\magnification=\magstep1
\font\s=cmr5
\font\a=cmbx6 scaled \magstep 4
\loadbold
%\input /home/descartes/weiss/tex/macros
\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\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}
\def \aa#1#2{\rlap{#1}\hfill\rlap{#2}\hfill\newline}
\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or
December\fi
\space\number\day, \number\year}
\def\now{\ifnum\time<60 %check to see if it's just after
midnight
12:\ifnum\time<10 0\fi\number\time am %and act accordingly.
\else
\ifnum\time>719\chardef\a=`p\else\chardef\a=`a\fi
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\minute=\time
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\number\hour:%
\multiply\hour by 60 %Use is made of the integer divide here.
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\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}
\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.
\endabstract
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\endtopmatter
\document
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%\input draft
\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 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 all disjoint, and
$\Delta_{i_1 \cdots i_n j} \subset \Delta_{i_1 \cdots i_n} $, for $j
= 1, \cdots, p$. Moreover, the interior and exterior diameters of the
set $ \Delta_{i_1 \cdots i_n j } $ are less than those of the set
$\Delta_{i_1 \cdots i_n} $, with ratios uniformly bounded from below
and above by numbers $\underline \lambda_j$ and $\overline \lambda_j$
respectively. These numbers are called the ratio coefficients of the process. 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.
\midspace{3truein} \caption{Figure 1. \quad Simple Geometric Process}
We also consider a much broader class of geometric constructions that
includes
\roster
\item Markov constructions
where admissible basic sets $\Delta_{i_1 \cdots i_n} $ are determined by a
transfer matrix $A$,
\smallskip
\item general symbolic constructions where admissible basic sets $\Delta_{i_1 \cdots i_n} $ are determined by
a compact shift-invariant subset of the full shift on $p$ symbols,
\smallskip
\item constructions where the numbers $\underline
\lambda_j$ and $\overline \lambda_j$ may vary from step to step but
have a well defined limiting behavior,
\smallskip
\item random constructions where the numbers
$\underline \lambda_j$ and $\overline \lambda_j$ are chosen randomly at
each step of the construction from a very general class of
distributions,
\smallskip
\item combinations of the above constructions.
\endroster
Our main results deal with the general types of geometric constructions
and provide below and above estimates for the Hausdorff dimension and
the box dimension of the limit Cantor-like set. In addition, we prove strict
positivity and boundedness of the Hausdorff measure of the limit set
for a broad class of constructions.
There are three main methods for obtaining a lower bound for the
Hausdorff dimension of a set: the mass distribution principle, the
potential principle, and the nonuniform mass distribution principle.
See Appendix 2. The 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 $x \in F$. Then $s$
produces a lower bound for $\dHF$, the Hausdorff dimension of $F$. We
stress that (1) must hold for {\it all} $x \in F$ and the constant $C$
is {\it independent} of $x$ and $r$.
The (uniform) mass distribution principle is the strongest of the three
methods, and it is thus quite surprising that it can be used to obtain
a good lower estimate for the Hausdorff dimension of the limit set for
many of the different geometric constructions that we consider.
One particular and important case is when a geometric construction
has sets on the $n$th step that are strictly
geometrically similar to the corresponding sets on 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
$$
\Delta_{i_1 \cdots i_n } = h_{i_1} \circ \cdots \circ h_{i_n}(\Delta)
$$
(see Section 4), where $\Delta$ denotes a ball in $\Bbb R^n$.
This construction is called a similarity
construction since it exposes a self-similar character of the geometric
process. These special constructions
were 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
sets with constant ratio coefficients $\underline \lambda_j = \overline \lambda_j = \lambda_j$, for $j =1, \cdots, p$. We call this {\it a Moran process}. For the Moran process the sets on the $n$th step {\it need not} be strictly
geometrically similar to the corresponding sets on the $(n-1)$th step.
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 sets $ \{ \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 including general symbolic constructions, non-stationary constructions, and random constructions as well as generalize his method to study local pointwise dimension and box dimension of the limit set.
The symbolic representation of a Cantor-like geometric process is
the key point in our study. 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
$m$ (see Appendix 2). To construct this measure,
we consider the symbolic space associated with the process. The
measure $m$ is the pullback of the equilibrium state
for the function $s \phi (x)=s \log \lambda_{i_1}$ 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 geometric constructions provided $\ll_i = \ol_i = \lam_i$. See Corollary 1.
\comment
In more general cases when the contraction ratios
depend on the directions, the Hausdorff dimension and the box dimension
need not coincide. See Example 6 in Section 6. However, the above
symbolic dynamics approach still works and allows us to obtain lower
and upper estimates.
\endcomment
The 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).
We emphasize that for the general symbolic geometric constructions the measure $m$ is an equilibrium measure and admits the nonuniform mass distribution principle. There is a crucial
difference between Gibbs measures and equilibrium measures in
statistical physics (\cite{R} and Appendix 3; see also Section 2, 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 equivalent to $m$ (see Proposition 2). If $m$ is an equilibrium
state, then a nonuniform mass distribution principle can be applied.
See Theorem 1.
Theorem 2 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 above. In Section 5, we provide an example of a simple
geometric construction for which the Hausdorff dimension $\dHF$,
the lower box dimension $ \dBFU$, 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.
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. This
limit $ d_m(x)$ is called the pointwise or local dimension at $x$. In
this case $m$ is called 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 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
nonstationary simple geometric construction. If the map is smooth, then $m$ is exact dimensional and ergodic, and
hence $d_m(x) = const$ 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 process in the Markov case was studied by
Stella \cite{St} with some additional strong assumptions. Using the potential principle, 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.
An essential part of our work consists of analyzing geometric
constructions where the ratio coefficients depend on steps of the
construction and admit {\it nonstationary} lower and upper estimates given
by $\underline \lam_{i_n}, \overline \lam_{i_n}, i_n=1, \cdots, p, \,
n=1, 2, \cdots$. In order to control the Hausdorff dimension and box
dimension, one requires some kind of good asymptotic behavior
when $n$ tends to infinity. We introduce two conditions, {\it strong
asymptotic and weak asymptotic}, under which information about
dimension can be obtained. Geometric constructions satisfying the
strong asymptotic condition can be treated by using the uniform mass
distribution principle while geometric constructions satisfying the
weak asymptotic condition need not admit the uniform mass distribution
principle. See Example 4. In the weak asymptotic case, a lower bound for the Hausdorff dimension can
be obtained using the nonuniform mass distribution principle, and in
some cases an upper estimation for the upper box dimension can be found.
Moreover, the Hausdorff measure at dimension $s$ can be either zero or
infinite. In order to obtain more information about Hausdorff measure
one can use a {\it gauge function} (see Appendix 4). In Section 5, we exhibit
a family of simple geometric constructions that satisfy the weak
asymptotic condition and 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.
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 $\Sigma^+_p$ with $p^n$ ratio coefficients at step
$n$ chosen randomly, essentially independently and with the same
distribution on $(0, 1)$. Furthermore, they assume some independence
conditions over $n$. In this paper, we consider branching processes
that are associated with compact shift-invariant subsets in
$\Sigma^+_p$. 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. {\it Our major idea in studying geometric constructions of
random type is to reduce the study to geometric constructions
satisfying the weak asymptotic condition.}
A good general reference for dimension theory is \cite{F1}.
\medskip \centerline{\bf Acknowledgement}
\medskip
The authors would like to
thank H. Furstenberg, A. Manning, and D. Mauldin for helpful discussions and would like to especially thank F. Ledrappier and Y. Peres for their
helpful comments and examples. The first author wishes to thank IHES
for their support and hospitality during his visit in Fall 1992 which
enabled him to think about some problems in this paper.
\bigskip
\head{\bf Section 1: Geometric Constructions} \endhead
\bigskip We define several different types of geometric constructions
of Cantor-like subsets of $\Bbb R^n$ where the ration coefficients may
vary from step to step but admit uniform estimates from below and above.
We begin with the simplest construction.
\medskip
\item{\bf 1)} A {\bf simple geometric construction} is defined by \itemitem{a)} $2 p$ numbers $\underline \lam_i, \overline \lam_i, \, i=1, \cdots, p$ such
that $0 < \underline \lam_i \leq \overline \lam_i < 1$.
\smallskip
\itemitem{b)}
a family of closed sets $ \{ \Delta_{i_1 \cdots i_n} \} \subset \Bbb
R^d $ \, for \, $ i_j = 1, 2, \dots, p, $ and $ n \in \Bbb N$, which
satisfy
$$
D ( C_1 \prod_{j=1}^n \underline \lambda_{i_j}) \subset
\Delta_{i_1 \cdots i_n} \subset D ( C_2 \prod_{j=1}^n \overline
\lambda_{i_j}) \quad \tag 4
$$
where $D(r)$ denotes a closed ball of
radius $r$ and $C_1$ and $C_2$ are positive constants. For any 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
$$
\Delta_{i_1 \cdots i_n} \cap \Delta_{i'_1 \cdots i'_n} = \emptyset
\quad \text{ if } \, (i_1 \cdots i_n) \neq (i'_1,\cdots,i'_n). \quad
\tag 5 $$
\medskip Let $$
F = \bigcap_{n=1}^{\infty} \bigcup_{(i_1 \cdots i_n)}\Delta_{i_1
\cdots i_n}. $$
$F$ is called the {\it limit set} and 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. They need not be open or closed, but they
should contain balls and be contained in balls. The interior balls
obviously do not overlap, but the exterior balls may overlap. See
Figure 1. \bigskip
The simple geometric construction has a symbolic description in the
space of all one-sided infinite sequences $(i_1 i_2 \cdots)$. We denote this
space by $\Sigma^+_p$ and endow it with its usual
topology (see Appendix 3). Namely, 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 $\chi: F \ra \Sigma_p^+$ defined by $x \ra (i_1 i_2
\cdots)$ is a homeomorphism from $F$ onto $\Sigma^+_p$. More general constructions are determined
by closed shift-invariant subsets of $\Sigma^+_p$. The associated
symbolic dynamics is our main tool to compute the Hausdorff dimension
of the limit set $F$.
\bigskip
\item{\bf 2)} A {\bf symbolic geometric construction} is
defined by
\itemitem{a)} $2 p$ numbers $\underline \lam_i, \overline
\lam_i, \, i=1, \cdots, p$ such that $0 < \underline \lam_i \leq
\overline \lam_i < 1$.
\itemitem{b)} a compact set $Q \subset
\Sigma_p^+$ invariant under the shift $\sigma$ (i.e.,
$\sigma(Q)=Q$).
\itemitem{c)} a family of closed 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 (4) and (5).
The {\it limit set} $F$ for this construction is defined by
$$
F = \bigcap_{n=1}^{\infty} \bigcup_{\underset \text{admissible} \to
{(i_1 \cdots i_n)}} \Delta_{i_1 \cdots i_n}.
$$
We again 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} \}$.
In the particular case when the set $Q = \Sigma_A^+$ the
construction is called a {\bf Markov geometric construction}. Here $A$
denotes a
$p \times p$ transfer matrix with entries $ A(i,j) = 0$ or $ 1$ \,
and $\Sigma_A^+$ 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
$). In a simple geometric construction the {\it actual size} of the sets
$\Delta_{i_1 \cdots i_n}$ may vary from step to step and in general
depends on the {\it past}, (i.e., numbers $\{i_1 \cdots i_n \}$).
In the multi-dimensional case, the sets $\Delta_{i_1 \cdots i_n}$ are
{\it non-isotropic} in the sense that they may have different {\it sizes} in different directions. The analysis in such a general case is quite complicated. We shall deal with estimates that are isotropic, i.e., there are estimates on the radii of inscribed and circumscribed balls for the sets $\Delta_{i_1 \cdots i_n}$. Condition (4) provides isotropic lower and upper bounds that are also {\it
stationary} over $n$. This will enable us to obtain crude lower and
upper estimates for the Hausdorff dimension and box dimension of the
limit set. More refined estimates can be obtained when more
information about the size of the sets $\Delta_{i_1 \cdots i_n}$ is
available. We wish to consider processes with ratio coefficients
admitting {\it nonstationary} lower and upper estimates given by
sequences of numbers $\ll_{i,n}, \ol_{i,n}, \, i =1, \cdots, p,$ and $n
\in \Bbb N$.
\medskip
\item{\bf 3)} A {\bf symbolic geometric construction with nonstationary
bounds} is given by
\itemitem{a)} sequences of numbers $\ll_{i,n},
\ol_{i,n}, \, i = 1, \cdots, p,$ and $n \in \Bbb N$ such that $0 <
\alpha \leq \ll_{i,n} \leq \ol_{i,n} \leq \beta < 1$.
\itemitem{b)} a
compact set $Q \subset \Sigma_p^+$ invariant under the
shift $\sigma$ (i.e., $\sigma(Q)=Q$).
\itemitem{c)} a family of sets $ \{ \Delta_{i_1 \cdots i_n}\}
\subset \Bbb R^n$
\, 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 $Q$ and these sets satisfy
$$
D ( C_1 \prod_{j=1}^n \underline \lambda_{i_j, j}) \subset
\Delta_{i_1 \cdots i_n} \subset D ( C_2 \prod_{j=1}^n \overline
\lambda_{i_j, j}) \quad \tag 6
$$
where $C_1, C_2$ are positive constants. We also require (5). \medskip
The {\it limit set} $F$ is defined by
$$
F = \bigcap_{n=1}^{\infty} \bigcup_{\underset \text{admissible} \to
{(i_1 \cdots i_n)}} \Delta_{i_1 \cdots i_n}.
$$
\medskip
In the particular case when the set $Q = \Sigma_A^+$, the construction
is called a {\bf Markov geometric construction with nonstationary
bounds } and if $Q=\Sigma^+_p$ the construction is called a {\bf
simple geometric construction with nonstationary bounds.} \bigskip
\item{\bf 4)} A {\bf random symbolic geometric construction} is
defined by
\itemitem{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 $0 < \alpha \leq \underline \lambda_{j,n} \leq
\overline \lambda _{j,n} \leq \beta < 1, \, \frak F$ denotes the
$\sigma-$algebra of Borel sets in $\Lambda$, and $\nu$ is an arbitrary
stationary shift-invariant ergodic Borel probability measure on
$\Lambda$.
\itemitem{b)} a compact set $Q \subset \Sigma^+_p$
invariant under the shift (i.e., $\sigma(Q)=Q$).
\itemitem{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 $Q$. We assume that these sets satisfy
$$
D ( C_1 \prod_{j=1}^n \underline \lambda_{i_j, j}(\vec
\lambda)) \subset
\Delta_{i_1 \cdots i_n}(\vec \lambda) \subset D ( C_2 \prod_{j=1}^n
\overline \lambda_{i_j, j}(\vec \lambda))
$$
where $C_1, C_2$ are positive constants.
\itemitem{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
$$
\quad \Delta_{i_1 \cdots i_n}(\vec \lambda) \cap \Delta_{i'_1 \cdots
i'_n}(\vec \lambda) = \emptyset, \, \text{ if } \,(i_1\cdots i_n) \neq
(i'_1,\cdots i'_n).
$$
For every $\vec \lambda \in \Lambda$, the {\it 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.
\bigskip
\centerline{}
\bigskip
\head{\bf Section 2: Main Results: geometric constructions with
stationary bounds} \endhead
\bigskip
We begin with the symbolic geometric construction. Consider the symbolic dynamical system $(Q, \sigma)$, where $Q \subset \Sigma^+_p$. There
exist two numbers $\underline s, \, \overline s$, that are uniquely defined, such that
$P(\underline s \log \underline \lam_{i_1}) = 0$ and $P(\overline s
\log \overline \lam_{i_1}) = 0$, where $P$ denotes the thermodynamic
pressure (see Appendix 3).
Theorem 1 below provides estimates for the Hausdorff dimension and box
dimension of the limit set $F$ for a symbolic geometric construction
while Theorem 2 deals with the dimensions of special equilibrium measures
concentrated on $F$ for which a (non-uniform) mass distribution principle
holds.
\proclaim{\bf Theorem 1} Let $F$ be the limit set
specified by a symbolic geometric construction. Then
\roster
\item $\underline s \leq \dHF \leq \dBFL \leq \dBFU \leq \overline s $,
\item $0 < \text{m}_{\text{H}}(\underline s, F) $ and
$\text{m}_{\text{H}}(\overline s, F) < \infty $, \, where $\dHF,
\dBFL, \, \text{and } \dBFU$ are respectively the Hausdorff dimension
and the lower and upper box dimensions of the set $F$, and
$\text{m}_{\text{H}}(t, F)$ denotes the $t-$dimensional Hausdorff
measure of $F$ (see Appendix 1).
\endroster
\endproclaim
\medskip
The second statement in Theorem 1 is nontrivial only when $\ls = \dHF$ or $\os = \dHF$. Otherwise, $m_H(\ls, F) = \infty$ or $m_H(\os, F) = 0$. See Corollary 1. If $\ls < \dHF < \os$, then the Haudorff measure at dimension may be zero or infinite. In example 4 of Section 5 we present simple geometric constructions where the Hausdorff measure of the limit set at dimension is zero.
The lower and upper estimates $\ls$ and $\os$ in Theorem 1 clearly do
not depend on the placement of the sets $\{ \Delta_{i_1 \cdots i_n}
\}$. However, the actual value of the Hausdorff dimension and box
dimensions can be any number between $\ls$ and $\os$ and {\it a priori}, may
depend on the placement of the sets. See \cite{BT} and \cite{Tr} for
details in the one-dimensional case. In Example 6 of Section 5, we show this for the lower box dimension.
The next theorem uses the notion of pointwise dimension of a measure
(see Appendix 2). Let $\underline \mu$ and $\overline \mu$ denote the
equilibrium measures for the functions $\underline s \log \underline
\lam_{i_1}$ $\overline s \log \overline \lam_{i_1}$ on $Q$,
and $\underline m, \, \overline m$ be the pull back measures under the
coding map $\chi$.
\proclaim{\bf Theorem 2}Let $F$ be the limit set
specified by a symbolic geometric construction. Then
\roster
\item $\underline s \leq {\underline
d}_{\underline m}(x)$ for $\underline m-$ almost every $x \in F$ and
$\overline s \geq {\overline d}_{\om}(x)$ for $\overline m-$ almost
every $x \in F$,
\item If $\underline \lambda_i = \overline \lambda_i =
\lambda_i$ for $1 \leq i \leq p$ then $\underline m = \overline m = m, \, \underline s = \overline s = s,$ and
$ {\underline d}_{m}(x) = {\overline d}_{m}(x) = s $ for $m-$almost every $x \in F$. In particular,
$$ \dim_H m = \underline \dim_B m = \overline \dim_B
m = \dim_HF = \underline \dim_B F = \overline \dim_B F = s.
$$
\endroster
\endproclaim
\medskip
In general, one can not expect the lower and upper pointwise dimensions
${\underline d}_{\lm}(x)$ and $ {\overline d}_{\lm}(x)$ (as well as
${\underline d}_{\om}(x)$ and $ {\overline d}_{\om}(x)$ ) to coincide
almost everywhere. See Sections 6.2 and 6.3 and also Remark (4).
\medskip
\proclaim{\bf Corollary 1} Let $F$ be the limit set
specified by a symbolic geometric construction with $\underline \lam_i =
\overline \lam_i = \lam_i$ for $i =1, \cdots, p$. Then
$$
s = \dHF = \dBFL = \dBFU
$$
where $s$ is the unique root of the equation $P( s \log \lam_{i_1}) = 0$. \endproclaim
\medskip
The lower bound for the Hausdorff dimension of the limit set follows from
the lower bound for the lower pointwise dimension of $\underline m$ in Theorem 2. To prove this we first observe that the equilibrium measure $\lm$ satisfies the nonuniform mass distribution principle and then apply the Moran method \cite{Mo}. 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 F. Ledrappier.
The coincidence of Hausdorff dimension and box dimension was established by Falconer \cite{F3} in a very special case when the symbolic construction is defined by similarity maps. However, he did not compute the actual value of the dimension. Peres has observed that some results in the general case can be easily obtained by studying the symbolic geometric construction defined by similarity maps (see Section 4). This enables us to obtain lower and upper estimates for the Hausdorff measure of the limit set at the dimension (see Proposition 4 below).
\medskip
We now study the Markov geometric construction which is a special type of symbolic geometric constructions. In this case we have our most refined statements about the Hausdorff dimension and Hausdorf measure. Consider
the 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
$M_t = \text{diag}(\alpha_1^t, \cdots,
\alpha_p^t)$. Let $\rho(B)$ denote the spectral radius of the matrix
$B$.
\medskip
\proclaim{\bf Proposition 1} The equation $P(s \log
\alpha_{i_1}) = 0$ is equivalent to the equation $\rho(A^*M_s) =1$,
where $A^*$ denotes the transpose of the matrix $A$. \endproclaim
This proposition immediately implies that the numbers $\underline s$
and $\overline s$ are the unique solutions of the equations $\rho(A^*
\underline \Lambda_t) = 1$ and $\rho(A^* \overline \Lambda_t) = 1$,
where $\underline \Lambda_t = \text{diag}(\underline \lambda_1^t,
\cdots, \underline \lambda_p^t)$ and $\overline \Lambda_t =
\text{diag}(\overline \lambda_1^t, \cdots, \overline \lambda_p^t)$.
>From now on 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 (3) below.
\medskip
The main difference between the Markov and general symbolic cases is that the
measures $\underline \mu$ and $\overline \mu$ are Gibbs measures, i.e., they
satisfy (22), see \cite{R}. A manifestation of this is that the pull-back measure $\underline m$ satisfies the uniform mass distribution principle.
The next proposition gives a detailed description of the dimension of the limit set for general symbolic geometric constructions where the measures $\lmu$ and $\omu$ are Gibbs measures.
\proclaim{\bf Proposition 2 } Let $F$ be the limit set
specified by a symbolic geometric construction. Assume that the measures $\lmu$ and $\omu$ are Gibbs measures.
\roster
\item The measure
$\lm$ satisfies the uniform mass distribution principle.
\item For {\bf every}
$x \in F, \, \ls \leq {\underline d}_{\underline m}(x) $ and $
\overline s \geq {\overline d}_{\om}(x)$.
\item If $\lambda_i= \lli = \oli,$ for $i = 1, \cdots, p$, then $\underline
\mu = \overline \mu \overset \text{def} \to \equiv \mu$ and hence $
\underline m = \overline m \overset \text{def} \to \equiv m =
\chi^* \mu$ and $ s= {\underline d}_{m}(x) = {\overline d}_{m}(x)
$ for {\bf every} $x \in F$. Moreover, the Hausdorff measure $m_H(s, \cdot)$ is equivalent to the measure $m(\cdot)$.
\item If $\underline \lam_i = \overline \lam_i = \lambda_i,$ for $i =1, \cdots, p$, then
$$
s = \dHF = \dBFL = \dBFU.
$$
\endroster
\endproclaim
The next corollaries give a detailed description of the dimension of the limit set for Markov and simple geometric constructions and immediately follows from Proposition 2.
\medskip
\proclaim{\bf Corollary 2 }
Let $F$ be the limit set specified by a Markov geometric construction. Then statements (1) to (4) of Proposition 2 hold, and in addition, $s = \frac{\log \rho(A) }{\log (\frac{1}{\lambda})}$. \endproclaim
\medskip
\proclaim{\bf Corollary 3 }
Let $F$ be the limit set
specified by a simple geometric construction.
\roster
\item $ \underline s
\leq \dHF \leq \dBFL \leq \dBFU \leq \overline s, $ where
$\underline s$ and $ \overline s$ are the unique roots of the
equations
$$
\sum_{i=1}^{p} \underline \lam_i^{\underline t} = 1 \quad
\text{ and } \quad \sum_{i=1}^{p} \overline \lam_i^{\overline t} = 1
$$
respectively. Moreover, the Gibbs measures $\lmu, \omu$ on $\Sigma^+_p$
satisfy
$$
\lmu(C_{i_1 \cdots i_n}) = \prod_{j=1}^n \ll_{i_j}^{\ls} \, \text{ and } \,
\omu(C_{i_1 \cdots i_n}) = \prod_{j=1}^n \ol_{i_j}^{\os}
$$
where $C_{i_1 \cdots i_n}$ is a cylinder set.
\item If $\underline \lam_i = \overline
\lam_i = \lam_i,$ for $i =1, \cdots, p$, then
$$
s = \dHF = \dBFL = \dBFU
$$
where $s$ is the unique root of the
equation $\sum_{i=1}^p \lam_i^t = 1$. Moreover $0 <
\text{m}_{\text{H}}(s, F) < \infty$. In particular, if $\lam_i =
\lambda$ for $i =1, \cdots, p$ then $s = \log p /\log (1/\lambda)$ (see
\cite{Mo}).
\endroster
\endproclaim
\medskip
The next proposition establishes upper estimates for the upper box dimension for symbolic geometric constructions. In particular, we show that if the topological entropy of the shift is zero, then the dimension of the limit set is also zero. In this case the measures $\lmu$ and $\omu$ are measures of maximal entropy and are not Gibbs measures.
\proclaim{\bf Proposition 3} \endproclaim Let $F$ be the limit
set specified by a symbolic geometric construction.
\roster
\item We have
$$
\overline s \leq \frac{h(\sigma \, | \, Q)}{ - \log \overline \lambda_{\max}} $$
where $ \overline \lambda_{\max} = \max_{k=1}^p \{ \overline \lambda_k
\}$, and $h(\sigma \, | \, Q) $ denotes the topological entropy. Equality occurs if
$\overline \lambda_i = \overline \lambda$ for $i=1, \cdots, p$.
\item See \cite{Fu}. If $\underline \lam_i = \overline \lam_i = \lambda$ for $i =1, \cdots, p$, then
$$
\underline s = \overline s= \dHF = \dBFL = \dBFU =
\frac{h(\sigma \, | \, Q)}{-\log \lambda}.
$$
\item If ${h(\sigma \, | \, Q)}
= 0$, then $\underline s = \overline s= \dHF = \dBFL = \dBFU = 0$.
\endroster
\bigskip
\proclaim{\bf Remarks} \endproclaim
\roster
\item The basic fact
underlying the proofs of Theorems 1 and 2 is that the equilibrium measure
$\lm$ satisfies the nonuniform mass distribution principle. If this measure is a Gibbs measure, then as we have mentioned it satisfies the uniform mass distribution principle.
Little is known about the existence of Gibbs measures for general symbolic dynamical systems. We believe that one can
find a compact shift-invariant subset $Q \subset \Sigma^+_p$ with
positive topological entropy of $\sigma \, | \, Q $ and can build a
symbolic geometric construction with some $\ll_i = \ol_i = \lam_i$ such that the measure $m = \lm = \om$ is not a Gibbs measure and is not equivalent to the $s-$Hausdorff measure.
If the topological entropy of $\sigma \, | \, Q$ is zero, then by
Proposition 3, the Hausdorff dimension and box dimension coincide and
are zero, but the measures $\underline m$ and $\overline m$ are
not Gibbs measures.
\medskip
\item Two closed sets are called non-overlapping if
their intersection consists of only boundary points.
By examining the proofs of Theorems 1, 2 and Propositions 2, 3, one can show that they
are valid for geometric constructions on the line with non-overlapping
sets. This is due to the use of the mass distribution principle. See
\cite{Mo} and \cite{MW1}.
\medskip
\item In the Markov geometric 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 also 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. See \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 geometric construction defined by the transitive matrix $A_i$
and hence admits lower and upper estimates for the Hausdorff and box
dimensions stated in Theorems 1, 2 and Propositions 1, 2, 3. In \cite{MW1}, the
authors discuss the effect of the wandering set $Q_1$ on the dimension
of the limit set $F$.
\item Let $F$ be the limit set specified by a symbolic geometric
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. In
particular, the measures $\underline m$ and $\overline m$ are
$G-$invariant. Moreover, they satisfy
$$
P(\underline s \underline
\phi)= \sup_m \left (h_m(G) + \underline s \int_F \underline \phi d m
\right) = h_{\underline m}(G) + \underline s \int_F \underline \phi d
\underline m = 0
$$
$$
P(\overline s \overline \phi)= \sup_m
\left(h_m(G) + \overline s \int_F \overline \phi d m \right) =
h_{\overline m}(G) + \overline s \int_F \overline \phi d \overline m =
0
$$
where $\underline \phi(x)=\log \underline \lambda_k, \overline
\phi(x)=\log \overline \lambda_k $ if $x \in \Delta_k$, $h_m(G)$ is the
Kolmogorov-Sinai entropy of $G$ and $P(\psi)$ is the topological
pressure of a function $\psi$ on $F$ with respect to $G$ (see Appendix
3). In other words, the measures $\lm$ and $\om$ are equilibrium
measures corresponding to the functions $ \ls \underline \phi$ and $\os
\overline \phi$. If the geometric construction is Markov and is given by
a transitive transfer matrix $A$, then $G$ is ergodic, and moreover has
the Bernoulli property with respect to $\lm$ and $\om$. As we will see
in Section 6.2, it may happen that $\underline d_{\lm}(x) \neq
\overline d_{\lm}(x)$ for $\lm-$almost every $x \in F$ and $\underline
d_{\om}(x) \neq \overline d_{\om}(x)$ for $\om-$almost every $x \in
F$. Such a map $G$ is an example of a continuous but not smooth map
possessing an equilibrium measure which is not exact dimensional. Note
that $G$ is not invertible. In Section 6.3 we will construct a homeomorphism
that is not exact dimensional.
If $\underline \lambda_i = \overline \lambda_i = \lambda_i$ for $i=1,
\cdots, p$, then according to Theorem 2 $\lm = \om =m$ and $\underline
s = \overline s = s$. The latter can be expressed as $$
s=-\frac{h_m(G)}{\int_F \phi dm} $$ where $\phi(x) = \log \lambda_k$ if
$x \in \Delta_k$. In this case, $\underline d_m(x) = \overline d_m(x) =
s$.
\item Let us 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 simple geometric construction with $\underline \lambda_i
= \overline \lambda_i = \lambda_i$ for $1 \leq i \leq p$. Let
$\mu_{\phi}$ be the unique Gibbs measure corresponding to the function
$s \phi(\omega)=s \log \lambda_{\omega_1}$ with $s$ the unique root of
the equation $P( s \phi) = 0$. We have that $h_{\mu_{\phi}}(\sigma) + s
\int \log \lambda_{\omega_1} d \mu_{\phi} = 0$, which gives that
$$
s
= - \frac{h_{\mu_{\phi}}(\sigma)}{\int \phi d\mu_{\phi} } = \frac{
{h_{\mu_{\phi}}}(\sigma) }{ \log \beta}
$$
if
$$
\beta = \exp \left( -
\int \phi d \mu_{\phi} \right). \quad \tag 7
$$
Thus the Hausdorff
dimension of $\mu_{\phi}$ 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 $\rho$
$$
\dim_H F = s = \frac{ {h_{\mu_{\phi}}}(\sigma)}{ \log \beta}
$$
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 $h_{\mu_{\phi}}(\sigma) = h(\sigma)$ if and only if
$\lambda_1 = \lambda_2 = \cdots = \lambda_p = \lambda$. \endroster
\bigskip
\head{\bf Section 3: Main Results: geometric constructions with
nonstationary bounds and random processes} \endhead
We now consider symbolic geometric constructions with nonstationary
bounds. We assume that $\ll_{i,n}, \ol_{i,n}$ admit an asymptotic
behavior, {\it but not necessarily convergent}. We can write $$
\ll_{i, n} = \lli \exp(\underline a_{i, n}), \quad \ol_{i,n} = \oli
\exp(\overline a_{i, n}) $$ where $0 < \lli \leq \oli < 1, \, i=1,
\cdots, p$ are some numbers and $ \underline a_{i, n}, \overline a_{i,
n}$ are sequences of numbers.
We recall that the collection of numbers $(\ll_1, \cdots, \ll_p)$ and
$(\ol_1, \cdots, \ol_p)$ generate two measures $\underline \mu$ and
$\overline \mu$ on $Q \subset \Sigma^+_p$ that are equilibrium
measures associated to the functions $ s \underline \phi(\omega) =
\underline s \log \ll_{i_1}, \, s \overline \phi( \omega) = \overline s
\log \ol_{i_1}, \, \omega \in Q$ and $\ls, \os$ are the roots of the
equations $P( s \underline \phi)=0$ and $P( s \overline \phi)=0$ (see
Appendix 5). Also let $\lm, \om$ denote the pull back of the measures
$\underline \mu, \overline \mu$ under the mapping $\chi$.
We consider the following two types of asymptotic behavior of the
sequence of numbers $\underline a_{i, n}, \overline a_{i, n}$:
\medskip
\item{(\bf 1)} {\bf strong asymptotic condition}
There exist constants $K_1 < 0, K_2 > 0$ such that for any admissible
$n-$tuple $(i_1 \cdots i_n)$, $$ K_1 \leq \sum_{j=1}^n \underline
a_{i_j, j} \leq K_2 \quad \text{ and } K_1 \leq \sum_{j=1}^n \overline
a_{i_j, j} \leq K_2 $$ \medskip \item {(\bf 2)} {\bf weak asymptotic
condition} \itemitem{a)} for $\lm-$ almost every $x \in F$ with
$\chi(x) = (i_1 i_2 \cdots)$, $$ \frac1n \sum_{j=1}^n \underline
a_{i_j, j} \ra 0 \quad \text{ as } n \ra \infty $$
\itemitem{b)} for $\om-$almost every $x \in F$ with $\chi(x) = (i_1 i_2
\cdots)$, $$ \frac1n \sum_{j=1}^n \overline a_{i_j, j} \ra 0 \quad
\text{ as } n \ra \infty. $$
It is obvious that the strong asymptotic condition implies the weak
asymptotic condition. Moreover, if $\{\ll_{i,n}, \ol_{i,n} \}$ satisfy
the strong asymptotic condition then we have convergence $\ll_{i, n}
\to \ll_i$ and $\ol_{i, n} \to \ol_i$. It is easy to construct a
nonstationary symbolic geometric construction satisfying the weak
asymptotic condition 2a) but not 2b) and vice-versa. If $\ll_i =
\ol_i$ for $i=1, \cdots, p$ then $\underline a_{i, n} = \overline a_{i,
n}, n=1, 2,\cdots, \lm = \om, \ls = \os,$ and conditions 2a) and 2b)
coincide.
The difference between the two asymptotic types of behavior is very
crucial. In the case of a simple geometric construction
with nonstationary bounds satisfying the strong asymptotic condition the measure $\lm$ is Gibbs and satisfies the {\it uniform} mass distribution principle while in
the case of a simple geometric construction with nonstationary bounds satisfying
the weak asymptotic condition, one can expect this measure to satisfy
only the {\bf nonuniform} mass distribution principle.
It immediately follows from the strong asymptotic condition that there
exist constants $K_3 > 0, K_4>0$ such that for any admissible $n-$tuple
$(i_1 i_2 \cdots)$,
$$
K_3 \prod_{j=1}^n \underline \lambda_{i_j} \leq
\prod_{j=1}^n \underline \lambda_{i_j,j} \leq K_4 \prod_{j=1}^n
\underline \lambda_{i_j}
$$
$$ K_3 \prod_{j=1}^n \overline
\lambda_{i_j} \leq \prod_{j=1}^n \overline \lambda_{i_j,j} \leq K_4
\prod_{j=1}^n \overline \lambda_{i_j}. \quad \tag 8
$$
These relations make it possible to reduce the study of symbolic
geometric constructions admitting nonstationary bounds satisfying the strong asymptotic condition to those
admitting stationary bounds. The next theorem follows immediately from
the proof of Theorems 1, 2, Propositions 2, 3, 4, and relation (8).
\proclaim{\bf Theorem 3} Let $F$ be the limit set specified by a
symbolic geometric construction with nonstationary bounds satisfying
the strong asymptotic condition. Then the statements of Theorems 1, 2,
Propositions 1, 2, 3, and Corollaries 1, 2, 3 hold.
\endproclaim
\medskip
In the case of symbolic geometric constructions with
nonstationary bounds satisfying the weak asymptotic condition, the
following weaker results can be shown:
\proclaim{\bf Theorem 4}
\roster
\item If the construction satisfies
the weak asymptotic condition 2a) then $\ls \leq {\underline
d}_{\lm}(x)$ for $\lm-$almost every $x \in F$ and hence $\ls \leq
\dHF$.
\item If the construction satisfies the weak asymptotic
condition 2b) then $ {\overline d}_{\om}(x) \leq \os$ for $\om-$
almost every $x \in F$ and hence $\overline{\text{dim}}_B \om \leq \os$.
\item If the construction satisfies the weak asymptotic conditions 2a) and 2b), and $\ll_i = \oli, \, i=1, \cdots, p$ then $\lm = \om = m, \, \ls
= \os = s , \, $ and $ s = \text{dim}_Hm = \underline {\text{dim}}_B m
= \overline{\text{dim}}_B m$.
\endroster
\endproclaim
In Section 5 we present an example of a simple geometric construction
with nonstationary bounds satisfying the weak asymptotic but not the
strong asymptotic condition (see example 4) and for which $\ll_{i, n} = \ol_{i,n}, \, i=1, \cdots, p$ and $n=1,2, \cdots$. For this construction we
show that the Hausdorff measure of the limit set can be $0$. This violates the uniform mass distribution principle. We believe that such an example can be constructed such that
$\overline s$ does not provide an upper bound for the upper box
dimension of the limit set.
To further investigate this situation one
can consider a gauge function $h(t)$ (see Appendix 4) such that the
corresponding $h-$measure is finite and positive. See also example 4.
We now consider random geometric constructions.
\proclaim{\bf Theorem 5} 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 \item{(1)} For $\nu-$almost every $\vec
\lambda \in \Lambda$ the following limits exist: \itemitem{a)} for
$\underline \mu-$almost every sequence $(i_1 i_2 \cdots) \in Q(\vec
\lambda)$ $$ \lim_{n \ra \infty} \frac1n \sum_{k=1}^{n}\underline
a_{i_k,k} = 0 $$ \itemitem{b)} for $\overline \mu-$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 $\underline
a_{i, j} = \log \left ( \frac{\underline \lambda_{i, j}}{\ll_i } \right
), \overline a_{i, j} = \log \left ( \frac{\overline \lambda_{i,
j}}{\ol_i } \right )$ and $ \underline \mu, \overline \mu$ are the
Gibbs measures corresponding to the functions $\ls \log \ll_{i_1}$ and
$\os \log \ol_{i_1}$. \item{(2)} For $\nu-$almost every $\vec \lambda
\in \Lambda$ $$ \ls \leq \dHF(\vec \lambda).
$$
\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., $ d(h_i(x), h_i(y)) \leq L_i d(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:
\roster
\item {\it The geometric construction can not be described by
any continuous maps}, i.e., there are no continuous maps satisfying
(9). This is the case 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 \item {\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 phenomenon 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 location of the
intervals $\Delta_{i_1 \cdots i_n}$ 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
\item {\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 $h_1$ can not be a
contraction.
\medskip \item {\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 $h_1$ can not be Lipschitz. \medskip
\item {\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 $x \in \Delta$).
\endroster \medskip
As we have seen in (4) the Lipschitz constants for the inverse maps
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
lower Hausdorff dimension of the limit set.
\bigskip
We point out that the coding we employ in our definitions of the
geometric constructions is not the obvious generalization of the {\it
natural} coding for a similarity process. The natural coding of a
similarity process 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 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 Remark (4) 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
Peres has observed that some dimension results in the general symbolic case can be obtained by studying the symbolic geometric construction defined by similarity maps. We now show how this remark enables us to obtain lower and upper estimates for the Hausdorff measure of the limit set at the dimension.
Let $F$ be the limit set for a symbolic geometric construction with ratio coefficients $\underline \lam_i, \overline \lam_i, \, i=1, \cdots, p$ that is determined by the set $Q \subset \Sigma^+_p$.
\proclaim{\bf Proposition 4 }
\roster
\item There exist similarity maps \, $\overline h_1, \cdots, \allowmathbreak \overline h_p:
\Delta \to \Delta$ with similarity coefficients $\overline \lambda_1,
\cdots, \allowmathbreak \overline \lambda_p$
and a Lipschitz map $\pi_1 : \overline F_{sym} \to F$ where $ \overline F_{sym}$ denotes the limit set for the symbolic geometric construction built using similarity maps $\overline h_i$. In particular,
$\dBFU \leq \overline {\text{dim}}_{B} \overline F_{sym}$ and $m_H (\overline s,F) \le \text {const } m_H (\overline s, \overline F_{sym})$.
\smallskip
\item There exist similarity maps $\underline h_1, \cdots, \underline h_p: \Delta \to \Delta$ with similarity coefficients $\underline \lambda_1, \cdots, \allowmathbreak \underline \lambda_p$
and a Lipschitz map $\pi_2 : F \to \underline F_{sym}$ where $\underline F_{sym}$ denotes the limit set for the similarity construction and built using the similarity maps $\underline h_i$. In particular,
${\text{dim}}_{H} \underline F_{sym} \leq \dHF$ and $m_H (\underline s, \underline F_{sym}) \le \text {const } m_H (\underline s, F)$.
\smallskip
\item If the ratio coefficients of the symbolic geometric construction satisfy $\underline \lam_i = \overline \lam_i = \lambda_i, \, i=1, \cdots, p$, then there exist similarity maps $h_1, \cdots, h_p: \Delta \to \Delta$ with similarity coefficients $\lambda_1, \cdots, \lambda_p$ such that
$$
\dim_HF = \underline \dim_B F = \overline \dim_B F = \dim_HF_{sym} = \underline \dim_B F_{sym} = \overline \dim_B F_{sym} = s,
$$
where $s$ is the root of the equation $P(s \log \lambda_{i_1})=0$.
Moreover, $0 < m_H(s, F) < \infty$.
\endroster
\endproclaim
\bigskip
\head{\bf Section 5: Examples}
\endhead
\midspace{3truein}
\caption{
Figure 2. \quad Sierpi\'nski Gaskets \quad a), \, b), \, c)} \item{\bf
1)} {\bf Sierpi\'nski Gaskets}
\itemitem{a)} It is well known that the
Hausdorff dimension of the Sierpi\'nski gasket (see Figure 2a)
coincides with the box dimension and is $ \frac{\log3}{\log2}$. This
immediately follows from 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 have a $12$ in it
(see 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 $ \dHF =\frac{\log
(\frac{3+\sqrt{5}}{2})}{\log 2} \allowmathbreak \approx 1.38848$.
\itemitem{c)} Figure 2c) illustrates a simple geometric construction
with nonstationary bounds of the Sierpi\'nski gasket with $\underline
\lambda_i = \overline \lambda_i = \frac13$ \, for $i =1, 2, 3$. 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} which become {\it asymptotically straight.} As
long as the approximation is sufficiently fast and uniform, meaning
that the construction satisfies the hypotheses of Theorem 3, 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
\item{\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{3truein}
\caption{Figure 3. \quad Smale-Williams Solenoid \quad A), B)}
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). It
immediately follows from Theorem 1 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
$\dim_H \Delta = \frac{\log 2 }{ \log (\frac{1}{\lambda})} + 1$, the
result obtained by Falconer \cite{F1}.
\medskip
\item{\bf 3)} In \cite{PW}, the authors construct families of affine
horseshoes (diffeomorphisms) of $\Bbb R^3$, for which the box
dimension exists (the lower box dimensions coincide with the upper box
dimensions) but sometimes does not coincide with the Hausdorff
dimension. They first construct a family of similarity processes in
the plane with limit sets having this property and then they build
horseshoes whose cross sections coincide with these limit sets. The
authors find that number theoretic properties of the ratio
coefficients, related to Pisot numbers, is a mechanism to cause
noncoincidence of box dimension and Hausdorff dimension.
\medskip
\item{\bf 4)} {\bf A Simple Geometric Construction With Ratio
Coefficients Satisfying the Weak Asymptotic, But Not Strong Asymptotic,
Condition}.
This example shows that in the second statement of Theorem 1, the measure of the limit set at dimension 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
$$
|\Delta_{i_1 \cdots i_n}| = \prod_{j=1}^n \lambda_{i_j, j} =\lambda^n
\exp (A_n),
$$
where $A_n = \sum_{k=1}^n a_k$ and $|\Delta_{i_1 \cdots
i_n}|$ denotes the diameter of the set $\Delta_{i_1 \cdots i_n}$. For
fixed $n$, the sets $\{ \Delta_{i_1 \cdots i_n}\}$ give a cover of $F$
with $$ \sum_{i_1 \cdots i_n} |\Delta_{i_1 \cdots i_n}|^s = 2^n
\lambda^{ns} \exp (s A_n) = \exp (s A_n). $$ 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 the strong asymptotic condition
and $s= \dHF = \dBF = \frac{\log 2}{\log (\frac{1}{\lambda})}$.
\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 weak asymptotic condition if and
only if $\frac1n A_n \ra 0$. For example, we can 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 should have $- \infty < s A_n + \phi(\lambda^n \exp(A_n)) <
\infty$. We 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 \item{\bf 5)} {\bf Random version of Example 4}
The following example is a special case of a family of random
constructions that were explained 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 $\liminf A_n \to -
\infty$. The law of large numbers shows that the process satisfies the
weak asymptotic condition, i.e., $\frac{A_n}{n} \ra 0$. 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 \item{\bf 6)} {\bf Simple Geometric Construction With a 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 geometric construction 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 (A) and horizontal (B). See Figure 4.
\midinsert
\vspace{3truein} \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 side 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
onto the two coordinate axes either coincide or are disjoint.
\medskip We need the following three lemmas:
\proclaim{\bf Lemma 1} Let $\frak U$ be a covering of the limit set $F$
by disjoint closed balls $B(x_i, r), i=1, \cdots, p$, and $r$ 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{\bf 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
$F \cap B(x'_j, r) \subset \cup_{k=1}^6 B(x_k, r)$. \enddemo \qed
\medskip
\proclaim{\bf Lemma 2} For any $n > 0, \, \text{dist}(\Delta_{i_1'
\cdots i_n' }, \Delta_{i_1'' \cdots i_n''}) \geq \ll^n$.
\endproclaim
\demo{\bf Proof} The proof is obvious. \quad \qed \enddemo
\medskip
\proclaim{\bf Lemma 3} 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 $\ol^{3k+1}$ 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 2, $\text{dist}(\Delta_{i_1'
\cdots i'_{n_{3k}} }, \Delta_{i_1'' \cdots i''_{n_{3k}}}) \geq
\ll^{n_{3k}}$. By Lemma 1, $N(r_k) \geq \frac16 2^{n_{3k+1}}$ 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 3, $\dBFU \leq \frac{\log 2}{- \log \ol}$. It follows that
$\dBFU = \frac{\log 2}{- \log \ol}$.
\comment Each rectangle $\Delta_{i_1 \cdots i_{n_{3k+1}}}$ can be
covered by a ball of radius $\leq 2 \ol^{n_{3k+1}}$ which is centered
at some point $x \in \Delta_{i_1 \cdots i_{n_{3k+1}}}$ and these balls
are disjoint since $\ol^{n_{3k+1}}= \ol^{[\alpha n_{3k} ] + 2} \leq
\frac13 \ll^{n_{3k}}$. By Lemma 1, $N(r_k) \geq \frac16 2^{n_{3k+1}}$
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 Theorem 1, $\dBFU \leq \frac{\log 2}{- \log \ol}$. It follows that
$\dBFU = \frac{\log 2}{- \log \ol}$. \endcomment
\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 $A = \Delta_{i_1 \cdots
i_{n_{3k+2}}} \cap F$ 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 $A$ 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 Corollary 2, 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
$2 \ll^{3k+1}$. 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 $$ \dBFL=
\lim_{m \to \infty} \frac{\log N(\epsilon_m)}{- \log \epsilon_m}
\geq \limsup_{m \to \infty} \frac{\log N(r_k)}{-\log r_k} = \gamma
\frac{\log 2}{ - \log \ol}.
$$
\comment This implies that
$$
\dBFL=
\lim_{m \to \infty} \frac{\log N(\epsilon_m)}{- \log \epsilon_m} \geq
\overline \lim_{m \to \infty} \frac{\log
(N(r_{k(m)})(\frac{\epsilon_m}{r_{k(m)}})^d)}{- \log \epsilon_m}
$$
$$
= \frac{ \overline \lim_{m \to \infty} \left( \frac{\log
(N(r_{k(m)}))}{- \log r_{k(m)} } \right) + d \, \overline \lim_{m \to
\infty} \left( \frac{\log (\frac{\epsilon_m}{r_k(m)})}{- \log r_{k(m)}
} \right)}{ \overline \lim_{m \to \infty} \left( \frac{- \log
\epsilon_m}{- \log r_{k(m)} } \right)}
$$
$$
= \frac{\gamma \,
\frac{\log 2}{- \log \ol}+ d \, (a-1) }{a} = \frac{b+ d \, (a-1) }{a}
$$
where $a \equiv \overline \lim_{m \to \infty} \left( \frac{\log
\epsilon_m}{- \log r_{k(m)}} \right) > 0$ and $b \equiv \gamma \,
\frac{\log 2}{- \log \ll} > 0$.
Since $\ol < \frac13$ and $d \geq 1$, the direct calculation shows
that $\frac{b+ d \, (a-1) }{a} \geq b$ \, and hence \, $\dBFL =
\gamma \, \frac{\log 2}{- \log \ll}$.
\endcomment
\bigskip
\head{\bf Section 6: Pointwise Dimension of Measures Concentrated on General
Cantor-like Sets}
\endhead
\bigskip {\bf 6.1} Let $F$ be the limit set
for a symbolic geometric construction with nonstationary bounds
satisfying $\underline \lambda_{i, n}=\overline \lambda_{i, n}
=\lambda_{i,n}$. 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 6}
\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 Theorems 4 and 6.
\proclaim{\bf Corollary 4} Let $F$ be the limit set for a symbolic
geometric construction with nonstationary bounds satisfying $
\underline \lambda_{i, n}=\overline \lambda_{i, n} $. 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
{\bf 6.2} 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 $x
\in F$ \itemitem{} $\underline d_{\om}(x) = \ls$ for $\om-$almost
every $x \in F$
\smallskip
and
\smallskip
\itemitem{b)} $\overline d_{\lm}(x) =\overline d_{\om}(x)= \os$
for every $x \in F$. }
\rm
\medskip
The fact $\dHF = \ls$ immediately
implies that $$ \underline d_{\lm}(x) \leq \ls \enspace \text{for }
\lm-\text{almost every } x \in F $$ and
$$ \underline d_{\om}(x) \leq
\ls \enspace \text{for } \om-\text{almoot every } x \in F.
$$
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 nonuniform mass distribution principle would
then imply that $\dHF \geq \text{dim}_{H}A \geq \ls + \epsilon$.
Statement a) follows from the first statement in Theorem 2.
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 and (12) imply that for all $x
\in F$, $$ \om(B(x, r_k)\bigcap F) \leq \om(\Delta_k(x) \bigcap F) \leq
D_1 \ol^{\os n_{3k+1}} = D_1 r_k^{\os} $$ and hence $$ \overline
d_{\om}(x) \geq \limsup_{k \to \infty} \frac{\log \om(B(x, r_k) \bigcap
F)}{\log r_k} \geq \os. $$
Similarly, using (10) we have that for all $x \in F$ $$ \lm(B(x, r_k)
\bigcap F) \leq \lm(\Delta_k(x) \bigcap F) \leq D_1 \ll^{\ls n_{3k+1}}
$$ and hence,
$$
\overline d_{\lm}(x) \geq \limsup_{k \to \infty}
\frac{\log \lm(B(x, r_k) \bigcap F)}{\log r_k} = \ls \frac{\log
\ll}{\log \ol} = \os.
$$
Statement (b) now follows from the first
statement of Theorem 2. \quad \qed \medskip {\bf 6.3} 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 4). 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 $\tilde \chi: \tilde F \to \Sigma_p$ (where $\Sigma_p$
denotes the space of two-sided sequences $( \cdots i_{-1} i_0 i_1
\cdots), \, i_j =1 \cdots p$) defined by
$$
\tilde \chi(x, y) = (\cdots
i_{-1} i_0 i_1 \cdots)
$$
where $\chi(x) = (\cdots i_{-1}), \, \chi(y)=
(i_0 i_1 \cdots)$. 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 3, $ \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 but not smooth map possessing a Gibbs measure that has
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
{\bf 6.4} We
will construct a simple geometric construction for which the map $G$
(see Remark (4)) 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 that the numbers $\lam_{i, n}, \, i=1,2$ satisfy the weak asymptotic condition.
Let $s$ be the unique root of the equation
$$
2(\sqrt{\alpha \beta})^s
+ (\sqrt{\gamma \delta})^s =1.
$$
According to Theorem 4 we have that
$$
s \leq \dHF, \enspace \underline {\text{dim}}_{H}m =
\underline {\text{dim}}_{B}m =\overline {\text{dim}}_{B}m = s,
$$
and
$$ \underline d_m(x) = \overline d_m(x) = s \, \text{ for $m-$almost
every $x \in F$}, $$ where $m$ is the Gibbs measure corresponding to
the function $s \log \lambda_{i_1}$.
\comment
Now consider the two sets $$ A = \left \{ x \in F \, : \, \chi(x) =
(i_1 i_2 \cdots) \text{ and } \cases i_j = 2, & \text{if $j$ is even},
\\ i_j = 2 \text{ or } 3, & \text{if $j$ is odd} \endcases \right \}
$$ and $$ B= \left \{ x \in F \, : \, \chi(x) = (i_1 i_2 \cdots) \text{
and } \cases i_j = 1, & \text{if $j$ is even}, \\ i_j = 2 \text{ or }
3, & \text{if $j$ is odd} \endcases \right \}. $$ We remark that the
sets $A$ and $B$ are not invariant under $G$. We have that for $x \in
A$, $$ \lim_{n \to \infty} \frac{\log \diam \Delta_n(x) }{n } = \frac12
\log (\alpha \beta) $$ and for $x \in B$, $$\lim_{n \to \infty}
\frac{\log \diam \Delta_n(x) }{n } =\log \alpha. $$ It is not
difficult to construct a measure $\mu$ on $\Sigma^+_3$ such that $\mu$
is invariant under the shift, $\mu$ is ergodic, $\mu(A) >0,\, \mu(B) >
0$ and $h_{\mu}(\sigma) > 0$. Thus for the measure $\nu = \chi^* \mu$,
$$
\underline d_{\nu}(x) = \overline d_{\nu}(x) = \frac{2
h_{\mu}(\sigma)}{\log (\alpha \beta)} \qquad \text{if $x \in A$}
$$
and
$$
\underline d_{\nu}(x) = \overline d_{\nu}(x) =
\frac{h_{\mu}(\sigma)}{\log \alpha} \qquad \text{if $x \in B$}.
$$
\endcomment
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 $d_{\tilde \nu }(x, y)$ is not
essentially constant.
\bigskip \head{\bf Section 7: Proofs} \endhead
\bigskip
We first consider the Markov geometric construction and provide proofs in this case.
\medskip
\demo {\bf Proof of Proposition 1} \enddemo
\medskip
We begin with the following general lemma:
\medskip
\proclaim {\bf Lemma 4} 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} 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 \enddemo
Proposition 1 follows by applying Lemma 4 to the function $f(\omega) =
f(\omega_1) = t \log \alpha_{\omega_1}$. \quad \qed
\bigskip
\demo{\bf Proof of Proposition 2}
\enddemo
We first show that $\underline s \leq
\dHF$. Fix $0 < r < 1$. For any $x \in F$ let $n(x)$ denote the unique
positive integer such that $\underline \lambda_{i_1} \underline
\lambda_{i_2} \cdots \underline \lambda_{i_{n(x)}} > r$ and
$\underline \lambda_{i_1} \underline \lambda_{i_2} \cdots \underline
\lambda_{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 in the construction of $F$ containing $x$ with the property
that for any $y \in \Delta(x), \enspace \Delta_{i_1 \cdots i_{n(y)}}
\subset \Delta(x)$ and there exists $z \in \Delta(x)$ such that
$\Delta(x) = \Delta_{i_1 \cdots i_{n(z)}}$. The sets $\Delta(x)$
corresponding to different $x \in F$ either coincide or are disjoint.
We denote these sets by $\Delta^{(j)}, \, j=1, \cdots, N$. There exist
points $x_j$ such that $\Delta^{(j)} = \Delta_{i_1 \cdots i_{n(x_j)}}
$. Note that the sets $\Delta^{(j)}$ depend on $r$. Each of these sets
has diameter $> r$, but contains at the next level of subdivision at
least one set of diameter $\leq r$.
By the definition of a Gibbs measure (see Appendix 3) there exist
positive constants $D_1$ and $D_2$ such that for $j=1, \cdots, N$
$$
D_1 \leq \frac{ \underline m(\Delta_{{i_1} \cdots i_{n(x_j)}})
}{\prod_{k=1}^{n(x_j)} \underline \lambda_{i_k}^{\underline s}} \leq
D_2 \quad \tag 10
$$
where $\underline m = \chi^* \underline
\mu$ is the pull back of the Gibbs measure $\underline \mu$ corresponding to the
function $\underline \phi(i_1 i_2 \cdots) = \underline s \log
\underline \lambda_{i_1}$ on $Q$. One can easily estimate the
number $N(x, r)$ of sets $\Delta^{(j)}$ that intersect the ball
$B(x,r)$ for sufficiently small $r$ as follows
$$
N(x, r) \leq
\frac{\text{vol} (B(x,r))}{\min_{1 \leq j \leq N} \text{vol}
(\Delta_{i_1 i_2 \cdots i_{n(x_j)}})} \leq \frac{\text{vol} (B(x, r))
}{ \min_{1 \leq j \leq N} \text{vol}(D (C_1 \prod_{k=1}^{n(x_j)}
\underline \lambda_{i_k }))}
$$
$$
\leq \frac{L_1 r^d}{ \min_{1 \leq j
\leq N} L_2 (\prod_{k=1}^{n(x_j)} \underline \lambda_{i_k})^d} \leq L_3
$$
where $L_1, L_2, L_3 > 0$ are constants and $\text{vol}$ denotes the
Euclidean volume in $\Bbb R^d$. Now by (10), we have that
$$
\underline m(B(x, r)) \leq \sum_{j=1}^{N(x,r)} \underline
m(\Delta^{(j)}) \leq \sum_{j=1}^{N(x,r)} D_2 \prod_{k=1}^{n(x_j)}
\underline \lambda_{i_k}^{\underline s}
$$
$$ \leq L_4
\sum_{j=1}^{N(x,r)} \prod_{k=1}^{n(x_j)+1} \underline
\lambda_{i_k}^{\underline s} \leq L_4 N(x, r) r^{\underline s} \leq L_5
r^{\underline s}, \quad \tag 11
$$ where $L_4, L_5 > 0$ are constants. Hence the measure $\underline m$ satisfies the uniform mass distribution principle. This proves statement (1). Moreover, (11) implies that $\underline s \leq {\underline d}_{\underline m}(x)$ for every $x \in F$. It follows that
$\underline s \leq \dHF$ and $\text{m}_{\text{H}}(\underline s, F) >
0$.
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 $\ol_{i_1} \ol_{i_2} \cdots
\ol_{i_{n(x)}} > r$ and $\ol_{i_1} \ol_{i_2} \cdots \ol_{i_{n(x)}}
\ol_{i_{n(x)+1}} \leq r$. By the definition of a Gibbs measure,
there exist constants $D_1, D_2 > 0$ such that for any $n \in \Bbb N,$
$$
D_1 \leq \frac{\overline m (\Delta_{i_1 \cdots i_n})}{\prod_{k=1}^n
\overline \lambda_{i_k}^{\overline s} } \leq D_2.
$$
It follows from the definition of $r$ 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 $x \in F,$
$$
\overline m (B(x, 2 C_2 r)) \geq \overline m(\Delta_{i_1 \cdots i_{n(x)
+ 1}}) \geq D_1 \prod_{k=1}^{n(x)+1} \overline
\lambda_{i_k}^{\overline s} \geq L_1 r^{\os}
$$ where $L_1> 0$ is a
constant. It follows that for all $x \in F,$
$$
{\overline d}_{\om}(x) = \limsup_{r \to 0} \frac{\log \overline
m(B(x,r))}{ \log r} \leq \overline s.
$$
This completes the proof of statement (2). The first part of statement (3) follows immediately from statements (1) and (2). We now prove the equivalence the measures $m_H(s, \cdot)$ and $m(\cdot)$.
Let $Z \subset F$ be a closed subset. Given $\gamma > 0$, 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
$$
m_H(s, Z) \leq \sum_{U^{(k)} \in \frak U} (\diam \, U^{(k)})^s + \gamma.
$$
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
$$
\sum_{\Delta^{(k)} \in \frak U } m(\Delta^{(k)}) \leq m(Z) + \gamma.
$$
By (4) and (10) it follows that
$$
m_H(s, Z) \leq \sum_{ \Delta^{(k)} \in \frak U} (\diam \, \Delta^{(k)})^s + \gamma \leq C_2 \sum_{ \Delta^{(k)} \in \frak U} \prod_{j=1}^{n(k)} \lambda^s_{i_j} + \gamma
$$
$$
\leq C_2 D_2 \sum_{ \Delta^{(k)} \in \frak U} m(\Delta^{(k)}) + \gamma \leq C_2 D_2 m(Z) + (C_2 D_2 + 1) \gamma.
$$
Since $\gamma$ is chosen arbitrarily this implies that $m_H(s, Z) \leq \text{ const } m(Z)$.
Given $\gamma > 0$, 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
$$
\sum_{U^{(k)} \in \frak U} (\diam \, U^{(k)})^s \leq m_H(s, Z) + \epsilon.
$$
By (11) it follows that
$$
m(Z) \leq \sum_{U^{(k)} \in \frak U} m(U^{(k)}) \leq L_5 \sum_{U^{(k)} \in \frak U} (\diam \, U^{(k)})^s \leq L_5 m_H(s, Z) + L_5 \gamma.
$$
Since $\gamma$ is chosen arbitrarily this implies that $m(Z) \leq m_H(s, Z)$.
\medskip
To prove statement (4) we show that $\dBFU \leq \overline s$.
This statment is true for general symbolic constructions but we provide an independent proof for our case which exploits the fact that $\om$ is a Gibbs measure. For the general case see the proof of Theorem 1.
Let $\overline m =
\chi^* \overline \mu$ be the pull back of the Gibbs measure
$\overline \mu$ corresponding to the function $(i_1 i_2 \cdots ) \to
\overline s \log \overline \lambda_{{i_1}} $ on $Q$. Fix $0
< r < 1$. For each $ x $ with $\chi(x)= (i_1 i_2 \cdots )$, choose the
unique $n(x)$ such that $\overline \lambda_{i_1} \overline
\lambda_{i_2} \cdots \overline \lambda_{i_{n(x)}} > r$ and $\overline
\lambda_{i_1} \overline \lambda_{i_2} \cdots \overline
\lambda_{i_{n(x)}} \overline \lambda_{i_{n(x)+1}} \leq r$. Repeating
the above arguments one can show that there exist points $x_j, \, j=1,
\cdots, N$ such that the sets $\Delta^{(j)}=\Delta_{i_1 \cdots
i_{n(x_j)}}$ are disjoint.
By the definition of a Gibbs measure, there exist constants $D_1, D_2
> 0$ such that for $j=1, \cdots, N$
$$
D_1 \leq \frac{\overline m
(\Delta_{i_1 \cdots i_{n(x_j)}})}{\prod_{k=1}^{n(x_j)} \overline
\lambda_{i_k}^{\overline s} } \leq D_2. \quad \tag 12
$$
By (12) we have that for $j=1, \cdots, N$
$$
\overline m(\Delta_{{i_1} \cdots
i_{n(x_j)}}) \geq D_1 \prod_{k=1}^{n(x_j)} \overline
\lambda_{i_k}^{\overline s} \geq D_1 r^{\overline s}.
$$
This implies that $N_{r}(F) \leq C r^{-\overline s} $, for some constant $C > 0$, where $N_r(F)$
denotes the minimum number of balls of radius $r$ required to cover the
limit set $F$. By (6) and (7), we have that
$$
\diam(\Delta_{i_1 \cdots {n(x_j)}}) \leq C_2 \prod_{k=1}^{n(x_j)}
\overline \lambda_{i_k} \leq L_6 \prod_{k=1}^{n(x_j)+1} \overline
\lambda_{i_k} \leq L_6 r,
$$
where $L_6$ is a positive constant. It follows that
$$
\dBFU = \limsup_{r \to 0} \frac{\log(N_{2r}(F))
}{\log(\frac1r ) } \leq \os.
$$
By repeating the above arguments one can easily show that $
m_H(\overline s, F) < \infty$.
\quad \qed
\bigskip
\demo{\bf Proof of Theorems 1 and 2} \enddemo
Given numbers $0 <
\lambda_1, \lambda_2, \cdots, \lambda_p < 1$, let $\mu$ be an
equilibrium measure on $Q$ corresponding to the function $s \log
\lambda_{i_1}$. By definition, $$
h_{\mu}(\sigma | Q) + s \int_Q \log \lambda_{i_1} d \mu = 0, \quad \tag
13 $$ where $h_{\mu}(\sigma | Q) \overset \text{def} \to = h$ is the
Kolmogorov-Sinai entropy. Let us first assume that $\mu$ is ergodic.
For fixed $\epsilon > 0$, it follows from the Shannon-McMillan-Breiman
theorem that for $\mu-$almost every $\omega \in Q$ one can find
$N_1(\omega) > 0$ such that for any $n \geq N_1(\omega)$,
$$
\exp(-(h + \epsilon) n) \leq \mu(C_{i_1 \cdots i_n(\omega)}) \leq
\exp(-(h - \epsilon) n) \quad \tag 14
$$
where $C_{i_1 \cdots i_n(\omega)}
$ is the cylinder set containing $\omega$. It follows from the Birkhoff
ergodic theorem that for $\mu-$almost every $\omega \in Q$ there exists
$N_2(\omega)$ such that for any $n \geq N_2(\omega)$,
$$
\frac1n \log
\prod_{j=1}^n \lambda_{i_j}^s - \epsilon \leq s \int_Q \log \lambda_{i_1} d
\mu \leq \frac1n \log \prod_{j=1}^n \lambda_{i_j}^s + \epsilon. \quad
\tag 15
$$
Combining (13), (14), and (15) we have that for $\mu-$almost
every $\omega \in Q$ and $n$ sufficiently large,
$$
\prod_{j=1}^n
\lambda_{i_j}^{s+ \alpha} \leq \prod_{j=1}^n \lambda_{i_j}^s
\exp(- 2 \epsilon n) \leq \mu(C_{i_1 \cdots i_n(\omega)}) \leq \prod_{j=1}^n \lambda_{i_j}^s \exp(2 \epsilon n) \leq \prod_{j=1}^n \lambda_{i_j}^{s- \alpha}, $$ where $\alpha=\frac{2
\epsilon}{ \min(\log \frac{1 }{\lambda_j}, 1 \leq j \leq p)} > 0$.
This implies that for $\mu-$almost every $\omega \in Q$ and any $n \geq \max \{N_1(\omega), N_2(\omega)\}$,
$$
\prod_{j=1}^n \lambda_{i_j}^{s+ \alpha} \leq
\mu(C_{i_1 \cdots i_n(\omega)}) \leq \prod_{j=1}^n
\lambda_{i_j}^{s- \alpha}. \quad \tag 16
$$
If $\mu$ 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 $x \in \chi^{-1}(Q_l))$
$$
\underline d_{\underline m}^l (x) \overset \text{def} \to \equiv \liminf_{r \to 0} \frac{\log \underline m(B(x, r) \cap \chi^{-1}(Q_l))}{\log r} \geq \ls - \alpha.
$$
It follows that for $x \in \chi^{-1}(Q_l)$,
$$
\underline d_{\underline m}(x) = \inf_{l > 0} \underline d^l_{\underline m}(x) \geq \underline s - \alpha,
$$
and thus this is true for $\underline m-$almost every $x \in F$. Since $\alpha$ is arbitrary, this implies that $\underline d_{\underline m}(x) \geq \underline s$.
We now show that ${\overline d}_{\om}(x) \leq \overline s$ for
$\overline m-$almost every $x \in F$. Fix $0 < r < 1$. For each $x$
with $\chi(x) = (i_1 i_2 \cdots)$, choose the unique $n(x)$ such that
$\ol_{i_1} \ol_{i_2} \cdots \ol_{i_{n(x)}} > r$ and $\ol_{i_1}
\ol_{i_2} \cdots \ol_{i_{n(x)+ 1}} \leq r$. It is easy to see that
$n(x) \to \infty$ as $r \to 0$ uniformly in $x$. Fix $l > 0$ for
which $\omu(Q_l) > 0$. 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 Proposition
2 and using (16) one can show that for any $x \in \chi^{-1}(Q_l)$ and
any $r > 0$ sufficiently small,
$$
\om(B(x, r)) \geq K r^{\os + \alpha}
$$ where $K = K(l) > 0$ is a constant. This implies that ${\overline
d}_{\om}(x) \geq \overline s + \alpha$ and hence, ${\overline
d}_{\om}(x) \geq \overline s$ for any $x \in \chi^{-1}(Q_l)$. This
completes the proof of statement (1) of Theorem 2.
\medskip
We now show that $\dBFU \leq \os$. The following arguments, essentially due to Bowen, are a slight modification of arguments shown to us by F. Ledrappier.
\medskip
\demo{\bf Proof} It is sufficient to prove that $P(\dBFU \log \ol_{x_0}) \geq 0$, since the map $t \to P(t \log \ol_{x_0})$ is a decreasing function \cite{Bo1}.
Given $\delta > 0$, it follows from the definition of $\dBFU$ that there exists $\epsilon > 0$ such that $N_{\epsilon}(F) \geq \epsilon^{\delta - \dBFU}$. Let $\{\Delta^{(j)} \} = \{ \Delta_{i_1, \cdots, i_{n(x_j)}} \},\, j=1, \cdots, N^{\epsilon}(F)$ denote the covering by basic sets constructed in Proposition 2. 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}{\ol_{\max}} )} $ and $C_2 = \frac{1}{\log(\frac{1}{\ol_{\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 $\phi(x) = (\dBFU - 2 \delta) \log \ll_{x_0}$. Then
$$
(S_n \phi)(x) = \sum_{k=0}^{n-1} \phi(\sigma^kx) = (\dBFU - 2 \delta) \log \prod_{k=0}^{n-1} \ol_{x_k}
$$
and hence $\exp(S_n \phi)(x) = (\prod_{k=0}^{n-1}
\ol_{x_k})^{\dBFU - 2 \delta}$.
It follows that
$$
P_{\alpha}( \dBFU - 2 \delta) \log \ol_{x_0}) \overset \text{def} \to \equiv \frac{1}{\alpha}
\log \sum_{ \underset \text{admissible} \to {(i_0 \cdots i_{\alpha - 1})}} \inf_{x \in \Delta_{(i_0 \cdots i_{\alpha -1)}}} (\prod_{k=0}^{n-1} \ol_{x_k})^{\dBFU - 2 \delta}
$$
$$
\geq \frac{1}{\alpha} \log \sum_{\underset \text{in covering } \{\Delta^{(j)} \} \to {(i_0 \cdots i_{\alpha - 1})}} (\frac{A}{\epsilon})^{\dBFU - 2\delta}
$$
$$\geq A^{\dBFU - 2 \delta} \frac{1}{\alpha} \log(\epsilon^{\dBFU - 2 \delta} \cdot \epsilon^{2 \delta - \dBFU}) \geq 0.
$$
Hence $P((\dBFU - 2 \delta ) \log \ol_{x_0}) = \lim_{\alpha \to \infty} P_{\alpha}((\dBFU - 2 \delta ) \log \ol_{x_0}) \geq 0$. \quad \qed
\enddemo
\comment
Given $r>0$, let $\Delta^{(j)}=\Delta_{i_1 \cdots
i_{n(x_j)}}, j=1, \cdots, N_r$ be the covering of $F$ constructed in thge proof of Proposition * MARKOV CASE. It follows from Lemma 1 in Section 5.6 that $N(r) \ge \test{const} N_r$, where $_r$ denotes the smallest number of balls of radius $r$ required to cover $F$. This implies that
$$
\dBFU \le -\limsup_{r \to \infty} \frac{\log N_r}{\log r }.
$$
Since $P( \os \log \lam_{i_1}}) =0)$, the definiton of Pressure (see Appendix 3) implies that
$$
0 = \lim_{n \rightarrow \infty} \frac1n \log \sum \Sb (i_1 \cdots i_n)
\ \text{admissible} \endSb \prod_{j=1}^n \olij^{\os}.
$$
\medskip
This completes the proof of statement (1) of Theorem 1. Statement (2) of Theorem 2 follows immediately. Statement (2) of Theorem 1 was proved for Markov geometric constructions in Proposition * (MARKOV). The proof in the general case follows from Proposition 5.
\quad \qed
\endcomment
\medskip
\proclaim{\bf Proof of Proposition 3 } \endproclaim
Let $\overline \mu$ denote an equilibrium measure for $\overline s \log
\overline \lambda_{i_1}$. It immediately follows from the variational
principle that
$$
\overline s = \frac{h_{\overline \mu}(\sigma \, | \,
Q)}{- \int \log \overline \lambda_{i_1} d \overline \mu} \leq \frac{
h(\sigma \, | \, Q)}{-\log \max \overline \lambda_k }.
$$
The case of equality is obvious.
If $\ll_i = \ol_i = \lambda$ for $i=1, \cdots, p$, then this immediately implies that $\overline s = \frac{h(\sigma \, | \,
Q) }{ - \log \lambda}$. This proves statements (2) and (3). \quad \qed
\medskip
\proclaim{\bf Proof of Theorem 4} \endproclaim
{\bf 1)} The proof combines arguments in the proofs of Theorems 1, 2 and Proposition 2. The weak asymptotic condition 2a) implies that for any $\epsilon >
0$ and $\lmu-$almost every $\omega \in Q$ 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
$$
\prod_{j=1}^n \ll_{i_j}\exp(- \epsilon n) \leq \prod_{j=1}^n \ll_{i_j,
j} \leq \prod_{j=1}^n \ll_{i_j} \exp(\epsilon n). \quad \tag 18 $$ It
is sufficient to consider only the case when $\lmu$ is ergodic with
respect to $\sigma$. Then for $\lmu-$almost every $\omega=(i_1 i_2
\cdots) \in Q$, the following limit exists:
$$
\lim_{n \to \infty}
\sum_{j=1}^n \log \ll_{i_j} = \int_Q \log \ll_{\omega_1}d \lmu(\omega)
\equiv a < 0.
$$
This implies that for any $\epsilon > 0$ and
$\lmu-$almost every $\omega \in Q$ there exists
$N_4(\omega)=N_4(\omega, \epsilon)$ such that for any $n \geq
N_4(\omega)$,
$$ \left |\frac1n \sum_{j=1}^n \log \ll_{i_j} - a \right
| \leq \epsilon. \quad \tag 19
$$
Given $l > 0$ denote
$$
Q_l =
\{\omega \in Q: N_i(\omega) \leq l, i=1,2, 3, 4 \}
$$
where $N_1(\omega), N_2(\omega)$ are the two functions constructed in the proof of Theorems 1, 2 (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 $\lmu(Q_l) > 0$ 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
$\underline \lambda_{i_1} \underline \lambda_{i_2} \cdots \underline
\lambda_{i_{n(\omega)}} > r$ and $\underline \lambda_{i_1} \underline
\lambda_{i_2} \cdots \underline \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 $n(\omega) \geq
\max\{N_3(\omega, \epsilon), N_4(\omega, \epsilon) \}$. Applying (19)
to $n=n(\omega)$ we have that $$ n(\omega) \leq \log r(a + \epsilon).
\quad \tag 20 $$ 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 that $$
l^{-1} \prod_{k=1}^{n(\omega_j) } \ll_{i_k}^{\ls} \leq \lmu(C^{(j)})
\leq l \prod_{k=1}^{n(\omega_j)} \ll_{i_k}^{\ls}. $$ 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 $r$. By virtue of (19) and (20), $$ N(x,
r) \leq \frac{ \text{vol}(B(x, r) \cap \chi^{-1}(Q_l) )}{\min_{1 \leq
j\leq N} \text{vol}(\chi^{-1}(C^{(j)})} \leq \frac{\text{vol}(B(x, r))
}{\min_{1 \leq j \leq N} \text{vol}(D(C_1 \prod_{k=1}^{n(\omega_j)}
\ll_{i_k, k} ) } $$ $$ \leq \frac{L_1 r^d }{\min_{1 \leq j \leq N} L_2
( \prod_{k=1}^{n(\omega_j)} \ll_{i_k, k} )^d } \leq \frac{L_1 r^d
}{\min_{1 \leq j \leq N} L_3 ( \prod_{k=1}^{n(\omega_j)} \ll_{i_k} )^d
\exp(- \epsilon d n(\omega_j)) } $$ $$ \leq \frac{L_1 r^d }{ L_4 r^d
r^{-\frac{\epsilon d}{a + \epsilon} }} \leq L_5 r^{\frac{\epsilon d}{a
+ \epsilon} } $$ where $L_i, i=1, \cdots, 5$ are positive constants and
$\text{vol}$ denotes the Euclidean volume in $\Bbb R^d$. Applying (16)
to the measure $\lmu$ we have that $$ \lmu(B(x, r) \cap \chi^{-1}(Q))
\leq \sum_{j=1}^{N(x,r)} \lmu(C^{(j)}) $$ $$ \leq \sum_{j=1}^{N(x,r)}
l \prod_{k=1}^{n(\omega_j)} \ll_{i_k}^{\ls- \alpha} \leq l N(x, r)
r^{\ls - \alpha} \leq L_6 r^{\gamma}, $$ where $L_6 > 0$ is a constant
and $\gamma = \ls - \alpha + \frac{\epsilon d}{a + \epsilon}$. This
implies that the lower pointwise dimension at $x$ calculated with
respect to the set $Q_l$ is not less than $\gamma$. Since this is true
for all $l > 0$ we have that ${\underline d}_{\lm}(x) \geq \ls - \alpha
+ \frac{\epsilon d}{a + \epsilon}$. Since $\epsilon$ can be chosen
arbitrarily small, $\alpha$ is arbitrarily small and hence
${\underline d}_{\lm}(x) \geq \ls$. Therefore $\dHF \geq \ls$ (see
Appendix 1). \medskip
{\bf 2)} \quad Using the weak asymptotic condition 2b) and repeating
the above arguments applied to the numbers $\ol_i, \ol_{i, n}$ instead
of $\ll_i, \ll_{i, n}$ one can show that they satisfy (18) and (19).
This fact and (16) applied to the measure $\om$ produce the sequence of
sets $Q_l, l > 0$ such that $Q_l \subset Q_{l+1}$ and $Q=
\cup_{l=1}^{\infty}Q_l \, \mod(0)$. Let us fix $Q_l$ with $\mu(Q_l) >
0$. Consider $ x \in F$ with $\chi(x) = \omega = (i_1 i_2 \cdots) \in
Q_l$. Choose the unique $n(\omega)$ such that $\overline \lambda_{i_1}
\overline \lambda_{i_2} \cdots \overline \lambda_{i_{n(\omega)}} > r$
and $\overline \lambda_{i_1} \overline \lambda_{i_2} \cdots \overline
\lambda_{i_{n(\omega)+1}} \leq r $. It follows from (19) that
$n(\omega)$ satisfies (20). By virtue of (18) and (20) we have that
$$ \diam (\Delta_{i_1 \cdots i_{n(\omega)}}) \leq C_2
\prod_{k=1}^{n(\omega)} \ol_{i_k,k} \leq C_2 \prod_{k=1}^{n(\omega)}
\ol_{i_k} \exp(\epsilon n) \leq L_7 r r^{\frac{\epsilon}{a + \epsilon}}
\leq L_7 r^{\beta} $$ where $L_7 > 0$ is a constant and $\beta= 1 +
\frac{\epsilon}{a + \epsilon}, \, 0 < \beta < 1$. This means that $$
\Delta_{i_1 \cdots i_{n(\omega)}} \subset B(x, 2 L_7 r^{\beta}). $$
Therefore, applying (16) to the measure $\om$ we have that
$$
\om( B(x,2 L_7 r^{\beta})) \geq \om( \Delta_{i_1 \cdots i_{n(\omega)}})
\geq \prod_{j=1}^{n(\omega)} \ll_{i_j}^{\os + \alpha} \geq L_8
r^{\os + \alpha}
$$
where $L_8 > 0$ is a constant. This leads to $$
{\overline d}_{\om}(x) = \limsup_{r \to 0} \frac{\log \om(B(x,
r))}{\log r} \leq \frac{\os + \alpha}{\beta}. $$ Since $\epsilon$ can
be taken arbitrarily small, $\alpha$ becomes arbitrarily small, and
$\beta$ arbitrarily close to 1. We thus obtain that ${\overline
d}_{\om}(x) \leq \os.$ \medskip
{\bf 3)} The proof follows from {\bf 2}). \quad \qed
\medskip
\demo{\bf Proof of Theorem 5}
We first prove part 1a). 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 $x \in X$.
\enddemo
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
$\{ 1 \leq k \leq N, \, i_k = j \} = \{ 1 \leq k \leq N, \, \sigma^k
\omega \in C_j\}$. Part 1a) follows immediately. The proof of 1b)
follows analogously. The proof of 2) now follows immediately from
Theorem 5a. \quad \qed \enddemo
\bigskip
\demo{\bf Proof of Proposition 4 } \enddemo
We define a homeomorphism $\pi_1: \overline F_{sym} \to F$ by setting $\pi_1 (x)=y$ where $x \in \overline F_{sym}, \, y \in F$ and $ \chi (x)= \chi (y)= (i_1,
i_2, \cdots)$. We shall show that this map is Lipschitz. Suppose $x, y \in \overline F_{sym}$. Then there exists $N \in \Bbb N$ such that $x$ and $y$ belong to the same basic set $\{\Delta_{i_1 \cdots i_N} \}$ but $x$ and $y$ do not belong to the same basic set at step $N+1$ of the construction. Hence $d(x, y) \geq \frac{1}{\max_{1 \leq i \leq p} \ol_{i}} \diam \{\Delta_{i_1 \cdots i_N} \}$. By construction, the diameter of a basic set for $\overline F_{sym}$ is greater than or equal to the diameter of the corresponding basic set for $F$. Hence
$$
d(\pi_1(x), \pi_1(y)) \leq \diam(\pi_1(\Delta_{i_1 \cdots i_N} )) \leq \diam(\Delta_{i_1 \cdots i_N}) \leq \frac{1}{\max_{1 \leq i \leq p} \ol_{i}} d(x, y).
$$
It follows that the map $\pi_1$ is Lipschitz with the Lipschitz constant given by $ \frac{1}{\max_{1 \leq i \leq p} \ol_{i}}$. This implies statement (1), see \cite {F1}. The second statement can be proven in the similar way. The third statement follows immediately from the first two,
the positivity and finiteness of the Hausdorff measure for similarity constructions are proven in \cite {F2}.
\medskip
\demo{\bf Proof of Theorem 6}
\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 that $$ \frac{\log \nu(B(x, r))}{\log r } \leq
\frac{\log \nu(\Delta_{n_r(x)}(x))}{\log r }. $$ 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 proceeding
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 $|U| = \sup\{|x-y|:x,y \in U\}$. 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 $$ \text{m}_{\text{H}}(s, Z) =\lim_{\delta \ra 0}
\inf_{\{U_i\}} \left\{ \sum_i |U_i|^{s}: \{U_i\} \text{ is a }
\delta\text{-cover of } Z \right\}. $$
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
upper and lower 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: Three Methods of Obtaining Lower
Bounds for $ \text{\bf dim}_{\bold H} \bold F $}
\proclaim{\bf 1. 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 $\mu(U) \leq c |U|^s$ for all sets $U$ with $|U|
\leq \delta$. Then $\text{m}_{\text{H}}(s,Z) \geq \mu(F)/c$ and $s \leq
\text{dim}_H Z$.
\endproclaim
\medskip
\proclaim{\bf 2. Potential
Principle\, \cite{Fr} } \endproclaim For $s \geq 0$ the {\it
s-potential} at a point $x \in \BbbR^n$ with respect to the measure
$\mu$ is defined by $\phi_s(x) =
\int_{\BbbR^n}\frac{d\mu(y)}{|x-y|^s}$. The {\it s-energy of $\mu$}
is $I_s(\mu) = \int_{\BbbR^n}\phi_s(x) d\mu(x) = \int_{\BbbR^n}
\int_{\BbbR^n} \frac{d\mu(x) d\mu(y)}{|x-y|^s}$.
\proclaim{\bf Proposition 1} Let $Z \subset \BbbR^n$.
\roster
\item If there is a
measure $\mu$ on $Z$ with $I_s(\mu) < \infty$, then
$\text{m}_{\text{H}}(s,Z) \allowmathbreak = \infty$ and
$\text{dim}_H Z \geq s$. \item $ \text{dim}_H Z = \sup\{t \, | \,
\text{there exists } \mu \text{ such that } \mu(Z) > 0, \, I_t(\mu)< \infty\}$. \endroster
\endproclaim
\medskip
\proclaim{\bf 3. 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 mass distribution principle implies the
potential method that, in turn, implies the nonuniform mass distribution
principle. In fact, $d_{\mu}^L(x) \geq d \Rightarrow \forall \alpha >
0, \enspace \mu(B(x,r)) \leq C( x, \alpha) r^{d- \alpha}$.
\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 $0s$ and $1s$,
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 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 $$ P(\phi) =
\lim_{n \rightarrow \infty} \frac1n \log \sum \Sb (i_1 \cdots i_n)
\ \text{admissible} \endSb \inf_{x \in \Delta_{(i_1 \cdots i_n)}} \exp (S_n
\phi(x)), $$
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 role 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 systems is
expansive, the supremum in the variational principle is attained by
some measure \cite{W}. This measure need not be unique.
%ok up to here
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 2 \cite{PP}} Let $ (\Sigma_A^+, \sigma) $ be mixing. 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 3 \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
$$ \text{m}^h_{\text{H}}(Z) =\lim_{\delta \ra 0} \inf_{\{U_i\}}
\left\{ \sum_i h(|U_i|): \{\delta\text{-cover of } Z \} \right \}. $$
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
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\enddocument