ is
a Gibbs measure and $m_{\lam_2}$ is not).
\medskip
{\bf (2)} There exists a weaker version of regularity of a construction which enables one to
prove lower estimates for the Haudorff dimension of the limit set. This weaker condition is
sufficient to prove Theorem 1.
Given $0 < r < 1$ and a vector of numbers $\g = (\g_1, \cdots, \g_p), \, 0 < \g_i < 1, \, i=1,
\dots, p$, \, consider
the special cover $\frak U_r = \frak U_r (\g)$ of the limit set $F$ by basic sets
$\Delta_r^{(j)} = \Delta_{i_1 \cdots i_{n(x)}}$. Let $N(x, r) $ denote the number of sets
$\Delta_r^{(j)}$ that have non-empty intersection with $B(x, r)$. We call a vector $\g$ {\bf
weak l-estimating}
if for every $\epsilon > 0$ there exists a $m-$measurable function $N(x, \epsilon)$ such that
for all $x \in F$ and $r>0$
$$
N(x, r) \leq N(x, \epsilon)\, r^{- \epsilon}. \quad \tag 6
$$
We call a symbolic construction {\bf weakly regular} if it admits a weakly l-estimating vector.
Obviously, a regular symbolic construction is weakly regular.
\proclaim {\bf Theorem 1'} Let $F$ be the limit set for a { weakly} regular symbolic
construction. Then $ s_{\g} \leq \dHF $ for any weakly l-estimating vector $\g$. Hence
$ \sup s_{\g} \leq \dHF $ where the supremum is taken over all weakly l-estimating vectors $\g$.
\endproclaim
\medskip
One can also obtain an approriate version of Theorem 5 under this
weaker hypothesis.
\medskip
{\bf (3)} Two closed sets are called {\it non-overlapping} if
their intersection consists of only boundary points.
By examining the proofs of Theorems 1--5, one can show that they are valid for
one-dimensional constructions with non-overlapping basic
sets. See \cite{Mo, MW1}.
\medskip
{\bf (4)} In the Markov constructions above, we assumed that the
transfer matrix $A$ was transitive. For an arbitrary transfer matrix
$A$, one can decompose the set $\Sigma^+_A$ into two shift-invariant
subsets: the wandering set $Q_1$ and the non-wandering set $Q_2$. The
latter can be further partitioned into finitely many shift-invariant
subsets of the form $\Sigma^+_{A_i}$ where each matrix $A_i$ is
transitive and corresponds to a class of equivalent recurrent
states \cite{AJ}. The limit set $F$ contains disjoint sets $F_i
= \chi^{-1}(\Sigma^+_{A_i})$. Each set $F_i$ is the limit set for a
Markov construction defined by the transitive matrix $A_i$
and hence admits lower and upper estimates for the Hausdorff and box
dimensions stated in Corollary 1. In \cite{MW1}, the authors
discuss the effect of the wandering set $Q_1$ on the dimension
of the limit set $F$.
\medskip
{\bf (5)} Let $F$ be the limit set specified by a symbolic
construction. Define the map $G:F \to F $ by $G(x) = \chi^{-1} \circ
\sigma \circ \chi(x)$. It is easy to see that $G$ is a continuous
endomorphism such that the set $G^{-1}(x), x \in F$ consists of
finitely many points. If $\mu$ is a $\sigma-$invariant measure then
its pull back measure $m=\chi^* \mu$ is $G-$invariant and has the same ergodic
properties as $\mu$.
\medskip
{\bf (6)} Consider the full shift $\sigma$ on $p$ symbols with the
standard metric $d_{\beta}, \, \beta > 1$. Let $\mu$ be a
$\sigma-$invariant ergodic measure, and let $r = \beta^{-n}$. Since
the ball centered at $\omega$ with radius $r$, $B(\omega, r)$, is a
cylinder set, we have by a theorem of Brin and Katok \cite{BK} that
for $\mu$ almost every $\omega$,
$$
\frac{\log \mu(B(\omega
,r))}{\log r} = - \frac{\log \mu(B(\omega,\beta^{-n}))}{n \log \beta}
\overset { n \ra \infty} \to \longrightarrow
\frac{h_{\mu}(\sigma)}{\log \beta},
$$ hence
$$
\frac{h_{\mu}(\sigma)}{\log \beta} = \underline d_{\mu}(\omega)=
\overline d_{\mu}(\omega)= \dim_H \mu = \underline
{\text{dim}}_{B}{\mu} = \overline {\text{dim}}_{B} {\mu},
$$
where $h_{\mu}(\sigma)$ denotes the Kolmogorov-Sinai entropy of the
shift map. It is evident that the common value depends on $\beta$,
which is not surprising since the two metrics $d_{\beta_1}$ and
$d_{\beta_2}$ are not equivalent for $\beta_1 \neq \beta_2$. Now
consider the Moran simple construction with parameters $\lam = (\lambda_1, \cdots, \lam_p)$. Since
$\mu_{\lam}$ is Gibbs we have that $h_{\mu_{\lam}}(\sigma) + s
\int \log \lambda_{\omega_1} d \mu_{\lam} = 0$, which gives
$$
s
= - \frac{h_{\mu_{\lam}}(\sigma)}{\int \log \lam_{i_1} d\mu_{\lam} } = \frac{
{h_{\mu_{\lam}}}(\sigma) }{ \log \beta}
$$
if
$$
\beta = \exp \left( - \int \log \lam_{i_1} d \mu_{\lam} \right). \quad \tag 7
$$
Thus the Hausdorff
dimension of $\mu_{\lam}$ calculated with respect to the two metrics
$d_{\beta}$ and $\rho$ (the Euclidean metric on $\Bbb R^d$) coincide.
The two metrics $\chi^* \rho$ and $d_{\beta}$ with $\beta$ satisfying
(7) are not equivalent if not all the $\lambda_i$ coincide. To see
this, notice that $\chi: F \to \Sigma_p^+$ is onto. We have with
respect to the metric $\rho$
$$
\dim_H F = s = \frac{ {h_{\mu_{\lam}}}(\sigma)}{ \log \beta}
$$
and with respect to the metric $d_{\beta}$
$$
\dim_H \Sigma^+_p =
\frac{\log p}{\log \beta} = \frac{h(\sigma)}{\log \beta}
$$
where $h(\sigma)$ denotes the topological entropy of $\sigma$. It
is easy to see that $h_{\mu_{\lam}}(\sigma) = h(\sigma)$ if and only if
$\lambda_1 = \lambda_2 = \cdots = \lambda_p = \lambda$.
\bigskip
\head{\bf Section 2: Examples of Symbolic Constructions} \endhead
Here we consider several classes of symbolic constructions with restrictions on the shapes of
basic sets or restrictions on the gaps between basic sets.
\subhead{\bf 1: Moran constructions} \endsubhead
The simpliest case is when the basic sets are
{\it essentially} balls. Such constructions were considered by Moran \cite{Mo}. A {\bf Moran
symbolic construction} is a symbolic construction
such that each basic set $\Delta_{i_1\cdots i_n} $ satisfies
$$
D ( C_1 \prod_{j=1}^n \lambda_{i_j}) \subset
\Delta_{i_1 \cdots i_n} \subset D ( C_2 \prod_{j=1}^n
\lambda_{i_j}) \quad \tag 8
$$
where $0 < \lam_i < 1, \, i=1, \cdots, p$ \, and $C_1, C_2$ are positive constants.
\medskip
\proclaim{\bf Proposition 3 } Let $F$ be the limit set for a Moran symbolic construction. Then
\roster
\item the construction is regular and bounded with l-estimating vector and u-estimating vector
equal to $\lam = (\lam_1, \dots, \lam_p)$
\item $s=s_{\lam} = \dHF = \dBFL = \dBFU $.
Moreover, if the measure $m_{\lam} $ is Gibbs, then $ m_H(s, *)$ is equivalent to $m_{\lam}$,
$ 0 < m_H(s, F) < \infty$, and
$ s= {\underline d}_{m}(x) = {\overline d}_{m}(x) $ for every $x \in F$. \endroster
\endproclaim
\medskip
The first statement of Proposition 3 is obvious, and the second statement follows from Theorem 5.
\subhead{\bf 2: Constructions with rectangles} \endsubhead
We now consider geometric constructions where the basic sets are (multi-dimensional) rectangles.
More precisely, we call a symbolic construction a {\bf construction with rectangles} if there
exist $2 p$ numbers
$\underline \lam_i, \overline \lam_i,
\, i=1, \cdots, p, \, 0 < \underline \lam_i \leq
\overline \lam_i < 1$ such that the basic set $ \Delta_{i_1 \cdots i_n}
\subset \Bbb R^d$ is a rectangle (the direct product of intervals, called sides, lying on $n$ orthogonal lines) with the largest side equal to $ C_1 \prod_{j
=1}^n \overline \lambda_{i_j}$ and the smallest side equal to $C_2 \prod_{j=1}^n
\underline \lambda_{i_j}$ where $C_1, C_2$ are positive constants.
\medskip
\proclaim{\bf Proposition 4 } Let $F$ be the limit set for a symbolic construction with rectangles.
Then
\roster
\item the construction is regular and bounded with l-estimating vector $\ll= (\ll_1, \dots, \ll_p)$ and u-estimating vector $\ol= (\ol_1, \dots, \ol_p)$
\item $s_{\ll} \leq \dHF \leq \dBFL \leq \dBFU \leq s_{\ol}$. Moreover, if the measures
$m_{\underline \lam}$ and $m_{\overline \lam}$ are Gibbs, then $s_{\ll} \leq
{\underline d}_{m_{\ll}}(x) $ and $ {\overline d}_{m_{\ol}}(x) \leq s_{\ol}$ for every $x \in F$.
\endroster
\endproclaim
\medskip
The first statment is obvious and the second statement follows from Theorems 1-4. In Section 5, Example 6, we will exhibit a simple construction with rectangles for which $ \dHF < \dBFL < \dBFU $.
\medskip
In \cite{Mc}, McMullen studied a special example of a geometric
construction in the plane with rectangles. Given positive integers $m$ and $ n$,\, $ m \leq n$, he considers the partition of the unit square into rectangles of size $\frac1m \times \frac1n$. One chooses any $r, \, 1 \leq r \leq m \cdot n$ of these rectangles and colors them. One then linearly contracts the unit square by a factor of $\frac1m \times \frac1n$ and inserts a copy of this set into each of the original colored rectangles. One keeps going and obtains a limit set $F$. This is not a similarity construction since one does not insert a copy of this set into all of the original rectangles, just the shaded ones. McMullen proves that $\dBFL = \dBFU \overset \text{def} \to \equiv \dBF$ and finds explicit formulars for
the box dimension and the Hausdorff dimension which use information about the initial configuration. It follows from his results that for {\it most} initial configurations, $\dHF < \dBF$.
The vectors $\g= (\frac1n, \frac1n)$ and $\lam = (\frac1m, \frac1m)$ are l-estimating and u-estimating vectors respectively. Proposition 4 is applicable to this example and gives the following estimate
$$
\frac{\log r}{- \log n} \leq \dim_HF \leq \dim_BF \leq \frac{\log r}{- \log m}.
$$
It follows from McMullen's formulas that for a most initial configuration, the three inequalities above are strict inequalities.
In \cite{PW}, the authors studied another type of simple geometric
constructions with rectangles in the plane. Their construction is the similarity construction with two rectangles, each of size $\lam_1 \times \lam_2, \, \lam_1 \leq \lam_2$, whose boundaries are alligned with the coordinate axes. The vectors $\g= (\lam_1, \lam_1)$ and $\lam = (\lam_2, \lam_2)$ are l-estimating and u-estimating vectors respectively. Proposition 4 is applicable to this example and gives the following estimate
$$
\frac{\log 2}{- \log \lam_1} \leq \dim_HF \leq \dBFL \leq \dBFU \leq \frac{\log 2}{- \log \lam_2}.
$$
The authors proved that: 1) $\dBFL = \dBFU \overset \text{def} \to \equiv \dBF$, 2) for ``almost all'' initial configurations with $\lam_2 \leq \frac12, \, \dim_HF = \dim_BF = \frac{\log 2}{- \log \lam_2}$, 3) for ``almost all'' initial configurations with $\lam_2 > \frac12,\, \dim_BF = \frac{\log(\frac{2 \lam_2}{\lam_1} )}{- \log \lam_1}$ and for a set of $\lam_2$ of positive measure, $\dHF = \dBF = \frac{\log( \frac{2 \lam_2}{\lam_1}) }{- \log \lam_1}$, 4) if $\lam_2$ is the reciprical of a Pisot number, then the Hausdorff dimension is strictly less than
$\frac{\log( \frac{2 \lam_2}{\lam_1}) }{- \log \lam_1}$. Hence the Hausdorff
dimension of the limit set may depend on delicate number theoretic properties of the ratio coefficients.
\medskip
\subhead{\bf 3: Constructions with exponentially large gaps} \endsubhead
\medskip
We now consider constructions where we have a strong control over the gaps, but no control over the shape and the size of the basic sets.
We call a symbolic construction a {\bf construction with exponentially large gaps} if there
exists a number $0 < \beta <1$ such that the (Euclidean) distance between any two basic sets
$ \Delta_{i_1 \cdots i_n}$ and $ \Delta_{j_1
\cdots j_n} $ is exponentially bounded away from $0$, i.e.,
$$
dist(\Delta_{i_1 \cdots i_n}, \Delta_{j_1 \cdots j_n} ) \geq C \beta^n
$$
where $C>0$ is a constant.
\medskip
\proclaim{\bf Proposition 5} Let $F$ be the limit set for a symbolic construction with exponentially large gaps. Then
\roster
\item the construction is regular with l-estimating vector $\g= (\beta, \dots, \beta)$.
\item $s_{\g} \leq \dHF$.
\endroster
\endproclaim
\medskip
Statement 1 is obvious and the proof of Statement 2 follows immediately from Theorem 1.
\medskip
\subhead{\bf 4: One-dimensional constructions} \endsubhead
We now consider one-dimensional symbolic constructions.
We will assume that each basic set $\Delta_{i_1\cdots i_n} $ is an interval that satisfies
$$
D ( C_1 \prod_{j=1}^n \underline \lambda_{i_j}) \subset
\Delta_{i_1 \cdots i_n} \subset D ( C_2 \prod_{j=1}^n
\overline \lambda_{i_j})
$$
where $0 < \ll_i < \ol_i < 1,\, i=1, \cdots, p$ and $C_1, C_2$ are positive constants.
\medskip
\proclaim{\bf Proposition 6} Let $F$ be the limit set for a one-dimensional
symbolic construction defined above. Then
\roster
\item the construction is regular with l-estimating vector $\ll = (\ll_1, \dots, \ll_p)$ and
bounded with u-estimating vector \, $\ol = (\ol_1, \dots, \ol_p)$.
\item $s_{\ll} \leq \dHF \leq \dBFL \leq \dBFU \leq s_{\ol}$.
\item if $\ll_i = \ol_i = \lam_i$, then $s=s_{\lam} = \dHF = \dBFL = \dBFU$.
Moreover, if the measure $m_{\lam} $ is Gibbs, then $ m_H(s, *)$ is equivalent to $m_{\lam}$,
$ 0 < m_H(s, F) < \infty$, and $ s= {\underline d}_{m}(x) = {\overline d}_{m}(x) $ for every $x
\in F$.
\endroster
\endproclaim
\medskip
The proof of Statement 1 is obvious and Statements 2 and 3 follow immediately from Theorems 1-5.
\medskip
\head{\bf Section 3: Asymptotic and Random Symbolic Constructions} \endhead
\medskip
\subhead{\bf 3.1} \endsubhead We now consider more general types of geometric constructions where the basic sets are
{\it asymptotically} balls. These constructions are not weakly regular, but are weakly regular
on each subset of an increasing sequence of subsets which exhaust the limit set up to a
set of measure zero. More precisely we say that a vector $\g$ is
a {\bf conditionally l-estimating} vector for a symbolic construction if there
exists a sequence of subsets $\{Q_l\}, l = 1, 2, \dots$ such that
\roster
\item $Q_l \subset Q_{l+1}$ and $\cup_l Q_l = Q$ up to a set of $\mu_{\g}-$measure zero
\item for every $l=1, 2, \dots$ and $\epsilon > 0$ there exists $N(l, \epsilon)$ such that for
all $x \in F$ with $\chi(x) \in Q_l$ and $r>0$
$$
N(x, r) \leq N(l, \epsilon)\, r^{- \epsilon},
$$
where $N(x, r) $ denotes the number of
sets in the special cover $\frak U_r (\g)$ of the limit set $F$ that have non-empty
intersection with $B(x, r) \cap Q_l$.
\endroster
We call a symbolic construction {\bf conditionally regular } if it admits a conditionally
l-estimating vector.
Note that a conditionally regular construction is weakly regular with respect to each set $Q_l$.
Thus by Theorem 1' one can obtain lower estimates for the Hausdorff dimension of the set $\chi^{-1}(Q_l)$ which is uniform over $l$. This immediately implies the following result:
\proclaim{Theorem 6} Let $F$ be the limit set for a conditionally regular symbolic
construction. Then $s_{\g} \leq \dHF$, $\g$ is a conditionally l-estimating vector.
\endproclaim
\medskip
We begin with an asymptotic version of the Moran symbolic constructions. We
will show that these constructions are conditionally regular.
\subhead{\bf 1: Asymptotic Moran constructions} \endsubhead
An {\bf asymptotic Moran symbolic construction} is a symbolic construction for which there exist
two sequences of numbers
$$
\ll_{i, n} = \lambda_i \exp(\underline a_{i, n}), \quad \ol_{i,n} = \lam_i \exp(\overline
a_{i, n})
$$
where $0 < \lam_i < 1, i=1, \cdots, p$ such that
\roster
\item"{\bf a)}" for $m_{\lam}-$almost every $x \in F$ with $\chi(x) = (i_1 i_2 \cdots)$ and $\lam=
(\lam_1, \dots, \lam_p)$,
$$
\frac1n \sum_{j=1}^n \underline a_{i_j, j} \ra 0 \quad
\text{ and } \quad \frac1n \sum_{j=1}^n \overline a_{i_j, j} \ra 0 \quad \text{ as }
n \ra \infty.
$$
\item"{\bf b)}" each basic set $ \Delta_{i_1, \cdots, i_n} $ satisfies
$$
D ( C_1 \prod_{j=1}^n \underline \lambda_{i_j, j}) \subset
\Delta_{i_1 \cdots i_n} \subset D ( C_2 \prod_{j=1}^n
\overline \lambda_{i_j, j})
$$
where $C_1, C_2$ are positive constants (compare with (8)).
\endroster
\medskip
\proclaim{\bf Proposition 7} Let $F$ be the limit set for an asymptotic Moran symbolic construction. Then construction is conditionally regular with the conditionally
l-estimating vector equal to $\lam = (\lam_1, \dots, \lam_p)$.
Hence $s_{\lam} \leq \dHF$.
\endproclaim
\medskip
Condition (a) in the definition of asymptotic Moran symbolic construction is quite weak; one can obtain more information about the Hausdorff and box dimension of the limit set if
the construction satisfies the following uniform version of (a)
\medskip
\roster
\item"{\bf a1)}" $$
\sup_{(i_1, \dots, i_n)} \frac1n \sum_{j=1}^n \underline a_{i_j, j} \ra 0 \quad
\text{ as } n \ra \infty
$$
\item"{\bf a2)}" $$
\sup_{(i_1, \dots, i_n)} \frac1n \sum_{j=1}^n \overline a_{i_j, j} \ra 0 \quad
\text{ as } n \ra \infty.
$$
\endroster
\medskip
\proclaim{Proposition 8} Let $F$ be the limit set for an asymptotic Moran
symbolic construction. Assume that the construction satisfies condition (a2). Then
$s=s_{\lam} = \dHF = \dBFL = \dBFU$.
\endproclaim
\medskip
Unlike the Moran symbolic construction, the limit set for an asymptotic Moran symbolic construction may have zero Hausdorff measure (even in the case when the construction satisfies the strong asymptotic conditions (a1) and (a2) and the measure $\mu_{\lam}$ is Gibbs, see Section 5, Example 4 and
compare with Statement 2 of Proposition 3).
\medskip
\subhead{\bf 2: Asymptotic one-dimensional symbolic construction} \endsubhead
We now consider an {\bf asymptotic one-dimensional symbolic construction}. This is a one-dimensional construction for which there exist two sequences of numbers
$$
\underline \lam_{i, n} = \underline \lambda_i \exp( \underline a_{i, n}), \quad \overline \lam_{i, n} = \overline \lambda_i \exp( \overline a_{i, n})
$$
where $0 < \underline \lam_i \leq \overline \lam_i < 1, \, i=1, \cdots, p$ such that
\roster
\item"{\bf a)}" for $m_{\underline \lam}-$almost every $x \in F$ with $\chi(x) = (i_1 i_2 \cdots)$ and $ \underline \lam=(\underline \lam_1, \dots, \underline \lam_p)$,
$$
\frac1n \sum_{j=1}^n \underline a_{i_j, j} \ra 0 \, \text{ as } \, n \ra \infty
$$
and
for $m_{\overline \lam}-$almost every $x \in F$ with $\chi(x) = (i_1 i_2 \cdots)$ and $ \overline \lam=(\overline \lam_1, \dots, \overline \lam_p)$,
$$
\frac1n \sum_{j=1}^n \overline a_{i_j, j} \ra 0 \, \text{ as } \, n \ra \infty
$$
\item"{\bf b)}" each basic set $\Delta_{i_1\cdots i_n} \subset \Bbb R$ satisfies
$$
D ( C_1 \prod_{j=1}^n \underline \lambda_{i_j, j}) \subset
\Delta_{i_1 \cdots i_n} \subset D ( C_2 \prod_{j=1}^n \overline \lambda_{i_j, j}) $$
where $C_1, C_2 > 0$ are positive constants.
\endroster
\medskip
\proclaim{\bf Proposition 9} Let $F$ be the limit set for an asymptotic one-dimensional
symbolic construction. Then the construction is conditionally regular with the conditionally
l-estimating vector $\underline \lam=(\underline \lam_1, \dots, \underline \lam_p)$. Hence $s_{\underline \lam} \leq \dHF$.
\endproclaim
\medskip
As in Proposition 8, one can obtain more information about the Hausdorff and box dimension of the limit set if the construction satisfies the uniform conditions (a1) and (a2).
\medskip
\proclaim{Proposition 10} Let $F$ be the limit set for an asymptotic one-dimensional symbolic construction. Assume that the construction satisfies condition (a2). Then
$s_{ \underline \lam} \leq \dHF \leq \dBFL \leq \dBFU \leq s_{ \overline \lam} $.
\endproclaim
\medskip
\subhead {\bf 3: Random symbolic construction} \endsubhead
We now consider a random version of the Moran symbolic construction. In this case the sizes
of the basic sets are chosen randomly with respect to some stationary ergodic distribution.
We will use an ergodic theorem in \cite{BFKO} to reduce the study of these constructions to the asymptotic
Moran constructions.
\medskip
A {\bf random symbolic construction} is defined by
\roster
\item"{\bf a)}" a stochastic vector process $(\Lambda, \frak F,
\nu)$ with \, $\Lambda = \{ \vec
\lambda = (\underline \lambda_{i,n}, \overline \lambda_{i,n}), \,
i = 1, \cdots, p$ and $\, n \in \Bbb N \}$ where $0 < \alpha
\leq \underline \lambda_{j,n} \leq \overline \lambda _{j,n} \leq
\beta < 1, \, \frak F$ is the $\sigma-$algebra of Borel sets in $\Lambda$, and $\nu$
is an arbitrary stationary, shift-invariant ergodic Borel
probability measure on $\Lambda$
\item"{\bf b)}" a compact set $Q \subset \Sigma^+_p$ invariant under the shift (i.e.,
$\sigma(Q)=Q$)
\item"{\bf c)}" for $\nu-$almost every $\vec \lambda \in \Lambda$,
a family of sets $ \{ \Delta_{i_1 \cdots i_n} \}(\vec
\lambda) \subset \Bbb R^n $ \, for \, $ i_j = 1, 2, \dots, p, $ \, where
the $n$-tuple $(i_1 \cdots i_n)$ is admissible with respect to
$Q$ and satisfies
$$
D ( C_1 \prod_{j=1}^n \underline \lambda_{i_j, j}(\vec \lambda))
\subset
\Delta_{i_1 \cdots i_n}(\vec
\lambda) \subset D ( C_2 \prod_{j=1}^n \overline \lambda_{i_j,
j}(\vec
\lambda))
$$
where $C_1 > 0$ and $C_2 > 0$.
\item"{\bf d)}" for any sequence $(i_1\cdots i_n) \in \{1,
\cdots p \}^n$, we
require that $ \Delta_{i_1 \cdots i_{n+1}}(\vec \lambda) \subset
\Delta_{i_1 \cdots i_n}(\vec \lambda) $ and
$$
\quad \Delta_{i_1 \cdots i_n}(\vec \lambda) \cap \Delta_{i'_1
\cdots
i'_n}(\vec \lambda) = \emptyset, \, \text{ if } \,(i_1\cdots
i_n) \neq
(i'_1,\cdots i'_n).
$$
\endroster
For every $\vec \lambda \in \Lambda$, the limit set
$$
F(\vec \lambda) = \bigcap_{n=1}^{\infty} \bigcup_{\underset
\text{admissible} \to {(i_1 \cdots i_n)}} \Delta_{i_1 \cdots
i_n}(\vec \lambda)
$$
is a perfect, nowhere dense, totally disconnected set.
The following Lemma describes the limiting behavior of the numbers $\underline
\lambda_{i,n}, \overline \lambda_{i,n}$ in the random symbolic construction:
\proclaim{\bf Lemma 1}
Let $F$ be the limit set specified by a random symbolic geometric construction.
Then there are numbers $\ll_i , \oli, 0 < \lli \leq \oli
< 1, i=1, \cdots, p$ such that for $\nu-$almost every $\vec
\lambda \in \Lambda$ the following limits exist:
\roster
\item for $\mu_{\underline \lam}-$almost every sequence $(i_1 i_2 \cdots) \in Q(\vec
\lambda)$
$$
\lim_{n \ra \infty} \frac1n \sum_{k=1}^{n}\underline a_{i_k,k} = 0
$$
\item for $\mu_{\overline \lam}-$almost every sequence $(i_1 i_2 \cdots) \in Q(\vec \lambda)$
$$
\lim_{n \ra \infty} \frac1n \sum_{k=1}^{n} \overline a_{i_k,k} = 0
$$
where $\underline a_{i, j} = \log \left ( \frac{\underline \lambda_{i, j}}{\ll_i } \right),
\overline a_{i, j} = \log \left ( \frac{\overline \lambda_{i,
j}}{\ol_i } \right ), \, \underline \lam = (\underline \lam_1, \dots, \underline \lam_p),
\overline \lam = (\overline \lam_1, \dots, \overline \lam_p)$.
\endroster
\endproclaim
\medskip
The next statement immediately follows from Lemma 1 and Propositions 7 and 9.
\medskip
\proclaim{\bf Proposition 11} Let $F$ be the limit set specified by a random symbolic
construction. Assume that either the construction is one-dimensional or $\ll_i = \ol_i$ for $i = 1,
\dots, p$. Then for $\nu-$almost every $\vec \lambda \in \Lambda$,
$$
s_{\ll} \leq \dHF(\vec \lambda).
$$
\endproclaim
\bigskip
\head{\bf Section 4: Generating Maps and Codings} \endhead
\bigskip
One particular but important case of a geometric construction is when
the sets $ \Delta_{i_1 \cdots i_n} $ are given by
$$
\Delta_{i_1
\cdots i_n} = h_{i_n} \circ h_{i_{n-1}} \circ \cdots \circ
h_{i_1}(\Delta) \quad \tag 9
$$
where $ h_1, \cdots, h_p: \Delta \to \Delta $ are contraction maps, i.e., $ dist(h_i(x), h_i(y))
\leq L_i dist(x, y)$ with $L_i < 1$ and $x,y \in \Delta$ (a ball in $\Bbb R^n)$. Most of the results
in the
literature in dimension theory require that the process be described in
this way. We stress that this is a very special case and that the
following situations can (and do) occur:
\medskip
{\bf (1)} {\it The construction can not be described by
any continuous maps}, i.e., there are no continuous maps satisfying
(9). This can occur if the boundary of a set $\Delta_{i_1 \cdots
i_n}$ is fractal. In the one-dimensional case the maps $h_j$ always
exist and are continuous, so the above mentioned pathology does not
occur. In this case the maps are well defined by
$$ h_j(\partial
\Delta_{i_1 \cdots i_n}) = \partial \Delta_{i_1 \cdots i_n j} $$ for
any admissible sequence $(i_1 \cdots i_n j)$, where $\partial
\Delta_{i_1 \cdots i_n j}$ denotes the boundary of the set $\Delta_{i_1
\cdots i_n j} $.
\medskip
{\bf (2)} {\it There are continuous maps
satisfying (9) that are not Lipschitz.} One obstruction is that the
boundary of a set $ \Delta_{i_1 \cdots i_n}$ can be a continuous but
not Lipschitz image of $\partial \Delta$. In the one-dimensional case,
this obstruction cannot occur. However, one can construct a
one-dimensional example where the process is defined by continuous but
not Lipschitz maps. Namely, there exists a simple geometric
construction on $[0,1]$ with $p=2, \, \underline \lambda_1 = \overline
\lambda_1 = \lambda_1, \, \underline \lambda_2 = \overline \lambda_2
= \lambda_2, \, 0 < \lambda_1 < \lambda_2 < 1$ and the locatios of the
intervals $\Delta_{i_1 \cdots i_n}$ are such that the map $h_1$ is not
Lipschitz. Choose
$$
\frac{d (\Delta{\underbrace{_{1 \cdots
1}}_{n_k}}, \Delta_{{\underbrace{\scriptstyle{1 \cdots 1}}_{n_k-1}},2}
) }{d (\Delta_{\underbrace{_{1, \cdots, 1}}_{n_k+1}},
\Delta_{{\underbrace{\scriptstyle{1 \cdots 1}}_{n_k-1}},2,1} )}
\overset {n_k \to \infty} \to \longrightarrow \infty.
$$
\medskip
{\bf (3)}{\it There are Lipschitz maps satisfying (9) but these maps are
not contractions.} To see this, choose intervals \, $\Delta_1,
\Delta_2$ and maps $h_1, h_2$ such that $\Delta_{11}= h_1(\Delta_1)$,
$\Delta_{21}=h_1(\Delta_2)$ and $d(\Delta_1, \Delta_2) <<
d(\Delta_{11}, \Delta_{21})$. Then the map $h_1$ is not a
contraction.
\medskip
{\bf (4)} {\it There are contraction maps satisfying (9) whose
inverse maps are not Lipschitz.} The Lipschitz constants for the
inverse maps depend on the gaps between the sets $\Delta_{i_1 \cdots
i_n}$. To see this, choose intervals $\Delta_{i_1 \cdots i_n}$ such
that $$ \frac{d (\Delta{\underbrace{_{1 \cdots 1}}_n},
\Delta_{{\underbrace{\scriptstyle{1 \cdots 1}}_{n-1}},2}
) }{d (\Delta_{\underbrace{_{1 \cdots 1}}_{n+1}},
\Delta_{{\underbrace{\scriptstyle{1 \cdots 1}}_{n-1}},2,1} )} \overset
{n \to \infty} \to \longrightarrow 0. $$
Then the inverse of the map $h_1$ is not Lipschitz.
\medskip
{\bf (5)} {\it There are contraction maps satisfying (9) whose inverse maps
are Lipschitz but the maps are not similarities} (i.e., $d(h_i(x),
h_i(y)) = L_i d(x,y)$ for all $x \in \Delta$).
\medskip
As we saw in case (4) the Lipschitz constants for the inverse maps
may depend on the gaps between the sets $\Delta_{i_1 \cdots i_n}$. For
this reason, even if the process can be described using contraction
maps with Lipschitz inverses, they may be of no use in estimating the
Hausdorff dimension of the limit set.
\bigskip
We point out that the coding we employ in our definitions of
geometric constructions is not the obvious generalization of the {\it
natural} coding for a similarity process. The natural coding is $\Delta_{i_1 \cdots i_n} = g_{i_n} \circ
g_{i_{n-1}} \circ \cdots\ g_{i_1} (\Delta)$ where the maps $g_k$ are
affine contractions. It follows that $\Delta_{i_1 \cdots i_n} \subset
\Delta_{i_2 \cdots i_n}$. This coding has several undesirable
properties which the coding that we adopted does not share. One
undesirable property is that for a given $x \in F$ with coding $(i_1 i_2 \cdots
)$, one can not determine which $ \Delta_k$ the point $x$ lies in.
The maps $h_k$ have an interesting manifestation on the level of
symbolic dynamics. The mapping $G: F \to F$ defined in Section 1, Remark (5) can
be described as $$ G(x) = h_k^{-1}(x) \quad \text{if } x \in \Delta_k.
$$ If the maps $h_k$ are all similarities, then the map $G$ is a
contraction.
\medskip
\head{\bf Section 5: More Examples} \endhead
\vskip 0.25in
%\centerline{\epsfxsize=4.0in \epsffile{pfig2.ps}}
\vskip 0.25in
\midinsert
%\vspace{4truein}
\botcaption{Figure 2. \quad Sierpi\'nski Gaskets \quad a), \, b), \, c)}
\endcaption
\endinsert
\medskip
{\bf 1)} {\bf Sierpi\'nski Gaskets}
\itemitem{a)} It is well known that the
Hausdorff dimension of the Sierpi\'nski gasket (Figure 2a)
coincides with the box dimension and is $ \frac{\log3}{\log2}$. This
immediately follows from Corollary 2 and Proposition 3 since $\lambda_1
= \lambda_2 =\lambda_3=\frac12$ and $p=3$.
\itemitem{b)} Suppose that in the construction of the Sierpi\'nski
gasket we forbid all configurations whose codings contain a $1$ followed by a $2$ (Figure 2b). The spectral radius of
$ \left( \matrix 1& 0& 1 \\ 1& 1 & 1\\ 1 & 1 & 1\endmatrix \right)$
is $\frac{3+\sqrt{5}}{2}$. Hence by Proposition 1 and Corollary 1, $ \dHF =\frac{\log
(\frac{3+\sqrt{5}}{2})}{\log 2} \allowmathbreak \approx 1.38848$.
\itemitem{c)} A simple construction
of the Sierpi\'nski gasket with $\lambda_i = \frac13$ \, for $i =1, 2, 3$ is
illustrated in Figure 2c). The
sets $\Delta_{i_1 \cdots i_n}$ are {\it asymptotically congruent} to
the corresponding triangles in the usual constructions and possess
{\it wiggly boundaries} that become {\it asymptotically straight.} As
long as the approximation is sufficiently fast and uniform, such
that the construction satisfies the hypotheses of Proposition 8, then the
Hausdorff dimension of the limit set is $ \frac{\log3}{\log2}$. One
can even choose the sets $\Delta_{i_1 \cdots i_n}$ in the construction
to have fractal boundaries.
\medskip
{\bf 2)} {\bf General Smale-Williams solenoid} \newline Let $P$ be a
solid torus embedded in
$\Bbb R^3$. We represent points on $P$ by means of coordinates
$(\theta, r ,s)$ where $\theta \in S^1 \text{the unit circle}, \, -1
\leq r, s \leq 1$ such that $r^2 + s^2 \leq 1$. The point $x$ with
coordinates $(\theta, r, s)$ belongs to the plane orthogonal to the
core of the torus through the point $\theta \in S^1$ having position
$(r, s)$ relative to the standard frame $(e_1, e_2)$. We define a
mapping $f: P \to P$ by $$ f(\theta, r, s) = ( 2 \, \theta, \lambda_1
r + \epsilon \cos \theta, \lambda_2 s + \epsilon \sin \theta) $$
where $p>0$ is an integer, $\epsilon$ is a small positive constant and
$0 < \lambda_1, \lambda_2 < 1$. The image $f(P)$ is contained in $P$
and wraps twice around $P$. See Figure 3a). The set $\Delta =
\cap_{n=1}^{\infty} f^n(P)$ is called a {\it solenoid}. See \cite{Sh}
for more details and nice pictures.
\midspace{2.0truein}
\midinsert
\botcaption{Figure 3. \quad Smale-Williams Solenoid \quad a), b)}
\endcaption
\endinsert
\medskip
Let $D_{\theta}$ be the section of $P$ determined by the plane
perpendicular to the core at $\theta$. The set $\Delta_{\theta} =
\Delta \cap D_{\theta}$ is the Cantor-like set obtained by the simple
geometric construction with $\underline \lambda_i= \lambda_1$ and
$\overline \lambda_i= \lambda_2$ for i = 1, 2. See Figure 3b.
Since the basic sets in the construction of $\Delta_{\theta}$ are
rectangles, it follows from Proposition 4 that
$$
\frac{\log 2}{ \log
(\frac{1}{\lambda_1})} \leq \dim_H \Delta_{\theta} \leq \frac{\log 2
}{\log( \frac{1}{\lambda_2})}.
$$
Appling Marstrand's theorem [F1],
we obtain
$$
1+ \frac{\log 2 }{ \log (\frac{1}{\lambda_1})} \leq
\dim_H \Delta.
$$
A very simple argument \cite{F1} shows that $ \dim_H
\Delta \leq 1 +\frac{\log 2 }{ \log( \frac{1}{\lambda_2})}$.
If $\lambda_1 = \lambda_2 = \lambda$, we have that $\dim_H
\Delta_{\theta} = \frac{\log 2 }{ \log (\frac{1}{\lambda})}$ and
$\dim_H \Delta = \frac{\log 2 }{ \log (\frac{1}{\lambda})} + 1$, which is the
result obtained by Falconer \cite{F1}.
\medskip
{\bf 3)} {\bf Geometric Construction With Rectangles in the Plane}.
The following simple example illustrates the fact that the regularity of the
construction depends not only on the sizes and shapes of the basic sets but also may depend on their spacing. Consider the two similarity constructions where all the basic sets at step
$n$ are congruent rectangles with width $\underline \lam^n$ and length $ \overline \lam^n$.
The rectangles are stacked horizontally in the first construction and vertically in the
second construction. The limit sets of both constructions are
one-dimensional Cantor sets. The first has Hausdorff dimension $\frac{\log 2}{- \log \overline \lam}$ and the second has Hausdorff dimension $\frac{\log 2}{- \log \underline \lam}$. The first construction is regular with l-estimating vector $\g = (\overline \lam, \overline \lam)$.
The vector $(\overline \lam, \overline \lam)$ is not l-estimating for
the first construction but is l-estimating for the second construction.
\medskip
{\bf 4)} {\bf A simple asymptotic Moran construction}.
This example shows that the second statement of Proposition 3 may fail for an asymptotic Moran construction, i.e., the measure of the limit set may be zero.
Let $p=2$ and suppose $ \lambda_{i, n} = \lambda_n = \lambda \exp(
a_n)$ for $i = 1, 2$. Let $F$ denote the limit set. Clearly
$$
\diam(\Delta_{i_1 \cdots i_n}) = \prod_{j=1}^n \lambda_{i_j, j} =\lambda^n
\exp (A_n),
$$
where $A_n = \sum_{k=1}^n a_k$. For
fixed $n$, the sets $\{ \Delta_{i_1 \cdots i_n}\}$ give a cover of $F$
with
$$
\sum_{i_1 \cdots i_n} \diam(\Delta_{i_1 \cdots i_n})^s = 2^n
\lambda^{ns} \exp (s A_n) = \exp (s A_n),
$$
where $s=\frac{\log
2}{\log (\frac{1}{\lambda})}$.
There is now a trichotomy:
\roster
\item if $\sum_{k=1}^\infty a_k < \infty$, then one can easily show
that the geometric process satisfies conditions a1) and a2), and by Proposition 8, $s= \dHF = \dBF = \frac{\log 2}{\log (\frac{1}{\lambda})}$.
\item if $\sum_{k=1}^\infty a_k = \infty$, then one can show that
$m_H(s, F) = \infty$.
\item if $\sum_{k=1}^\infty a_k = -\infty$,
then $m_H(s, F) = 0$.
\endroster
\medskip
The sequence $\{a_k\}$ satisfies the condition in the definition of asymptotic Moran construction if and
only if $\frac1n A_n \ra 0$. For example, choose $a_n =
\frac1n$ in the case (2) and $a_n = -\frac1n$ in case (3). One can show that $s$ is still the Hausdorff dimension of the limit set $F$. The mass
distribution principle does not hold in case 3) since otherwise we
would have $m_H(s, F) > 0$.
We now construct gauge functions for certain sequences $\{a_j\}$ where
$\sum_{k=1}^\infty a_j = - \infty$. Let $A_n = \sum_{k=1}^n a_k$. We
seek a function $h(t)$ such that $0 < 2^n h(\lambda^n \exp(A_n)) <
\infty$. We will find $h(t)$ in the form $h(t) = t^s \exp(\phi(t))$.
Then we would have $- \infty < s A_n + \phi(\lambda^n \exp(A_n)) <
\infty$. Define $t=\lambda^n \exp(A_n)$. Since $\lim_{n \to
\infty} \frac{A_n}{n} = 0$, then $t \asymp \lambda^n$, and hence we can
set $\phi(t) = -s A_{\frac{\log t}{\log \lambda}}$.
If $a_n = - \frac1n$ then $A_n= O(-\log n)$ and $\phi(t)= \log
(\frac{\log t}{\log \lambda})$. If $a_n = - \frac{1}{n \log n}$ then
$A_n = O(-\log(\log n))$ and $\phi(t)=\log (\log (\frac{\log t}{\log
\lambda}))$. More generally, if $a_n = - \frac{1}{n \log^{(i)} n}$,
where $\log^{(i)} n$ denotes the $i-$fold composition of $\log n$, then
$A_n = O(-\log^{(i+1)} n)$ and hence $\phi(t) = \log^{(i+1)}
(\frac{\log t}{\log \lambda})$. We can thus obtain gauge functions with
{\it arbitrarily many logs} from this basic construction. For these
different sequences, the coefficients $\lambda_n = \lambda \exp(a_n)$
converge to $\lambda$, but of course with different speeds, and hence,
require different gauge functions.
\medskip
{\bf 5)} {\bf Random version of Example 4}
The following example is a special case of random
constructions that was pointed out to the authors by Peres. We
consider the construction in Example 4 where the numbers $\{a_n\}$ are
independent and identically distributed random variables on the
interval $\infty < a \leq a_n \leq b < \log(\frac{1}{\lambda})$ having
mean $0$. If we define the random variable $A_n = \sum_{k=1}^n a_k$,
then the law of the iterated logarithm implies that $\liminf A_n = -
\infty$. The law of large numbers shows that $\frac{A_n}{n} \ra 0$. It immediately
follows from a simple calculation as in Example 4 that $s = \dHF =
\frac{\log 2}{- \log \lambda}$ and $m_H(s, F) = 0$.
\medskip
{\bf 6)} {\bf Simple geometric construction with rectangles having limit
set F for which ${\bold{dim}}_{\bold H} \bold F \boldkey < \underline {\bold{dim}}_{\bold B} \bold F \boldkey < \overline {\bold{dim}}_{\bold B} \bold F$ }
\medskip
We describe a simple construction with rectangles in $\Bbb R^2$ with $p=2$,
$\underline \lambda_1 = \underline \lambda_2 = \ll$, $\overline
\lambda_1 = \overline \lambda_2 = \ol$, $0 < \ll < \ol < \frac13$ such
that the limit set $F$ satisifes $$ \dHF = \frac{\log 2}{-\log \ll},
\qquad \dBFL = \gamma \frac{\log 2}{-\log \ll}, \qquad \dBFU =
\frac{\log 2}{-\log \ol}, $$ where $\gamma \in (1, \alpha)$ is an
arbitrary number and $\alpha = \left[\frac{\log \ll}{\log \ol}\right] >
1$.
Let $n_0=0$ and for $k = 0, 1, 2, \cdots$, let $n_{k+1} = [\alpha n_k]
+ 2 $ and $\beta_k = 2^{(\gamma - \alpha) n_{3k+1}}$. In order to
describe the $n$th step of the construction we use the basic types of
spacings: vertical stacking (A) and horizontal staking (B). See Figure 4.
\vskip 0.25in
%\centerline{\epsfxsize=4.0in \epsffile{pfig4.ps}}
\vskip 0.25in
\midinsert
\botcaption{Figure 4 \quad vertical stacking \, (a) \qquad horizontal stacking \, (b)}
\endcaption
\endinsert
\medskip
\roster \item We start with two horizontally stacked rectangles.
During steps $n_{3k} < n \leq n_{3k+1}$ we use (B).
\item We begin with $2^{n_{3k+1}}$ rectangles. Choose $\beta_k $
percent of these rectangles arbitrarily and paint them blue; paint the
others green. During steps $n_{3k+1} < n \leq n_{3k+2}$, we use
(B) in all blue rectangles and use (A) in all green rectangles.
\item During steps $n_{3k+2} < n \leq n_{3k+3}$, we use (A) in all
blue rectangles and use (B) in all green rectangles.
\item Repaint all $2^{n_{3k+3}} $ rectangles white; repeat steps 1 to
4.
\endroster
\medskip
The collection of rectangles at the $n$th step contains $2^n$
rectangles each with vertical and horizontal sides of size $\ll^n \times
\ol^n$ (the size in the vertical direction is $\ll^n$ and the size in
the horizontal direction is $\ol^n$). Any two subrectangles on step
$n+1$ that are contained in the same rectangle at step $n$ are stacked
either horizontally or vertically and the distance between them is at
least $\frac13 \ll^n$. The projections of any two distinct rectangles
on step $n$ onto the two coordinate axes either coincide or are disjoint.
\medskip We need the following three lemmas:
\proclaim{\bf Lemma 2} Let $\frak U$ be a covering of the limit set $F$
by disjoint closed balls $B(x_i, r), i=1, \cdots, p$, where $r$ is fixed.
Assume that every ball in the covering has a non-trivial intersection
with the limit set $F$. Then $N(r) \geq \frac16 \#(\frak U)$, where
$N(r)$ denotes the smallest number of balls of radius $r$ needed to
cover $F$. \endproclaim
\demo{ Proof} Let $\frak V = \{B(x'_i, r)
\}, i=1, \cdots, q$ be a covering of $F$. Then by elementary geometry,
for all $x'_j$, there are at most 6 points $x_1, \cdots, x_6$ such that
$F \cap B(x'_j, r) \subset \cup_{k=1}^6 B(x_k, r)$. \qed
\enddemo
\medskip
\proclaim{\bf Lemma 3} For any $n > 0, \, \text{dist}(\Delta_{i_1'
\cdots i_n' }, \Delta_{i_1'' \cdots i_n''}) \geq \frac13 \ll^n$.
\endproclaim
\demo{ Proof} The proof is obvious. \quad \qed
\enddemo
\medskip
\proclaim{\bf Lemma 4} If $0 < r < \epsilon$ and an optimal covering of the
limit set $F$ by balls of radius $r$ consists of disjoint balls
$B(x_i, r)$, then $N(\epsilon) \geq \frac14 N(r) \left(
\frac{r}{\epsilon} \right)^2$.
\endproclaim
\demo{\bf Proof} Let
$\{B(y_j, \epsilon )\}$ be a covering of $F$. For any point
$x_i$ one
can find a point $y_i$ such that $x_i \in B(y_i, \epsilon)$. Hence,
$B(x_i, r) \subset B(y_i, 2 \epsilon)$. Therefore $
\text{vol}( B(x_i, r)) \leq \text{vol}(B(y_i, 2 \epsilon))$. Since the balls $B(x_i, r)$ are disjoint,
this implies that $N(r) r^2 \leq N(\epsilon) (2 \epsilon)^2$. The
lemma immediately follows. \quad \qed
\enddemo
\medskip
We now calculate the Hausdorff dimension and the lower and
upper box dimensions of the limit set $F$.
\medskip \demo{\it a) Calculation of Upper Box Dimension} \enddemo
Let $r_k = \ol^{n_{3k+1}}$. Consider the covering of $F$ by rectangles
$\Delta_{i_1 \cdots i_{n_{3k+1}}}$. Each rectangle can be covered by a
square of side length less than or equal to $\ol^{n_{3k+1}}$ that is
centered at some point $x \in \Delta_{i_1 \cdots i_{n_{3k+1}}}$. These
squares are disjoint since $\ol^{n_{3k+1}} = \ol^{[\alpha n_{3k}]+ 2
}\leq \frac13 \ll^{n_{3k}}$, and by Lemma 3, $\text{dist}(\Delta_{i_1'
\cdots i'_{n_{3k}} }, \Delta_{i_1'' \cdots i''_{n_{3k}}}) \geq
\frac13 \ll^{n_{3k}}$. By Lemma 2, $N(r_k) \geq \frac16 2^{n_{3k+1}}$ and
hence
$$
\dBFU \geq \overline \lim_{k \to \infty} \frac{\log N(r_k)}{-
\log r_k} = \frac{\log 2}{- \log \ol}.
$$
On the other hand, by Proposition 4, $\dBFU \leq \frac{\log 2}{- \log \ol}$. It follows that
$\dBFU = \frac{\log 2}{- \log \ol}$.
\medskip
\demo{\it b) Calculation of Hausdorff Dimension} \enddemo \medskip
Given $\epsilon > 0$ choose $k > 0$ such that $\ll^{n_{3k+1}} \leq
\epsilon $. Consider the covering of $F$ consisting of green
rectangles for $n=n_{3k+1}$ and blue rectangles for $n=n_{3k+2}$.
Consider a green rectangle $\Delta_{i_1 \cdots i_{n_{3k+1}}}$. By
construction, the intersection $A = \Delta_{i_1 \cdots i_{n_{3k+1}}}
\allowmathbreak \cap F$ is contained in $2^{n_{3k+2}-{n_{3k+1}}}$ small green
rectangles corresponding to $n={n_{3k+2}}$. These rectangles are vertically
aligned and have size $\ll^{n_{3k+2} } \times \ol^{n_{3k+2}}$. Since
$\ol^{n_{3k+2}} = \ol^{[\alpha n_{3k+1} ] + 2} \leq \frac13
\ll^{n_{3k+1}}$, the set $A$ is contained in a square of size
$\ll^{n_{3k+1}} $. Thus we have $(1 - \beta_k) 2^{n_{3k+1}}$ green
squares of length $\ll^{n_{3k+1}}$.
Now consider a blue rectangle $\Delta_{i_1 \cdots i_{n_{3k+1}}}$. By
our construction, the intersection $B = \Delta_{i_1 \cdots
i_{n_{3k+2}}} \cap F$ is contained in $2^{n_{3k+3}-{n_{3k+2}}}$ small
blue rectangles corresponding to $n={n_{3k+3}}$. They are vertically
aligned and have size $\ll^{n_{3k+3} } \times \ol^{n_{3k+3}}$. Since
$\ol^{n_{3k+3}} \leq \frac13 \ll^{n_{3k+2}}$, the set $B$ is contained
in a square of size $\ll^{ n_{3k+2}} $. Thus we have $\beta_k
2^{n_{3k+1}}\, 2^{n_{3k+2}- n_{3k+1}} = \beta_k 2^{n_{3k+2}}$ blue
squares of length $ \ll^{n_{3k+1}}$.
The collection of green and blue rectangles comprise a covering $\frak
G = \{U_i\}$ of $F$ such that
$$
\sum_{U_i \in \frak G} (\text{diam } U_i)^{s} = (1-\beta_k)
2^{n_{3k+1}}(2 \ll^{n_{3k+1}})^{s} + \beta_k 2^{n_{3k+2}}(2
\ll^{n_{3k+2}})^{s} = 1 > 0 $$ if $s= \frac{\log 2}{- \log \ll}$. This
implies that $\dHF \leq \frac{\log 2}{- \log \ol}$. On the other hand,
by Proposition 4, we know that
$\dHF \geq \frac{\log 2}{- \log \ll}$, hence $\dHF = \frac{\log 2}{-
\log \ll}$.
\medskip \demo{\it c) Calculation of Lower Box Dimension} \enddemo
\medskip Consider $r_k = \ll^{n_{3k+1}}$ and the covering $ \frak
G_k$ of $F$ by green rectangles for $n=n_{3k+1}$ and blue rectangles
for $n=n_{3k+2}$.
There are $(1 - \beta_k) 2^{n_{3k+1}}$ green rectangles and the
intersection of each of them with $F$ is contained in a square of size
$2 \ll^{n_{3k+1}}$. There are $\beta_k \, 2^{n_{3k+1}} \, 2^{n_{3k+2}-
n_{3k+1}} = \beta_k 2^{n_{3k+2}}$ \, blue rectangles each of which is
contained in a square of size $ \ol^{n_{3k+2}}$. It is easy to see that
$$ L_1 \ll^{n_{3k+1}} \leq \ol^{n_{3k+2}} \leq L_2 \ll^{n_{3k+1}},
$$
where $L_1, L_2$ are positive constants. Therefore $\frak G_k$
induces a covering $\frak H_k= \{U_i\}$ of $F$ by squares of size $L
\ll^{3k+3}$, where $L > 0$ is a constant. The cardinality $N_k$ of
this covering is
$$
N_k= \left( (1-\beta_k) 2^{n_{3k+1}}+ \beta_k
2^{n_{3k+2}} \right) \leq ((1-\beta_k) 2^{n_{3k+1}} + 4 \beta_k
2^{\alpha n_{3k+1}}).
$$
Since $\alpha > \gamma$, we have that
$$
(1 - \beta_k) 2^{ (\alpha- \gamma) n_{3k+1}} \to 0
$$
as $k \to
\infty$ and by the definition of $\beta_k$,
$$
\beta_k 2^{ (\alpha -
\gamma) n_{3k+1}} = 1.
$$
Therefore
$$
\dBFL \leq \lim_{k \to \infty}
\frac{\log N(r_k)}{- \log r_k} = \gamma \frac{\log 2}{- \log \ll}. $$
\medskip Choose a sequence $\epsilon_m \to 0$. Given $m > 0$, one can
choose an integer $k = k(m)$ satisfying $\ll^{n_{3k+1}} \leq
\epsilon_m \leq L_3 \ll^{n_{3k+1}} $, where $L_3$ is a positive
constant that is independent of $m$. We wish to compare $N(\epsilon_m)$
and $N(r_k)$. Since the squares comprising the covering $\frak H_k$
are disjoint, Lemma 3 implies
$$
N(\epsilon_m) \geq \frac14 N(r_k)
\left (\frac{r_k}{\epsilon_m} \right)^2.
$$
It follows that $$ \dBFL=
\lim_{m \to \infty} \frac{\log N(\epsilon_m)}{- \log \epsilon_m}
\geq \limsup_{m \to \infty} \frac{\log N(r_k)}{-\log r_k} = \gamma
\frac{\log 2}{ - \log \ol}.
$$
\medskip
{\bf 8)} {\bf Example where diameters of inscribed balls in basic sets
is not an l-estimating vector}
Let $\g_1, \g_2, \g_3, \lam$ be any numbers in $(0, 1)$ and let $A(\g_i)$
denote a simple geometric construction on the interval
$[0, 1] \times \{i\}, \, i=-1, 0, 1$ with $2^n$ basic sets of size $\g_i^n$
at step $n$. We wish to use these three one-dimensional constructions
to define a simple geometric construction in the square $[0, 1] \times [-1, 1]$.
Since the $2^n$ intervals at step $n$ in each of the one-dimensional constructions are clearly ordered, we may refer to the $i$th subinterval at step $n, \, 1 \leq i \leq 2^n$ of these constructions. Consider the
$2^n$ polygons in $[0, 1] \times [-1, 1]$ having six vertices which consist of the two endpoints of the $i$th subinterval at step $n$ for all three constructions.
We define the $2^n$ basic sets of our construction at step $n$ by
intersecting these $2^n$ polygons with the square $[0, 1] \times [-\lam^n, \lam^n]$. See Figure 5.
\vskip 0.25in
%\centerline{\epsfysize=2.75in \epsffile{pfig5.ps}}
\vskip 0.25in
\midinsert
\botcaption{Figure 5.}
\endcaption
\endinsert
\medskip
It is easy to see that the limit set $F$ of this construction coincides with the
limit set of the construction $A(\g_2)$. Hence $\dim_HF =
\frac{\log 2}{-\log \g_2}$ and does not depend on
$\g_1, \g_3$ or $\lam$. If we choose $\g_2 < \g_1 = \g_3 < \lam$, then the
inscribed and circumscribed balls of the basic sets at step $n$ have radii $C_1 \g_1^n$ and $C_2 \lam^n$, where $C_1$ and $ C_2 $ are positive
constants that are independent of $n$. Thus these balls cannot be used to determine the Hausdorff dimension of the limit set.
\bigskip
\head{\bf Section 6: Pointwise Dimension of Measures Concentrated on General
Cantor-like Sets}
\endhead
\bigskip
\subhead{\bf 6.1} \endsubhead
Let $F$ be the limit set
for an asymptotic Moran or one-dimensional symbolic construction. We formulate a powerful criterion that allows one to
estimate the lower and upper pointwise dimensions with respect to a
Borel probability measure $\nu$ on $F$. Given $x \in F, n > 0$
consider the unique set $\Delta_{i_1 \cdots i_n} = \Delta_n(x)$ that
contains the point $x$. Denote $$ \underline d (x) = \liminf_{n \to
\infty} \frac{\log \nu(\Delta_n(x))}{\log \diam \Delta_n(x)}; $$ $$
\overline d (x) = \limsup_{n \to \infty} \frac{\log
\nu(\Delta_n(x))}{\log \diam \Delta_n(x)}. $$
The following theorem is related to a result in [C]:
\proclaim{\bf Theorem 7}
\roster
\item $\overline d_{\nu}(x) \leq
\overline d(x)$ and $\underline d(x) \leq \underline d_{\nu}(x)$ for
all $x \in F$.
\item If $\underline d(x) = \overline d(x) \overset
\text{def} \to \equiv d(x)$ for $\nu-$almost every $x \in F$, then $$
\underline d_{\nu}(x) = \overline d_{\nu}(x) = d(x) $$ for $\nu-$almost
every $x \in F$.
\endroster
\endproclaim
The next statement is an immediate corollary of Propositions 7 and 9.
\proclaim{\bf Corollary 3} Let $F$ be the limit set for an asymptotic
Moran or one-dimensional construction. Assume that
there is a Borel measure $\nu$ on $F$ such that $\underline d(x) =
\overline d(x) \overset \text{def} \to \equiv s$ for $\nu-$almost
every $x \in F$. Then $s \leq \dHF$. \endproclaim
\medskip
\subhead{\bf 6.2} \endsubhead We present several computations of pointwise dimension. We
start with Example 6 and show that:
\it{
\itemitem{a)} $\underline d_{\lm}(x) = \ls$ for $\lm-$almost every $x
\in F$ \, \text{ and } \, $\underline d_{\om}(x) = \ls$ for $\om-$almost
every $x \in F$, where $\underline s = \frac{\log 2 }{\ \log \underline \lambda}$
\smallskip
\itemitem{b)} $\overline d_{\lm}(x) =\overline d_{\om}(x)= \os$
for every $x \in F$, } where $\overline s = \frac{\log 2 }{\ \log \overline \lambda}$.
\rm
\medskip
The fact that $\dHF = \ls$ immediately
implies that
$$
\underline d_{\lm}(x) \leq \ls \enspace \text{for }
\lm-\text{almost every } x \in F
$$
and
$$
\underline d_{\om}(x) \leq
\ls \enspace \text{for } \om-\text{almost every } x \in F.
$$
Otherwise there would exist a set $A$ of positive $\lm-$measure
(respectively $\om-$measure) with $\underline d_{\lm}(x) \geq \ls +
\epsilon$ (respectively $\underline d_{\om}(x) \geq \ls + \epsilon$)
for any $x \in A$. The non-uniform mass distribution principle would
then imply that $\dHF \geq \text{dim}_{H}A \geq \ls + \epsilon$.
Statement (a) follows from Statement 2 in Proposition 4.
In order to prove (b), consider $r_k = \ol^{n_{3k+1}}$ and denote
$\Delta_k(x)$ the unique cylinder set $\Delta_{i_1 \cdots n_{3k+1}}$
that contains $x \in F$. It is easy to see that $B(x, r_k) \bigcap F
\subset \Delta_k(x) \bigcap F$. This implies that for all $x
\in F$,
$$
\om(B(x, r_k) \cap F) \leq \om(\Delta_k(x) \cap F) \leq
D_1 \ol^{\os n_{3k+1}} = D_1 r_k^{\os}
$$ and hence
$$
\overline
d_{\om}(x) \geq \limsup_{k \to \infty} \frac{\log \om(B(x, r_k) \cap
F)}{\log r_k} \geq \os.
$$
Similarly, we have that for all $x \in F$ $$ \lm(B(x, r_k)
\cap F) \leq \lm(\Delta_k(x) \cap F) \leq D_1 \ll^{\ls n_{3k+1}}
$$ and hence,
$$
\overline d_{\lm}(x) \geq \limsup_{k \to \infty}
\frac{\log \lm(B(x, r_k) \cap F)}{\log r_k} = \ls \frac{\log
\ll}{\log \ol} = \os.
$$
Statement (b) now follows from the second
statement of Proposition 4. \quad \qed
\medskip
\subhead {\bf 6.3} \endsubhead Let $F$ be the
limit set generated by the simple geometric construction in Example 6,
$G:F \to F$ the map $G=\chi^{-1} \circ \sigma\circ \chi$, where $\chi$
denotes the coding map and $\sigma:\Sigma_p^+ \to :\Sigma_p^+$ is the
full shift (see Remark 5). Consider the set $\tilde F= F \times F$
endowed with the metric
$$
\tilde \rho((x_1, y_1), (x_2, y_2) ) =
\rho(x_1, x_2) + \rho(y_1, y_2), \qquad x_1, x_2, y_1, y_2 \in F
$$
and
the coding map $\tilde \chi: \tilde F \to \Sigma_p$ defined by
$$
\tilde \chi(x, y) = (\cdots
i_{-1} i_0 i_1 \cdots)
$$
where $\Sigma_p$
denotes the space of two-sided sequences $( \cdots i_{-1} i_0 i_1
\cdots), \, i_j =1 \cdots p$ and $\chi(x) = (\cdots i_{-1}), \, \chi(y)=
(i_0 i_1 \cdots)$. Set $\tilde G = \tilde \chi^{-1} \circ \sigma \circ
\tilde \chi$. It is easy to see that $\tilde G$ is a homeomorphism and
that for any $(x, y) \in \tilde F$, $$ \pi_1 \tilde G(x, y) = G(x),
\quad \pi_2 \tilde G^{-1}(x, y) = G(y) $$ where $\pi_1, \pi_2$ are the
projections, $\pi_1(x, y) = x$ and $\pi_2(x, y) = y$. Consider the measure $\tilde {\lm} = \tilde {\chi}^* \tilde
{\lmu}$ where $ \tilde {\lmu}$ is the measure on $\Sigma_p^+$ defined by
$$
\tilde {\lmu}(\Delta_{i_k \cdots i_n}) = \ll^{(n-k) \ls}.
$$
By virtue of Corollary 2, $ \tilde {\lmu}$ is invariant under $\sigma$
and hence $ \tilde {\lm}$ is invariant under $\tilde G$. It is easy
to see that $ \tilde {\lm} = \lm \times \lm$. It follows that for
$\tilde {\lm}-$almost every $(x, y)$ $$ {\underline d}_{\tilde
{\underline m}}(x, y) = {\underline d}_{\underline m}(x) +
{\underline d}_{\underline m}(y) = \ls $$ $$ {\overline d}_{\tilde
{\underline m}}(x, y) = {\overline d}_{ \underline m}(x) +
{\overline d}_{ \underline m}(y) = \os.
$$
It is not difficult to
check that ${\tilde {\underline m}}$ is an equilibrium measure (and is
a Gibbs measure) corresponding to the function $\ls \underline \phi(x, y) =
2 \ls \log \ll$. Thus the map $\tilde G$ provides an example of a
homeomorphism that is not a smooth map but possesses a Gibbs measure with
different lower and upper pointwise dimension almost everywhere. The
same is also true with respect to the measure $\tilde {\om}= \tilde{
\chi}^* \tilde {\omu}$ where $\tilde {\omu}$ is defined by $\tilde
{\omu}(\Delta_{i_k \cdots i_n})= \ol^{(n-k) \os}$. As we mentioned,
this does not hold for smooth maps \cite{L, PY}.
\medskip
\subhead{\bf 6.4} \endsubhead We will construct a simple geometric construction for which the map $G$
(see Remark (5)) possesses an ergodic invariant measure with positive
entropy whose pointwise dimension exists almost everywhere, and is not
constant. Our approach follows [C].
Let $F$ be the limit set for simple geometric construction on $[0, 1]$
with $p=3$ and ratio coefficients $\lambda_{i, n}, \, i=1, 2, 3
$ and $ n=1, 2, 3, \cdots$ given by
$$
\lambda_{1,n}=\lambda_{2,n}= \cases \alpha &
\text{if $n$ is even}\\ \beta &\text{if $n$ is odd}, \endcases
$$
\smallskip
$$
\lambda_{3,n}= \cases \gamma, & \text{if $n$ is even}\\
\delta, &\text{if $n$ is odd}, \endcases
$$
where $0 < \alpha < \beta < 1, \, 0< \gamma < \delta < 1$ and $\alpha \beta \neq \gamma \delta$. It is easy to show that the
following limit exists:
$$
\log \lambda_i \overset \text{def} \to
\equiv \lim_{n \to \infty} \frac1n \sum_{k=1}^n \log \lambda_{i, k} =
\cases \frac12 \log (\alpha \beta), & \text{if $i=1,2$}\\ \frac12 \log (\delta \gamma), &\text{if $i=3$}, \endcases
$$
and thus the construction is a one-dimensional asymptotic construction.
Let $s$ be the unique root of the equation
$$
2(\sqrt{\alpha \beta})^s
+ (\sqrt{\gamma \delta})^s =1.
$$
According to Proposition 9 we have that $ s \leq \dHF$.
Consider the transitive stochastic matrix
$$
P = \left( \matrix
0 &1&0\\
\frac12 & 0 & \frac12 \\
0& 1 & 0
\endmatrix \right).
$$
Let $\mu$ be the Markov measure determined by $P$ and let $\nu = \chi^* \mu$.
Consider the sets
$$
A = \left \{ x \in F \, : \, \chi(x) =
(i_1 i_2 \cdots) \in \Sigma^+_P \text{ with } i_j = 1 \text{ or } 2 \right \}
$$
and
$$ B= \left \{ x \in F \, : \, \chi(x) = (i_1 i_2 \cdots) \in \Sigma^+_P \text{
with } i_1 = 3\right \}.
$$
The sets $A$ and $B$ are not invariant under $G$ and $\nu(A) > 0, \, \nu(B) > 0$. Noting that $P^3 = P$, and using arguments in \cite{C}, one can show that for $\nu-$almost every $x \in A$,
$$
\lim_{n \to \infty} \frac{\log \diam \Delta_n(x) }{n } = \frac12
\log (\alpha \beta)
$$
and for $\nu-$almost every $x \in B$,
$$
\lim_{n \to \infty} \frac{\log \diam \Delta_n(x) }{n } = \frac12
\log (\gamma \delta).
$$
Consider the map $G = \chi^{-1} \circ \sigma \circ
\chi$. Since $\chi(\Delta_n(x))$ is a cylinder set, the
Shanon-McMillan-Breiman Theorem implies that for $\nu-$almost every $x
\in F$,
$$ \lim_{n \to \infty} \frac{\log \nu(\Delta_n(x))}{n} =
h_{\nu}(G) = h_{\mu}(\sigma) > 0
$$ where $h_{\nu}(G)$ is the
Kolmogorov-Sinai entropy of the map $G$. Thus
$$
\underline d(x) = \overline d(x) = d(x) = \frac{h_{\mu}(\sigma)}{ \lim_{n \to \infty} \left( \frac{\log \diam
\Delta_n(x)}{n} \right) } = \frac{2 h_{\nu}(G)}{\log(\alpha \beta)}
\quad \text{ if } x \in A
$$
and
$$
\underline d(x) = \overline d(x) = d(x) = \frac{h_{\mu}(\sigma) }{ \lim_{n \to \infty} \left( \frac{\log \diam
\Delta_n(x)}{n} \right) } = \frac{2 h_{\nu}(G)}{\log(\gamma \delta)} \quad
\text{ if } x \in B.
$$
Repeating the construction in Section 6.3, one
can prove that the homeomorphism $\tilde G$ possesses an invariant
ergodic measure $\tilde \nu$ for which $d_{\tilde \nu }(x, y) =
\underline d_{\tilde \nu}(x, y) = \overline d_{\tilde \nu}(x, y)$ for
$\nu-$almost every $(x, y)$ and where $d_{\tilde \nu }(x, y)$ is not
essentially constant.
\bigskip \head{\bf Section 7: Proofs} \endhead
\bigskip
\demo{ Proof of Theorem 2}
\enddemo
Let $\g$ be a strongly l-estimating vector. Given $r, 0 < r < 1$, consider a cover $\frak U_r = \frak U_r (\g)$ of the limit set $F$ which consists of basic sets $\Delta^{(j)}, j=1, \dots, N$.
By the definition of a Gibbs measure (see Appendix 3) there exist
positive constants $D_1$ and $D_2$ such that for $j=1, \cdots, N$
$$
D_1 \leq \frac{ m_{\g}(\Delta_{{i_1} \cdots i_{n(x_j)}})
}{\prod_{k=1}^{n(x_j)} \g_{i_k}^{ s_{\g}}} \leq
D_2 \quad \tag 10
$$
where $m_{\g} = \chi^* \mu_{\g}$ is the pull back of the Gibbs measure $ \mu_{\g}$ corresponding to the
function $ \phi(i_1 i_2 \cdots) = s_{\g} \log
{\g}_{i_1}$ on $Q$. Since the vector $\g$ is strongly l-estimating, by (10), we have
$$
m_{\g}(B(x, r)) \leq \sum_{j=1}^{N(x,r)} m_{\g}(\Delta^{(j)}) \leq \sum_{j=1}^{N(x,r)} D_2 \prod_{k=1}^{n(x_j)} {\g}_{i_k}^{ s_{\g}}
$$
$$ \leq L_4
\sum_{j=1}^{N(x,r)} \prod_{k=1}^{n(x_j)+1}
{\g}_{i_k}^{s_{\g}} \leq L_4 N(x, r) r^{ s_{\g}} \leq L_5
r^{ s_{\g}}, \quad \tag 11
$$ where $L_4, L_5 > 0$ are constants. Hence the measure $m_{\g}$ satisfies the uniform mass distribution principle. This proves statement (1). Moreover, (11) implies that $ s_{\g} \leq {\underline d}_{m_{\g}}(x)$ for every $x \in F$. It follows that
$ s_{\g} \leq \dHF$ and $\text{m}_{\text{H}}( s_{\g}, F) >
0$.
We now prove that $m_{\g}(Z) \leq m_H(s, Z)$. Given $\delta > 0$, there exists $\epsilon > 0$ and a covering $\frak U = \{U^{(k)}\}$ of $Z$ by open sets $U^{(k)}$ with $\diam \, U^{(k)} \leq \epsilon$ satisfying
$$
\sum_{U^{(k)} \in \frak U} (\diam \, U^{(k)})^s \leq m_H(s, Z) + \delta.
$$
By (11) it follows that
$$
m_{\g}(Z) \leq \sum_{U^{(k)} \in \frak U} m_{\g}(U^{(k)}) \leq L_5 \sum_{U^{(k)} \in \frak U} (\diam \, U^{(k)})^s \leq L_5 m_H(s, Z) + L_5 \delta.
$$
Since $\delta$ is chosen arbitrarily this implies that $m_{\g}(Z) \leq m_H(s, Z)$. \quad \qed
\medskip
\demo { Proof of Proposition 2} \enddemo
\medskip
We begin with the following general lemma:
\medskip
\proclaim {\bf Lemma 5} Let $A= (A(i,j))$ be a transitive $(p
\times p)$ matrix of $0$s and $1$s, and consider the subshift of finite
type defined by $A$. Let $f: \Sigma^+_A \to \Bbb R$ be a continuous
function that depends only on the first coordinate. Let $F$ denote
the $(p \times p)$ diagonal matrix $\text{diag}(e^{f(1)}, e^{f(2)},
\cdots, e^{f(p)})$. Then $P_A(\log f(\omega)) = P_A(\log f(\omega_1))
=\log r$, where $r$ denotes the spectral radius of the $(p \times p) $
matrix \, $A^*F $. \endproclaim
\medskip
\demo{ Proof} \enddemo
In the proof, we exploit the fact that the
exponential of the pressure is the maximal eigenvalue of the transfer
operator. Let $\phi: \Sigma_A^+ \to \Bbb R$ be a continuous function.
Then the transfer operator $$ (L_f\phi)(x) \overset \text{def} \to =
\sum_k \exp(f(kx)) \phi(kx) A(k, x_1) = \sum_k \exp(f(k)) \phi(kx)
A(k,x_1), $$ where $x= (x_1 x_2 \cdots)$. The eigenvalue equation for
$L_f$ is $$ \sum_k \exp(f(k)) h(k) A(k,j) = \eta h(j). $$ According to
\cite{PP, page 24 (note normalization of the transfer operator)}, the
largest eigenvalue of $L_f$ is $\eta = \exp(P(f))$. Hence $\exp(P(f))$
is the spectral radius of the matrix $A^* F$. \quad \qed
Proposition 2 follows by applying Lemma 5 to the function $f(\omega) =
f(\omega_1) = t \log \alpha_{\omega_1}$. \quad \qed
\bigskip
\demo{ Proof of Theorem 4}
\enddemo
By the definition of a Gibbs measure,
there exist constants $D_1, D_2 > 0$ such that for any $n \in \Bbb N,$
$$
D_1 \leq \frac{m_{\lam} (\Delta_{i_1 \cdots i_n})}{\prod_{k=1}^n
\lambda_{i_k}^{s_{\lam}} } \leq D_2. \quad \tag 12
$$
Fix $0 < r < 1$. For each $ x \in F $ with $\chi(x)=(i_1 i_2 \cdots )$,
choose $1 \leq i_{n(x)} \leq p$ such that $ \lambda_{i_1} \lambda_{i_2} \cdots
\lambda_{i_{n(x)}} > r$ and $ \lambda_{i_1} \lambda_{i_2} \cdots \lambda_{i_{n(x)}}
\lambda_{i_{n(x)+1}} \leq r$.
It follows that $\Delta_{i_1 \cdots i_{n(x) + 1}} \subset B(x, 2 C_2 r)$, where $B(x,r)$ denotes the ball of radius $r$ around the point $x$. Hence for all $x \in F,$
$$
m_{\lam} (B(x, 2 C_2 r)) \geq m_{\lam}(\Delta_{i_1 \cdots i_{n(x)
+ 1}}) \geq D_1 \prod_{k=1}^{n(x)+1} \lambda_{i_k}^{s_{\lam}} \geq L_1 r^{s_{\lam}}
$$
where $L_1> 0$ is a
constant. It follows by (12) that for all $x \in F,$
$$
{\overline d}_{m_{\lam}}(x) = \limsup_{r \to 0} \frac{\log
m_{\lam}(B(x,r))}{ \log r} \leq s_{\lam}.
$$
We now prove that $m_H(s, \cdot) \leq const \, m_{\lam}(\cdot)$.
Let $Z \subset F$ be a closed subset. Given $\delta > 0$, there exists $\epsilon > 0$ such that for any covering $\frak U = \{U^{(k)}\}$ of $Z$ by open sets $U^{(k)}$ with $\diam \, U^{(k)} \leq \epsilon$, we have
$$
m_H(s, Z) \leq \sum_{U^{(k)} \in \frak U} (\diam \, U^{(k)})^s + \delta.
$$
There exists a covering $\frak U$ of $Z$ by basic sets $\Delta^{(k)} = \Delta_{i_1 \cdots i_{n(k)}}$ satisfying $\diam \, \Delta^{(k)} \leq \epsilon$ and
$$
\sum_{\Delta^{(k)} \in \frak U } m_{\lam}(\Delta^{(k)}) \leq m_{\lam}(Z) + \delta.
$$
Since $\lam$ is a u-estimating vector, it follows from (2) that
$$
m_H(s, Z) \leq \sum_{ \Delta^{(k)} \in \frak U} (\diam \, \Delta^{(k)})^s + \delta \leq C_2 \sum_{ \Delta^{(k)} \in \frak U} \prod_{j=1}^{n(k)} \lambda^s_{i_j} + \delta
$$
$$
\leq C_2 D_2 \sum_{ \Delta^{(k)} \in \frak U} m_{\lam}(\Delta^{(k)}) + \delta \leq C_2 D_2 m_{\lam}(Z) + (C_2 D_2 + 1) \delta.
$$
Since $\delta$ is chosen arbitrarily this implies the desired result.
\medskip
\demo{Proof of Theorem 1} \enddemo
Given numbers $0 <
\lambda_1, \lambda_2, \cdots, \lambda_p < 1$, let $\mu_{\lam}$ be an
equilibrium measure on $Q$ corresponding to the function $s \log
\lambda_{i_1}$. By definition,
$$
h_{\mu_{\lam}}(\sigma | Q) + s \int_Q \log \lambda_{i_1} d \mu_{\lam} = 0, \quad \tag
13 $$ where $h_{\mu}(\sigma | Q) \overset \text{def} \to = h$ is the
Kolmogorov-Sinai entropy. Let us first assume that $\mu_{\lam}$ is ergodic.
For fixed $\epsilon > 0$, it follows from the Shannon-McMillan-Breiman
theorem that for $\mu_{\lam}-$almost every $\omega \in Q$ one can find
$N_1(\omega) > 0$ such that for any $n \geq N_1(\omega)$,
$$
\exp(-(h + \epsilon) n) \leq \mu_{\lam}(C_{i_1 \cdots i_n(\omega)}) \leq
\exp(-(h - \epsilon) n) \quad \tag 14
$$
where $C_{i_1 \cdots i_n(\omega)}
$ is the cylinder set containing $\omega$. It follows from the Birkhoff
ergodic theorem that for $\mu_{\lam}-$almost every $\omega \in Q$ there exists
$N_2(\omega)$ such that for any $n \geq N_2(\omega)$,
$$
\frac1n \log
\prod_{j=1}^n \lambda_{i_j}^s - \epsilon \leq s \int_Q \log \lambda_{i_1} d
\mu_{\lam} \leq \frac1n \log \prod_{j=1}^n \lambda_{i_j}^s + \epsilon. \quad
\tag 15
$$
Combining (13), (14), and (15) we have that for $\mu_{\lam}-$almost
every $\omega \in Q$ and $n$ sufficiently large,
$$
\prod_{j=1}^n
\lambda_{i_j}^{s+ \alpha} \leq \prod_{j=1}^n \lambda_{i_j}^s
\exp(- 2 \epsilon n) \leq \mu_{\lam}(C_{i_1 \cdots i_n(\omega)}) \leq \prod_{j=1}^n \lambda_{i_j}^s \exp(2 \epsilon n) \leq \prod_{j=1}^n \lambda_{i_j}^{s- \alpha}, $$ where $\alpha=\frac{2
\epsilon}{ \min(\log \frac{1 }{\lambda_j}, 1 \leq j \leq p)} > 0$.
This implies that for $\mu_{\lam}-$almost every $\omega \in Q$ and any $n \geq \max \{N_1(\omega), N_2(\omega)\}$,
$$
\prod_{j=1}^n \lambda_{i_j}^{s+ \alpha} \leq
\mu_{\lam}(C_{i_1 \cdots i_n(\omega)}) \leq \prod_{j=1}^n
\lambda_{i_j}^{s- \alpha}. \quad \tag 16
$$
If $\mu_{\lam}$ is not ergodic, then (16) is still valid and can be shown by
decomposing $\mu$ into its ergodic components.
Given $l > 0$ denote $Q_l = \{ \omega \in Q : N_1(\omega) \leq l \,
\text{ and } N_2(\omega) \leq l \}$. It is easy to see that $Q_l
\subset Q_{l+1}$ and $Q = \cup_{l=1}^{\infty} Q_l \pmod{0}$. Thus there
exists $l_0 > 0$ such that $\mu(Q_l) > 0$ if $l \geq l_0$. Let $x \in
Q_l, l \geq l_0, 0 0$ is a constant and $x \in \chi^{-1}(Q_l))$ is any point.
This implies that for any $l > 0$ and $x \in \chi^{-1}(Q_l)$
$$
\underline d_{m_{\g}}^l (x) \overset \text{def} \to \equiv \liminf_{r \to 0} \frac{\log m_{\g}(B(x, r) \cap \chi^{-1}(Q_l))}{\log r} \geq s_{\g} - \alpha.
$$
It follows that for all $l \in \Bbb N, \, \dim_H(F) \geq s_{\g} - \alpha$.
This implies that $\dim_H(F) \geq s_{\g} - \alpha$. Since $\alpha$ can be arbitrarily small, this gives the desired result.
\medskip
\demo{Proof of Theorem 3} \enddemo
Fix $0 < r < 1$. For each $x$
with $\chi(x) = (i_1 i_2 \cdots)$, choose the unique $n(x)$ such that
$\lam_{i_1} \lam_{i_2} \cdots \lam_{i_{n(x)}} > r$ and $\lam_{i_1}
\lam_{i_2} \cdots \lam_{i_{n(x)+ 1}} \leq r$. It is easy to see that
$n(x) \to \infty$ as $r \to 0$ uniformly in $x$. Fix $l > 0$ for
which $\mu_{\lam}(Q_l) > 0$, where $Q_l$ are the sets constructed in the proof of Theorem 1. One can now choose $r = r(l) > 0$ sufficiently
small such that $n(\omega)$ becomes large enough to satisfy (16) for
any $\omega \in Q_l$. Repeating the arguments in the proof of Theorem 4 with $Q$ replaced by $Q_l$, and applying (16) to the numbers $(\lam_1, \dots, \lam_p)$ and the measure $\mu_{\lam}$, one can show that for any $x \in \chi^{-1}(Q_l)$ and
any $r > 0$ sufficiently small,
$$
m_{\lam}(B(x, r)) \geq K r^{\s_{\lam} + \alpha}
$$ where $K = K(l) > 0$ is a constant. This implies that ${\overline
d}_{m_{\lam}}(x) \leq s_{\lam} + \alpha$ and hence, ${\overline
d}_{m_{\lam}}(x) \leq s_{\lam} + \alpha$ for any $x \in F$. This
completes the proof of statement (2).
\medskip
We now show that $\dBFU \leq s_{\lam}$. It is sufficient to prove that $P(\dBFU \log \lam_{i_1}) \geq 0$, since the map $t \to P(t \log \lam_{i_1})$ is a decreasing function \cite{Bo1}.
Given $\delta > 0$, it follows from the definition of $\dBFU$ that there exists $\epsilon > 0$ such that $N_{\epsilon}(F) \geq \epsilon^{\delta - \dBFU}$. Consider the cover $\{\Delta^{(j)} \} = \{ \Delta_{i_1, \cdots, i_{n(x_j)}} \},\, j=1, \cdots, N^{\epsilon}(F)$. Note that this covering need not be the optimal covering, i.e., $N^{\epsilon}(F) \geq N_{\epsilon}(F)$. There clearly exists $A = A(\epsilon) > 0$ such that for $j = 1, \cdots, N^{\epsilon}(F)$,
$$
\frac{\epsilon}{A} \leq \prod_{k=1}^{n(x_j)} \overline \lambda_{i_k} \leq \epsilon
$$
and hence
$$
C_1 \log (\frac{1}{\epsilon}) \leq n(x_j) \leq C_2 \log(\frac{A}{\epsilon})
$$
where $C_1 = \frac{1}{\log(\frac{1}{\lam_{\max}} )} $ and $C_2 = \frac{1}{\log(\frac{1}{\lam_{\min}} )} $. This implies that $n(x_j)$ can take
on at most $C_2 \log (\frac{A}{\epsilon})- C_1 \log (\frac{1}{\epsilon})$ possible values.
We now think of having $N_{\epsilon}(F)$ balls and $C_2 \log (\frac{A}{\epsilon})- C_1 \log (\frac{1}{\epsilon})$ baskets. Then for
$N^{\epsilon}(F) \geq C_2 \log (\frac{A}{\epsilon})- C_1 \log
(\frac{1}{\epsilon})$, there exists a basket containing at least
$\frac{N^{\epsilon}(F)}{ C_2 \log (\frac{A}{\epsilon})- C_1 \log
(\frac{1}{\epsilon})}$ balls. This implies that
there
exists a positive integer $\alpha, \, \, C_1 \log (\frac{1}{\epsilon}) \leq \alpha
\leq C_2 \log (\frac{A}{\epsilon}) $ such that for $\epsilon$ sufficiently small,
$$
\# \{x_j \, \text { such that } n(x_j) = \alpha \} \geq
\frac{N^{\epsilon}(F)}{ C_2 \log (\frac{A}{\epsilon})- C_1 \log
(\frac{1}{\epsilon})}
$$
$$
\geq \frac{N_{\epsilon}(F)}{ C_2 \log (\frac{A}{\epsilon})- C_1 \log
(\frac{1}{\epsilon})} \geq \frac{\epsilon^{\delta - \dBFU} }{ C_3
\log(\frac{1}{\epsilon})} \geq \epsilon^{2 \delta - \dBFU}.
$$
Let $\phi(x) = (\dBFU - 2 \delta) \log \lam_{i_1}$, where $\chi(x) = (i_1 i_2 \dots)$. Then
$$
(S_n \phi)(x) = \sum_{k=1}^{n} \phi(\sigma^kx) = (\dBFU - 2 \delta) \log \prod_{k=1}^{n}
\lam_{i_k}
$$
and hence $\exp(S_n \phi)(x) = (\prod_{k=1}^{n}
\lam_{i_k})^{\dBFU - 2 \delta}$.
It follows that
$$
P_{\alpha}(( \dBFU - 2 \delta) \log \lam_{1_1}) \overset \text{def} \to \equiv \frac{1}{\alpha}
\log \sum_{ \underset \text{admissible} \to {(i_1 \cdots i_{\alpha })}}
\inf_{x \in \Delta_{(i_1 \cdots i_{\alpha)}}} (\prod_{k=1}^{n} \lam_{i_k})^{\dBFU - 2 \delta}
$$
$$
\geq \frac{1}{\alpha} \log \sum_{\underset \text{in covering } \{\Delta^{(j)} \} \to
{(i_1 \cdots i_{\alpha})}} \left( \frac{A}{\epsilon} \right)^{\dBFU - 2\delta}
$$
$$\geq A^{\dBFU - 2 \delta} \frac{1}{\alpha} \log(\epsilon^{\dBFU - 2 \delta} \cdot \epsilon^{2 \delta - \dBFU}) \geq 0.
$$
Hence $P((\dBFU - 2 \delta ) \log \lam_{i_1}) = \lim_{\alpha \to \infty} P_{\alpha}((\dBFU - 2 \delta ) \log \lam_{i_1}) = 0$. \quad \qed
\medskip
\demo{ Proof of Proposition 1 } \enddemo
Let $\mu_{\lam}$ denote an equilibrium measure for $ s_{\lam} \log
\lambda_{i_1}$. It immediately follows from the variational
principle that
$$
s_{\lam} = \frac{h_{\mu_{\lam}}(\sigma \, | \,Q)}{- \int \log \lambda_{i_1} d \mu_{\lam} }\leq \frac{h(\sigma \, | \, Q)}{-\log \max \lambda_k }.
$$
The case of equality is obvious.
If $\lam_i =\lambda$ for $i=1, \cdots, p$, then this immediately implies that $ s_{\lam} = \frac{h(\sigma \, | \,Q) }{ - \log \lambda}$. This proves the desired results. \quad \qed
\medskip
\demo{ Proof of Proposition 7} \enddemo
{\bf 1)} Condition a) implies that for any $\epsilon > 0$ and $\mu_{\lam}-$almost every $\omega \in Q, \, \lam=(\lam_1, \dots, \lam_p)$, there exists $N_3(\omega)
=N_3(\omega, \epsilon)$ such that for any $n \geq N_3(\omega)$,
$$
\left |\frac1n \sum_{j=1}^n \underline a_{i_j, j} \right | \leq
\epsilon. \quad \tag 17
$$
The inequality (17) is equivalent to
$$
\prod_{j=1}^n \lam_{i_j}\exp(- \epsilon n) \leq \prod_{j=1}^n \ll_{i_j,
j} \leq \prod_{j=1}^n \lam_{i_j} \exp(\epsilon n). \quad \tag 18
$$
It
is sufficient to consider only the case when $\mu_{\lam}$ is ergodic with
respect to $\sigma$. Then for $\mu_{\lam}-$almost every $\omega=(i_1 i_2
\cdots) \in Q$, the following limit exists:
$$
\lim_{n \to \infty}
\sum_{j=1}^n \log \lam_{i_j} = \int_Q \log \lam_{\omega_1}d \mu_{\lam}(\omega)
\equiv a < 0.
$$
This implies that for any $\epsilon > 0$ and
$\mu_{\lam}-$almost every $\omega \in Q$ there exists
$N_4(\omega)=N_4(\omega, \epsilon)$ such that for any $n \geq
N_4(\omega)$,
$$ \left |\frac1n \sum_{j=1}^n \log \lam_{i_j} - a \right
| \leq \epsilon. \quad \tag 19
$$
Given $l > 0$ denote
$$
Q_l =
\{\omega \in Q: N_i(\omega) \leq l, i=1,2, 3, 4, 5\}
$$
where $N_1(\omega), N_2(\omega)$ are the two functions constructed in the proof of Theorems 1 (see (16)). It is easy to see that $Q_l \subset Q_{l+1}$ and
$Q=\cup_{l=1}^{\infty} Q_l \, \mod(0)$. Thus there exists $l_0>0$ such
that $\mu_{\lam}(Q_l) > 0$ if $l \geq l_0$.
Consider $\omega=(i_1 i_2 \cdots) \in Q_l, l \geq l_0$ and $0 < r < 1$.
Let $n(\omega)$ denote the unique positive integer such that
$\lambda_{i_1} \lambda_{i_2} \cdots \lambda_{i_{n(\omega)}} > r$ and $ \lambda_{i_1}\lambda_{i_2} \cdots \lambda_{i_{n(\omega)+1}} \leq r $.
Obviously, $n(\omega) \ra \infty$ as $r \ra 0$ uniformly in $\omega$.
Hence we can assume $r=r(l)$ is so small that $n(\omega) \geq
\max\{N_3(\omega), N_4(\omega) \}$. Applying (19) to $n=n(\omega)$ we have that
$$
n(\omega) \leq \frac{\log r}{ (a + \epsilon)}.
\quad \tag 20
$$
Consider the cylinder set $C_{i_1 \cdots
i_{n(\omega)}}$. We have that $\omega \in C_{i_1 \cdots
i_{n(\omega)}}$ and if $\omega' \in C_{i_1 \cdots i_{n(\omega)}} \cap
Q_l$ and $n(\omega') \leq n(\omega)$ then
$$
C_{i_1 i_2 \cdots
i_{n(\omega')}} \supset C_{i_1 i_2 \cdots i_{n(\omega)}}.
$$
Let $C(\omega)$ be the largest cylinder set containing $\omega$ with the
property that for every $\omega' \in C(\omega) \cap Q_l$ we have
$C_{i_1 \cdots i_{n(\omega')}} \cap Q_l \subset C(\omega) \cap Q_l$
and there exists $\tilde \omega \in C(\omega)$ such that $ C(\omega) =
C_{i_1 \cdots i_{n(\tilde \omega)}}$. It is easy to see that the sets
$C(\omega) \cap Q_l$ corresponding to different $\omega \in Q_l$ either
coincide or are disjoint. We denote these sets by $C^{(j)}, j= 1,
\cdots N$. There exists points $\omega_j$ such that $C^{(j)}= C_{i_1
\cdots i_{n(\omega_j)}}$. For any $j=1, \cdots, N$ we have
$$
l^{-1} \prod_{k=1}^{n(\omega_j) } \lam_{i_k}^{s_{\lam}} \leq \mu_{\lam}(C^{(j)})
\leq l \prod_{k=1}^{n(\omega_j)} \lam_{i_k}^{s_{\lam}}.
$$
Consider a point $x \in F$ with $\chi(x) = \omega \in Q_l$. We estimate the number $N(x, r)$ of
sets $C^{(j)}$ that intersect the set $\chi(B(x, r)) \cap
Q_l$ for sufficiently small $r$. It follows from (19) and (20) that
$$
N(x,r) \leq \frac{ \text{vol}(B(x, r + \max_{1 \leq j \leq N} (\diam(\chi^{-1}(C^{(j)})) ))}
{\min_{1 \leq
j\leq N} \text{vol}(\chi^{-1}(C^{(j)} \boldsymbol \cap Q_l ))}
$$
$$
\leq \frac{ \max_{1 \leq j \leq N} \text{vol}(B(x, r + \frac{r}{\lam_{\min}} \exp(\epsilon n(w_j)) ))} {\min_{1 \leq
j\leq N} \text{vol} B(x, C_1 r^{\frac{a}{a+ \epsilon} })}
$$
$$
\leq C_2 \frac{ \left ( r( 1 + \frac{1}{\lam_{\min}} r^ \frac{\epsilon}{a + \epsilon} ) \right )^d }{ (r^{\frac{\alpha}{a + \epsilon}} )^d }
\leq C_3 r^{- b \epsilon},
$$
where $C_1, C_2, C_3, b$ are positive constants. This implies that the
construction is conditionally regular. The proof of Proposition 8 is a slight modification of the proof of Proposition 7.
\medskip
{\bf 3)} The proof follows from {\bf 2)}. \quad \qed
\medskip
\demo{ Proof of Lemma 1} \enddemo
We first prove Statement (1). The proof of Statement (2) is similar. Applying the Birkhoff ergodic theorem for the
stationary ergodic process to the functions $\underline f(\vec
\lambda) = (\log \underline \lambda_{1,1}, \cdots, \log \underline
\lambda_{p,1})$ and $\overline f(\vec \lambda) = (\log \overline
\lambda_{1,1}, \cdots, \log \overline \lambda_{p,1}) $ we obtain that
$$ \frac1n \sum_{k=1}^n \log \underline \lambda_{i,k} \to \log
\underline \lambda_i $$
for $\nu$ almost every $\vec \lambda$, where we define $\log
\underline \lambda_i $ as the limiting value. We need to show that for $\lmu$ almost every $\omega=(i_1 i_2 \cdots)
\in Q$, $$ \lim_{N \ra \infty} \frac1N \sum_{k=1}^N \log \left(
\frac{\underline \lambda_{i_k,k}}{\underline \lambda_{i_k}} \right ) =
0. $$
We break up the sum $$ \frac1N \sum_{k=1}^N \log \left(
\frac{\underline \lambda_{i_k,k}}{\underline \lambda_{i_k}} \right ) =
\frac1N \sum_{j=1}^p \sum \Sb i_k = j \\ 1 \leq k \leq N \endSb \log
\left( \frac{\underline \lambda_{j ,k}}{\underline \lambda_{j}} \right
) $$ $$ = \sum_{j=1}^p \frac{\#\{i_k =j\}}{N} \left(\frac{1}{\#\{i_k
=j\}} \sum \Sb i_k = j \\ 1 \leq k \leq N \endSb \log \left(
\frac{\underline \lambda_{j, k}}{\underline \lambda_{j}} \right )
\right). $$
We wish to show that for $\lmu-$almost every $\omega$, each of the
above terms in parentheses tends to zero as N tends to infinity. This
immediately follows from the following generalization of the Birkhofff
ergodic theorem for return times to a set.
\demo{\bf Theorem \cite{BFKO}} Let $(X, \frak B, \nu, T)$ and $(Y,
\frak C, \mu, S)$ be two ergodic measure theoretic dynamical systems.
Let $A \subset Y$ be of positive measure $\mu(A) > 0$. For every $y
\in Y$ let $A_y$ denote the return time sequence $\{n \in \Bbb N \, |
\, S^ny \in A \}$. Then for $\mu-$almost every $y \in Y$ and for $f
\in L^1(X)$, $$ \lim_{N \to \infty} \frac1N \sum \Sb 1 \leq k \leq N
\\ k \in A_y \endSb T^k f (x) = \int _X f d \nu $$ for $\nu-$almost
every $x \in X$.
We apply this theorem for $T$ the ergodic stationary process in the definition of
random symbolic geometric construction, $S=
\sigma: Q \to Q$ the shift map with invariant Gibbs measure $\lmu$, and
$A = C_j =\{ \omega \in Q \, | \, \omega_1 = j \}$. Clearly, the set
$\{ 1 \leq k \leq N, \, i_k = j \} = \{ 1 \leq k \leq N, \, \sigma^k
\omega \in C_j\}$.
\quad \qed \enddemo
\medskip
\demo{ Proof of Theorem 7}
\enddemo
Given $x \in F, \chi(x) = (i_1 i_2 \cdots)$ and $r > 0$, choose
$n_r(x)$ such that $\prod_{k=1}^{n_r(x)} \lambda_{i_k,k} < r$ and
$\prod_{k=1}^{n_r(x)-1} \lambda_{i_k,k} \geq r$. Since $x \in
\Delta_{n_r(x)}(x)$ we have that $\Delta_{n_r(x)}(x) \subset B(x,
r)$. This implies
$$ \frac{\log \nu(B(x, r))}{\log r } \leq
\frac{\log \nu(\Delta_{n_r(x)}(x))}{\log r }.
$$
We also have that
$$
r \leq \prod_{k=1}^{n_r(x)-1} \lambda_{i_k,k} = \prod_{k=1}^{n_r(x)}
\lambda_{i_k,k} \frac{1}{\lambda_{ i_{n_r(x)}}} \leq \frac{1}{\beta}
\diam \Delta_{n_r(x)}(x). $$ It follows that $$ \overline d_{\nu}(x) =
\limsup_{r \to 0} \frac{\log \nu(B(x, r))}{\log r } \leq \limsup_{r
\to 0} \frac{ \log \nu(\Delta_{n_r(x)}(x))}{\log (\frac{1}{\beta} \diam
\Delta_{n_r(x)}(x))} $$
$$ \leq \limsup_{r \to 0} \frac{ \log \nu(\Delta_n(x))}{\log \diam
\Delta_n(x)} = \overline d(x). $$
We now prove the second estimate in (1). Given $\alpha \geq 0, C > 0$,
define $F_{\alpha, C} = \{ x \in F \, : \, \nu(\Delta_n(x)) \leq C
(\diam \Delta_n(x))^{\alpha}$ for all $n \geq 0 \}$. Fix $x \in
F_{\alpha,C}$ and $r > 0$. By repeating the arguments in the proof of
Theorem 1, one can find points $x_j \in F_{\alpha, C}, j = 1, \cdots,
N$ (where $N$ is independent of $x$ and $r$) such that $x_j \in
\Delta^{(j)} = \Delta_{i_1 \cdots i_{n(x_j)}}$ and such that the
collection $\Delta^{(j)}$ are disjoint, $\prod_{k=1}^{n(x_j)}
\lambda_{i_k, k} \leq r$, and $B(x,r ) \cap F_{\alpha, C} \subset
\cup_{j=1}^N( \Delta^{(j)} \cap F_{\alpha, C})$. If follows that
$\nu(B(x, r)) \leq \sum_{j=1}^N \nu(\Delta^{(j)})$ and hence $$
\underline d_{\nu}(x) = \liminf_{r \to 0} \frac{\log \nu(B(x, r)) }{
\log r} \geq \liminf_{r \to 0} \frac{ \log \sum_{j=1}^N
\nu(\Delta^{(j)})}{\log r } \geq \alpha. $$ The result follows since
$F = \cup_{\alpha} \cup_{C > 0} F_{\alpha, C}$. \medskip
The last statement is a direct consequence of the preceeding
statements. \quad \qed
\newpage
\proclaim {\bf Appendices} \endproclaim
\bigskip
\head{\bf Appendix 1: Hausdorff Dimension and Box Dimension} \endhead
\medskip Let $\U \subset \BbbR^n$. The {\it diameter} of U is defined
as $\diam(U) = \sup\{|x-y|:x,y \in U\}$. If $\{U_i\}$ is a countable
collection of sets of diameter at most $\delta$ that cover Z, i.e., $Z
\subset \cup_i U_i$ with $0 < |U_i| \leq \delta$ for each $i$, we say
that $\{U_i\}$ is a $ \delta$-cover of Z.
Suppose that $Z \subset \BbbR^n$ and $s \geq 0$. For any $s > 0$ we
define
$$ \text{m}_{\text{H}}(s, Z) =\lim_{\delta \ra 0}
\inf_{\{U_i\}} \left\{ \sum_i \diam(U_i)^{s}: \{U_i\} \text{ is a }
\delta\text{-cover of } Z \right\}.
$$
We call $\text{m}_{\text{H}}(s, Z) $ the {\it s-dimensional Hausdorff
measure of Z}. There exists a unique critical value of $s$ at which
$\text{m}_{\text{H}}(s, Z)$ jumps
from $\infty$ to $0$. This critical value is called the {\it Hausdorff
dimension} of $Z$ and is written $\text{dim}_H Z$. If
$s=\text{dim}_H Z$, then $\text{m}_{\text{H}}(s,Z)$ may be $0,
\infty$, or finite. Hence $\text{dim}_H Z = \sup\{s:
\text{m}_{\text{H}}(s, Z) = \infty\} = \inf\{s:
\text{m}_{\text{H}}(s,Z) = 0\}$.
\medskip Let $N_{\delta}(Z)$ denote the minimum number of sets of
diameter precisely $\delta$ needed to cover the set $Z$. We define the
lower and upper box dimensions of $Z$ by
$$
\underline \dim_B Z = \liminf_{\delta \ra 0} \frac{\log
N_{\delta}(Z)}{\log(\frac{1}{\delta})}, \quad \text{ and} \quad
\overline \dim_B Z = \limsup_{\delta \ra 0} \frac{\log
N_{\delta}(Z)}{\log(\frac{1}{\delta})}.
$$
\medskip
It is easy to see that $\dim_H Z \leq \underline
{\text{dim}}_{B} Z \leq \overline {\text{dim}}_{B} Z$.
\bigskip \centerline{\bf Appendix 2: Two Methods of Obtaining Lower
Bounds for $ \text{\bf dim}_{\bold H} \bold F $}
\proclaim{ Uniform Mass Distribution Principle \cite{Fr}} Let $\mu$ be a probability measure
of $Z$ and suppose that for some $s$ there are numbers $c > 0$, and
$\delta > 0$ such that $\mu(U) \leq c \diam(U)^s$ for all sets $U$ with $\diam(U)^s
\leq \delta$. Then $\text{m}_{\text{H}}(s,Z) \geq \mu(F)/c$ and $s \leq
\text{dim}_H Z$.
\endproclaim
\medskip
\proclaim{ Non-Uniform Mass Distribution
Principle}
Let $\mu$ be a probability measure supported
on $Z \subset R^n$. We define the {\bf Hausdorff dimension of the
measure $\mu$,} \enspace $\text{dim}_H\mu = \inf \{ \text{dim}_HU, \,
\mu(U) \allowmathbreak = 1 \}, $ and the {\bf lower and upper pointwise
dimensions of $\mu$,} \, ${\underline d}_{\mu}(x) \allowmathbreak =
\liminf_{\epsilon \ra 0} \frac{\log \mu(B(x, \epsilon))}{\log
\epsilon}$ and ${\overline d}_{\mu}(x) = \limsup_{\epsilon \ra 0}
\frac{\log \mu(B(x, \epsilon))}{\log \epsilon}$.
% It is clear that
% $\text{dim}_H Z= \text{dim}_H \text{supp} \mu \geq \text{dim}_H\mu$.
\endproclaim
\medskip
\proclaim{\bf Proposition}
If ${\underline d}_{\mu}(x) \geq d \geq 0$ for a.e. $x \in Z$, then
$\text{dim}_H{\mu} \geq d$, and hence $\text{dim}_H Z \geq d$.
\endproclaim
\medskip
The uniform mass distribution principle implies the non-uniform mass distribution
principle. In fact, $d_{\mu}^L(x) \geq d$ implies that for any $ \alpha >
0, \, \mu(B(x,r)) \leq C( x, \alpha) r^{d- \alpha}$.
\bigskip
\head {\bf Appendix 3: Thermodynamic Formalism } \endhead
\medskip Good
references for this material are \cite{Bo1} and \cite{PP}. Given a
transitive $p \times p$ transfer matrix $A$ of $0's$ and $1's$,
consider the one-sided subshift of finite type $(\Sigma_A^+, \sigma)$
with $ \sigma: \Sigma_A^+ \rightarrow \Sigma_A^+$ the shift map. The
set $\Sigma_A^+$ consists of all admissible words, i.e., a word $x =
(x_1, x_2, \cdots) \in \{1, \cdots, p\}^{\Bbb N}$ is admissible if $
a_{x_i,x_{i+1}} = 1$ for all $i \in \Bbb N$. The space $\Sigma_A^+$
has a natural family of metrics defined by $d(x, y) =
\sum_{k=1}^{\infty} \frac{|x_k - y_k|}{\beta^k}, \,$ where $\beta$ is
any number satisfying $\beta > 1$. The set $ \Sigma_A^+$ is compact
with respect to the topology induced by $d$ and the shift map $\sigma:
\Sigma^+_A \ra \Sigma^+_A$ is a homeomorphism. We let $\Sigma_p^+ =
\{1, \cdots,p \}^{\Bbb N}$.
More generally, we consider a general symbolic system, i.e., a
compact subset $Q \subset \Sigma_p^+$ that is $\sigma-$invariant,
i.e., $\sigma(Q) = Q$.
Let $C^0( Q), \, C^{\alpha}( Q)$ denote the spaces of
continuous and $\alpha-$\Holder continuous functions on $ Q$. We
define a mapping $P: C^0( Q) \rightarrow \Bbb R$ by
$$
P(\phi) =
\lim_{n \rightarrow \infty} \frac1n \log \left( \sum \Sb (i_1 \cdots i_n)
\\ \text{admissible} \endSb \inf_{x \in \Delta_{(i_1 \cdots i_n)}} \exp (S_n
\phi(x)) \right),
$$
where $S_n \phi(x) = \sum_{i=0}^{n-1} \phi (\sigma^i x).$
We call $P(\phi)$ the {\it topological pressure} of $\phi$. \medskip The
following result is a variational characterization of pressure that is
valid for {\it all} topological dynamical systems. Let $\frak M (Q)$
denote the space of all shift-invariant Borel probability measures on
$Q$.
\medskip
\demo{\bf Variational Principle} Let $\phi \in C^{0}(
Q)$. Then $$
P(\phi) = \sup_{\mu \in \frak M (
\Sigma_A)} \left(
h_{\mu}(\sigma) + \int_{\Sigma_A} \phi d \mu \right).
$$
\enddemo
\medskip Measures that realize the variational principle for
topological pressure play crucial roles in the thermodynamical
formalism.
\demo{\bf Definition} A Borel probability measure $ \mu = \mu_{\phi}$ on $Q$
is called an {\bf equilibrium measure} for the potential $\phi$ if $$
P(\phi) =h_{\mu}(\sigma) + \int_{\Sigma_A} \phi d \mu. \quad \tag 21 $$
\enddemo
\medskip Since the shift map on a general symbolic system is
expansive, the supremum in the variational principle is attained by
some measure \cite{W}. This measure need not be unique.
Another important class of measures are Gibbs measures:
\medskip
\demo{\bf Definition} Let $\phi \in C^0( Q)$. A Borel
probability measure $ \mu = \mu_{\phi}$ on $Q$ is called a {\bf Gibbs
measure} for the potential $\phi$ if there exist constants $D_1, D_2 >
0$ such that $$ D_1 \leq \frac{\mu \{y: y_i = x_i, \,\, i=0, \cdots,
n-1 \} } { \exp(-n P(\phi) + \sum_{k=0}^{n-1} \phi(\sigma^k x))} \leq
D_2 \quad \tag 22 $$ for all $x =(x_1 x_2 \cdots) \in \Sigma_A^+$ and
$n \geq 0$.
\enddemo
\medskip
For subshifts of finite type, Gibbs
measures exist for any Hold\"er continuous potential $\phi$, are
unique, and coincide with the equilibrium measure for $\phi$. Little
is known about the existence of Gibbs measures for general symbolic
systems.
The main tool used in constructing and studying Gibbs measures for
subshifts of finite type is the (linear bounded) transfer operator
$L_{f}: C^{\alpha}(\Sigma_A^+) \to C^{\alpha}(\Sigma_A^+)$ defined by
$$ (L_{f}\phi)(x) = \sum_{y \in \sigma^{-1}(x)} \exp(f(y)) \phi(y) =
\sum_{k} \exp(f(k x)) \phi(k x). $$ along with its dual operator
$L_{f}^* : M( \Sigma_A^+) \ra M( \Sigma_A^+)$, where $M( \Sigma_A^+)$
denotes the space of Borel probability measures on $\Sigma_A^+$. The
following theorem of Ruelle constructs Gibbs measures using
the operator $L_{f}$:
\proclaim{Proposition \cite{PP}} Let $ (\Sigma_A^+, \sigma) $ be mixing subshift of finite type. There
exists $\lambda = \exp(P(\phi)) \geq 1, \, h \in C^0(\Sigma_A^+, \Bbb
R)$ with $h > 0$ and $\nu \in M(\Sigma_A^+)$ for which $L_{f}h =
\lambda h, L_{f}^* \nu = \lambda \nu$, and $\nu(h) = 1$. Then $\mu =
h \nu$ is a $\sigma-$invariant probability measure on $\Sigma_A^+$ and
is a Gibbs measure for $\phi$.
\endproclaim
\medskip
In this paper we deal exclusively with a special class of potentials
that depend only on the first coordinate, i.e., $\phi(x) = \phi(x_1)$.
In this case, the measures $\nu$ and $\mu$ are Markov and can be
described explicitly: the eigenfunction $h = h(x_1)$ satisfies $$
\sum_i h(i) A(i, j) \exp(\phi(i)) = \lambda h(j) $$ and the measure
$\nu$ is defined on cylinder sets by
$$
\nu[i_0, \cdots, i_n] = P(i_0, i_1) \cdots P(i_{n-1}, i_n) p(i_n)
\quad \text{ where } \quad P(i,j) = \frac{ A(i , j) h(i) \exp(\phi(i))}
{ \lambda h(j)}
$$
and $P p = p$ with $\sum_i p(i) = 1$. Finally, the
Gibbs measure $\mu$ is defined by $d \mu = h(x_1) d \nu$. \medskip The
Gibbs measure $\mu$ is unique provided the potential $f$ is H\"older
continuous.
\proclaim{\bf Proposition \cite{PP}} Given numbers $\lam_i, \, i=1,
\cdots, p,$ \, define the function
$$
\phi: \Sigma^+_A \ra \Sigma^+_A \quad \text{ by } \phi(x) = \phi(x_1,
x_2, \cdots ) = \log \lam_{x_1}.
$$
Then $\phi$ is \Holder continuous with respect to the standard metric.
Furthermore, there exists a unique $s, \, 0 \leq s \leq 1$ such that $P(s \phi)= 0$.
\endproclaim
\head{\bf Appendix 4: Gauge Functions} \endhead \medskip One can
generalize the definition of Hausdorff measure to give more refined
information about a set whose Hausdorff measure at the
dimension is zero.
Suppose $h(t)$ is a continuous increasing function defined on $(0,
\epsilon)$ with $\epsilon > 0$ such that $h(t) \to 0$ as $t \to 0$.
Using the notation in Appendix 1, we define
$$
\text{m}^h_{\text{H}}(Z) =\lim_{\delta \ra 0} \inf_{\{U_i\}}
\left\{ \sum_i h(\diam(U_i)): \{\delta\text{-cover of } Z \} \right \}.
$$
We call $\text{m}^h_{\text{H}}(Z) $ the {\it s-dimensional Hausdorff
measure of Z with respect to the gauge function $h$}. Clearly letting
$h(t) = t^s$ gives the $s-$dimensional Hausdorff measure defined in
Appendix 1.
\newpage
\Refs \widestnumber\key{BFKO0}
\ref \key AJ \by V. Alekseyev and M. Jakobson \jour Physics Reports \paper
Symbolic Dynamics and Hyperbolic Dynamical Systems \vol 75 \yr 1981
\pages 287--325 \endref
\ref \key AS \paper The Hausdorff Dimension of Attractors Appearing by
Saddle-Node Bifurcations \by V. Afraimovich and M. Shereshevsky \jour
Inter. Jour. of Bifurcation and Chaos \vol 1:2 \pages 309--325 \yr 1991
\endref
\ref \key Ba \by L. Barreira \paperinfo Personal Communication \endref
\ref \key BFKO \paper Pointwise Ergodic Theorems for Arithmetic Sets
\by J. Bourgain \paperinfo Appendix by J. Bourgain, H. Furstenberg, Y.
Katznelson, and D. Ornstein \jour Publ. Math. IHES \yr 1989 \vol 69
\pages 5--45 \endref
\ref \key BK \by M. Brin and A. Katok \paper On Local Entropy \jour
Geometric Dynamics \pages 30--38 \paperinfo Lecture Notes in
Mathematics 1007 \publ Springer Verlag \yr 1981 \endref
\ref \key Bo1 \by R. Bowen \book Equilibrium States and the Ergodic
Theory of Anosov Diffeomorphisms \publ Springer Verlag \bookinfo
Springer Lecture Notes \#470 \yr 1975 \endref
\ref \key Bo2 \by R. Bowen and C. Series \jour Publ. Math. IHES \paper
Hausdorff Dimension of Quasi-circles \yr 1979 \vol 50 \pages 11--25
\endref
\ref
\key BM
\by A. Bertrand-Mathis
\paper Questions Diverses Relatives aux Syst\`ems cod\'es: Applications au $\theta-$shift
\paperinfo preprint
\endref
\ref \key BU \by T. Bedford and M. Urba\'nski \paper The box and
Hausdorff Dimension of Self-Affine Sets \jour Ergod. Th. and Dynam.
Systems \vol 10 \yr 1990 \pages 627--644 \endref
\ref \key C \by C. D. Cutler \paper Connecting Ergodicity and Dimension
in Dynamical Systems \jour Ergod. Th. and Dynam. Systems \vol 10 \yr
1990 \pages 451--462 \endref
\ref \key ER \by J. P. Eckmann and D. Ruelle \paper Ergodic Theory of
Chaos and Strange Attractors \jour Rev. Mod. Phys. \vol 57 \yr 1985
\pages 617--656 \paperinfo 3 \endref
\ref \key F1 \by K. Falconer \book Fractal Geometry,
Mathematical Foundations and Applications \publ Cambridge Univ. Press
\yr 1990 \endref
\ref \key F2 \by K. Falconer \paper Random Fractals \jour Math. Proc.
Camb. Phil. Soc. \vol 100 \yr 1986 \pages 559--582 \endref
\ref
\key F3
\by K. Falconer
\paper Dimensions and Measures of Quasi Self-Similar Sets
\jour Proceedings of the AMS
\yr 1989
\vol 106
\pages 543--554
\endref
\ref \key Fr \by O. Frostman \paper Potential d'\'equilibre et
Capacit\'e des Ensembles Avec Quelques Applications \`a la Th\'eorie
des Fonctions \jour Meddel. Lunds Univ. Math. Sem. \vol 3 \yr 1935
\pages 1--118 \endref
\ref \key Fu \by H. Furstenberg \paper Disjointness in Ergodic Theory,
Minimal Sets, and a Problem in Diophantine Approximation \jour
Mathematical Systems Theory \vol 1 \yr 1967 \pages 1--49 \endref
\ref \key G \by S. Graf \paper Statistically Self-similar Fractals
\jour Prob. Theory and Related Fields \vol 74 \pages 357--397 \yr 1987
\endref
\ref \key GMW \by S. Graf, D. Mauldin and S. Williams \paper The Exact
Hausdorff Dimension in Random Recursive Constructions \jour Mem. Am.
Math. Soc. \vol 71 \paperinfo 381 \yr 1888 \endref
\ref \key K \by J. P. Kahane \paper Sur le Mod\'ele de Turbulence de
Benoit Mandelbrot \pages 621--623 \jour C.R. Acad. Sci. Paris \vol
278A \yr 1974 \endref
\ref \key L \by F. Ledrappier \paper Dimension of Invariant Measures
\paperinfo preprint \yr 1992 \endref
\ref \key LM \by F. Ledrappier and M. Misiurewicz \jour Ergod. Th. and
Dynam. Systems \vol 5 \yr 1985 \pages 595--610 \paper Dimension of Invariant Measures for Maps with Exponent Zero \endref
\ref \key LY \by F. Ledrappier and L. S. Young \jour Annals of Math.
\vol 122 \yr 1985 \pages 540--574 \paper The Metric Entropy of
Diffeomorphisms \paperinfo Part II \endref
\ref \key Mc \by C. McMullen \jour Nagoya Math. J. \yr 1984 \pages 1--9
\vol 96 \paper The Hausdorff Dimension of General Sierpi\'nski Carpets \endref
\ref \key MW1 \by R. Mauldin and S. Williams \paper Hausdorff Dimension
in Graph Directed Constructions \jour Transactions of the AMS \vol
309:2 \pages 811--829 \yr 1988 \endref
\ref \key MW2 \by R. Mauldin and S. Williams \paper Random Recursive
Constructions: Asymptotic, Geometric and Asymptotic Properties \jour
Trans. Am. Math. Soc. \vol 298 \pages 325--346 \yr 1986 \endref
\ref \key Mo \by P. Moran \paper Additive Functions of Intervals and
Hausdorff Dimension \jour Proceedings of the Cambridge Philosophical
Society \vol 42 \pages 15--23 \yr 1946 \endref
\ref \key PY \by Y. Pesin and C. B. Yue \paper Hausdorff Dimension of
Measures with Non-zero Lyapunov Exponents and Local Product Structure
\paperinfo PSU preprint \yr 1993 \endref
\ref \key PP \by W. Parry and M. Pollicott \paper Zeta Functions and
the Periodic Orbit Structures of Hyperbolic Dynamics \jour Ast\'erisque
\vol 187--188 \yr 1990 \endref
\ref \key PW \by M. Pollicott and H. Weiss \paper The Dimensions of
Some Self-affine Limit Sets in the Plane and Hyperbolic Sets \paperinfo
preprint \yr 1993 \endref
\ref \key PU \by F. Przytycki and M. Urba\'nski \paper On Hausdorff
Dimension of Some Fractal Sets \jour Studia Mathematica \vol 93 \yr
1989 \pages 155--186 \endref
\ref \key R \by D. Ruelle \book Thermodynamic Formalism \publ
Addison-Wesley \yr 1978 \endref
\ref \key S \by M. Shereshevsky \paper On the Hausdorff Dimension of a
Class of Non-Self-Similar Fractals \jour Math. Notes \vol 50 \pages
1184--1187 \yr 1991 \issue 5 \endref
\ref \key Sh \by M. Shub \book Global Stability of Dynamical Systems
\publ Springer Verlag \yr 1987 \endref
\ref \key St \by S. Stella \jour Proceedings of the American Math.
Soc. \paper On Hausdorff Dimension of Recurrent Net Fractals. \vol
116 \yr 1992 \pages 389--400 \endref
\ref \key Y \by L. S. Young \paper Dimension, Entropy, and Lyapunov
Exponents \jour Ergod. Th. and Dynam. Systems \vol 2 \yr 1982 \pages
109--124 \endref
\ref \key W \by P. Walters \book Introduction to Ergodic Theory \publ
Springer Verlag \yr 1982
\endref
\ref \key We \by B. Weiss \paper Subshifts of Finite Type and Sophic
Systems
\jour Monatschefte f\"ur Mathematik \vol 77 \pages 462--474 \yr 1973
\endref
\endRefs
\enddocument