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\topmatter
\title
DECAY OF CORRELATIONS FOR PIECEWISE EXPANDING MAPS
\endtitle
\author
Carlangelo Liverani
\endauthor
\affil University of Rome {\sl Tor Vergata}
\endaffil
\address
Liverani Carlangelo,
Mathematics Department,
University of Rome II, Tor Vergata,
00133 Rome, Italy.
\endaddress
\email
liverani\@mat.utovrm.it
\endemail
\abstract
This paper investigates the decay of correlations in
a large class of non-Markov one-dimensional expanding maps.
The method employed is a special version of
a general approach recently proposed
by the author. Explicit bounds on the rate of decay of correlations are
obtained.
\endabstract
\thanks
\bf I would like to thank M. Blank, C. Falcolini and
S. Vaienti for helpful discussions. In addition, it is
a pleasure to acknowledge the hospitality of the Centre de Physique
Th\'eorique of Marseille where part of this work was done.
\endthanks
\endtopmatter
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\centerline{\bf CONTENT}
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\roster
\item"0." Introduction\dotfill p. 2
\item"1." Operators and Invariant Cones\dotfill p. 3
\item"2." The Map and an invariant cone
\dotfill p. 5
\item"3." Decay of correlations \dotfill p. 7
\item"4." General results concerning expanding maps\dotfill
p. 11
\item"" Appendix (An example)\dotfill p. 14
\item"" References\dotfill p. 15
\endroster
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\vskip1cm
\subhead \S 0 INTRODUCTION
\endsubhead
\vskip1cm
In this paper we apply a new technique--introduced in \cite{L} to
study the decay of correlations in hyperbolic systems--to expanding
one-dimensional non-Markov maps (see section 2 for a precise definition).
A main feature of such a method is the possibility of
obtaining explicit bounds on the rate of decay.
The standard approach to
this problem \footnote{ See \cite {LY} where it is proven the existence of
an invariant measure absolutely continuous with respect to Lebesgue and
\cite{HK} where the decay of correlations is investigated.} is to study the
Perron--Frobenius (P--F) operator via Ionescu-Tulcea and Marinescu type
spectral theorems \cite{ITM}.
Unfortunately, this suffices to prove exponential decay of
correlations but does not provide any constructive bound on the rate of
decay. One can hope to obtain some bounds using the theory of Ruelle zeta
functions \cite{Ru}, but it is not clear in which generality this can be
accomplished.
Here, we will study the P-F operator as well, but from a different point of
view:
we will see that there exists a convex cone of functions that is mapped
``strictly" inside itself by the P-F operator. We will then be able to
take advantage of an idea by Garrett Birkhoff: he showed that one can
associate an Hilbert metric to the above mentioned cone and that such a
metric is contracted by the P-F operator. Once we obtain a contraction all
the wanted consequences are the result of straightforward arguments.
The possibility to obtain bounds on the rate of decay was recently
addressed also by P. Collet \cite{Co}. He used techniques developed
for the study
of markov chains \cite{DS}. Here we show that the approach proposed in
\cite{L} is more direct, produces substantially better bounds for the class
of maps studied in \cite{Co}, and we apply it to a larger class of maps.
If $T$ is a map of the type under consideration (see \S 2) and $\mu$ is
the associated invariant probability measure, absolutely continuous with
respect to the Lebesgue measure (i.e., there exists $\phi\in L^1([0,\,1])$
such that $d\mu=\phi dx$), then
I prove the following:
\proclaim{Theorem 0.1}If $\inf\limits_{x\in[0,\,1]}\phi(x)\geq\gamma>0$ and
the system
$(T,\,\mu)$ is mixing, then there exists $b,\,K>0$, $\Lambda\in (0,\,1)$
such that, for each function $f\in L^1([0,\,1])$
and $g\in BV$ (the space of function of bounded variation), $\int_0^1 g=1$,
$$
\left|\int_0^1g f\circ T^n-\int_0^1f d\mu\right|
\leq K\Lambda^{-n}\|f\|_1\left(1+b\bigvee_0^1g\right).
$$
\endproclaim
In addition, it is possible to state a concrete condition that insures
$\inf\phi\geq\gamma>0$ (see Lemma 4.2), and, most of all,
I will obtain explicit
estimates for the constants $K,\,b,\,\Lambda$ (although they may not be
optimal).
The paper is organized as follows: in section 1 I provide a brief review of
some basic facts on which the subsequent arguments rest. Section 2 contains
a detailed description of the class of maps under consideration,
and it
presents the main ingredient in the proof of Theorem 0.1: a cone of functions
invariant under the
action of the Perron-Frobenius operator. In section 3 the bound on the
decay of correlations is proven, but under apparently stronger hypotheses
than the ones in the statement of Theorem 0.1. In section 4 I show that
the hypotheses of Theorem 0.1 imply the assumptions used in section 3,
whereby concluding the argument. As a byproduct, I also answer to some
questions, concerning general properties of expanding maps, recently raised
by P. Collet \cite{Co}.
Finally, the paper includes an appendix where I give numerical bounds, for
all the constants involved, in a concrete example; whereby
emphasizing the constructive nature of the present approach.
\vskip1cm
\subhead \S 1 OPERATORS AND INVARIANT CONVEX CONES
\endsubhead
\vskip1cm
This section illustrates some results in lattice
theory originally due to
Garrett Birkhoff \cite{B2}. More details and the proofs of the following
results can be found in \cite{L}.
\par
Consider a topological vector space $\Bbb V$ with a partial
ordering ``$\preceq$," that is a vector lattice.\footnote{We are assuming the
partial order to be well behaved with respect to the algebraic structure:
for each $f,\,g\in\Bbb V$ $f\succeq g\Longleftrightarrow f-g\succeq 0$;
for each $f\in\Bbb V$, $\lambda\in\Bbb R^+\backslash \{0\}$
$f\succeq 0 \Longrightarrow
\lambda f\succeq 0$; for each $f\in \Bbb V$ $f\succeq 0$ and $f\preceq 0$ imply
$f=0$ (antisymmetry of the order relation).}
We require the partial order to be ``continuous,"
i.e. given $\{f_n\}\in \Bbb V$ $\lim\limits_{n\to \infty}f_n=f$, if
$f_n\succeq g$
for each $n$, then $f\succeq g$. We call such vector lattices ``integrally
closed." \footnote{To be precise, in the literature ``integrally closed"
is used in a weaker sense. First, $\Bbb V$ does not need a topology. Second,
it suffices that for $\{\alpha_n\}\in\Bbb R$, $\alpha_n\to\alpha$;
$f,\,g\in\Bbb V$, if $\alpha_n f\succeq g$, then $\alpha f\succeq g$. Here we
will ignore these and other
subtleties: our task is limited to a brief account of
the results relevant to the present context.}
\par
We define the closed convex cone \footnote{Here, by ``cone," we mean any
set such that, if $f$ belongs to the set, then $\lambda f$ belongs to it
as well, for each $\lambda>0$.}
$\Cal C=\{f\in\Bbb V\;|\;f\neq 0,\,f\succeq 0\}$
(hereafter, the term ``closed cone" $\Cal C$ will mean that $\Cal C\cup
\{0\}$ is closed), and the equivalence relation ``$\sim$":
$f\sim g$ iff there exists $\lambda\in\Bbb R^+\backslash\{0\}$
such that $f=\lambda g$. If we call $\widetilde{\Cal C}$ the quotient of
$\Cal C$ with respect to $\sim$, then $\widetilde{\Cal C}$ is a closed
convex set. Conversely, given a closed convex cone
$\Cal C\subset\Bbb V$, enjoying the property $\Cal C\cap -\Cal C=\emptyset$,
we can define an order relation by
$$
f\preceq g \iff g-f\in\Cal C \cup \{0\}.
$$
Henceforth, each time that we specify a convex cone we will assume the
corresponding order relation and vice versa.
\par
It is then possible to define a projective metric $\Theta$
(Hilbert metric),\footnote{In fact, we define a semi--metric, since
$f\sim g\Rightarrow \Theta(f,\,g)=0$. The metric that we describe
corresponds to the conventional Hilbert metric on $\wt{\Cal C}$.}
in $\Cal C$, by the construction:
$$
\aligned
\alpha (f,\,g)=&\sup\{\lambda\in\Bbb R^+\;|\; \lambda f\preceq g\}\\
\beta(f,\,g)=&\inf\{\mu\in\Bbb R^+\;|\;g\preceq \mu f\}\\
\Theta(f,\,g)=&\log\left[\frac{\beta(f,\,g)}{\alpha(f,\,g)}\right]
\endaligned
$$
where we take $\alpha=0$ and $\beta=\infty$ if the corresponding sets are
empty.
\par
The importance of the previous
constructions is due, in our context, to the following theorem.
\proclaim{Theorem 1.1} Let $\Bbb V_1$, and $\Bbb V_2$
be two integrally closed vector lattices;
$T:\Bbb V_1\to \Bbb V_2$ a linear map such that
$T(\Cal C_1)\subset \Cal C_2$, for the two corresponding closed convex
cones $\Cal C_1\subset\Bbb V_1$ and $\Cal C_2\subset\Bbb V_2$.
Let $\Theta_i$ be the Hilbert metric corresponding to the cone
$\Cal C_i$. Setting
$\Delta=\sup\limits_{f,\,g\in T(\Cal C_1)}\Theta_2(f,\,g)$
we have
$$
\Theta_2(Tf,\,Tg)
\leq\tanh\left(\frac \Delta 4\right)\Theta_1(f,\,g)
\qquad \forall f,\,g\in\Cal C_1
$$
($\tanh(\infty)=1$).
\endproclaim
\proclaim{Remark 1.2}If $T(\Cal C_1)\subset\Cal C_2$ then it follows that
$\Theta_2(Tf,\,Tg)\leq\Theta_1(f,\,g)$. However, a uniform rate of
contraction depends on the diameter of the image being finite.
\endproclaim
\par
In particular, if an operator maps a convex cone strictly inside itself
(in the sense that the diameter of the image is finite),
then it is a contraction in the Hilbert metric. This implies the
existence of a ``positive" eigenfunction (provided the cone
is complete with respect to the Hilbert metric), and,
with some additional work,
the existence of a gap in the spectrum of $T$ (see \cite{B1} for
details).
The relevance of this theorem for the study of
invariant measures and their ergodic properties is obvious.
\par
It is natural to wonder about the strength of the Hilbert metric compared
to other, more usual, metrics. While, in general, the answer
depends on the cone, it is nevertheless possible to state an interesting
result.
\proclaim{Lemma 1.3} Let $\|\cdot\|$ be a norm on the vector lattice
$\Bbb V$, and
suppose that, for each $f,\,g\in\Bbb V$,
$$
-f\preceq g\preceq f \Longrightarrow \|f\|\geq\|g\|.
$$
Then, given $f,\,g\in\Cal C\subset\Bbb V$ for which $\|f\|=\|g\|$,
$$
\|f-g\|\leq\left(\e^{\Theta(f,\,g)}-1\right)\|f\| .
$$
\endproclaim
\vskip1cm
\subhead \S 2 THE MAP AND AN INVARIANT CONE
\endsubhead
\vskip1cm
We consider a map $T$ from the interval $[0,\,1]$ into itself.
We assume that there exists a partition
(mod--0, with respect to the Lebesgue measure)
$\Cal A_0$ of $[0,\,1]$ into open intervals
such that, for each interval $I\in\Cal A_0$, the map $T$, restricted to
$I$,
can be extended to a $C^2$ map on an open interval containing the closure of
$I$. In addition, we assume that $|DT|\geq \lambda>1$ (expansivity).
The simplest examples, in this class of maps, occur when,
for each $I\in\Cal A_0$, $TI$ is
equal, mod-0, to the union of elements of $\Cal A_0$. This case is called
``Markov case," and it is well understood in the literature; in
particular, explicit bounds on the rate of decay of correlations are
available \cite{F}, \cite{H}, \cite{L}. If the above mentioned property
fails, the map is called ``non--Markov," this is the case addressed here.
Thanks to the work of Lasota and Yorke \cite{LY}
it is known that any piecewise smooth expanding map
has at least one invariant measure absolutely continuous with respect to
Lebesgue.\footnote{See \cite{BK} for a generalization of such a
result.} We will call $\phi$ the density, with respect to the Lebesgue
measure, of such an invariant measure $\mu$, and
assume that the dynamical systems $(T,\, \mu)$ is mixing.
The results of \cite{HK} and \cite{BK}
imply that the class of maps under consideration
exhibits exponential decay of correlations. But, no explicit bound
on the rate of decay is available in such a generality.
A first technical obstacle is that the set
$\{x\in[0,\,1]\;|\;\phi(x)=0\}$ may have positive Lebesgue measure.
This is a concrete possibility and it is very easy to construct examples
with this property. The simplest case of this behavior occurs when there exists
an attracting set; clearly such a
set supports all the invariant measures in the class under
consideration (i.e., measures that are
absolutely continuous with respect to the Lebesgue measure).
We will assume that this is not the case. In principle, such an assumption
does not imply any loss of generality:
we can alway study the restriction of the map
to an invariant set.\footnote{In fact, there is nothing sacred
about the interval $[0,\,1]$.} Yet, some more complex situations may be
possible, we are therefore forced to further reduce the class of maps under
consideration.
On the one hand, we assume that $|DT|\geq\lambda>2$
(here no loss of generality is
implied: this condition can always be satisfied replacing $T$ by an
appropriate power). On the other hand we ask that there exists $\gamma>0$
such that $\inf\limits_{x\in[0,\,1]}\phi(x)\geq\gamma$.
To study the decay of correlations it is useful to introduce the
Perron-Frobenius (P-F) operator $\wt T$. Such operator is defined by the
relation\footnote{If not otherwise stated, all the integrals are between 0
and 1, and taken with respect to the Lebesgue measure.}
$$
\int g f\circ T=\int f\wt T g
$$
for each $f\in L^1([0,\,1])$ and $g\in L^\infty([0,\,1])$.
A direct computation shows that
$$
\wt T g(x)=\sum_{y\in T^{-1}(x)}g(y)|D_yT|^{-1}.
$$
We will show that the P-F operator leaves invariant a cone of functions; the
decay of the correlations will then follow from the theory discussed in \S 1.
Let $BV$ be the space of functions of bounded variation on $[0,\,1]$.
The cones that we will use in this paper are:
$$
\Cal C_a=\left\{g\in BV \;\bigg|\;g(x)\geq 0\;\forall x\in[0,\,1];
\;\bigvee^1_0 g\leq a\int^1_0 g\right\},
$$
for $a>0$.\footnote{By $\bigvee\limits_0^1 g$ we mean the variation of the
function $g$ in the interval $[0,\,1]$.}
The main ingredient for studying the maps under consideration is the
following inequality (due to Lasota and Yorke) \cite {LY}: for each
$g\in BV$
$$
\bigvee_0^1\wt T g\leq 2\lambda^{-1}\bigvee_0^1 g+ A\int_0^1 |g|
\tag 2.1
$$
where $A=\lambda^{-2}\| D^2T\|_\infty+2b_0^{-1}\lambda^{-1}$;
$b_0=\inf\limits_{I\in\Cal A_0}|I|$.\footnote{The
alert reader has certainly noticed that we have defined $\wt T$ only on
$L^\infty([0,\,1])$, to define it on $BV$ it is necessary to specify the
value of $\wt T g$ at all points, that is also at the boundaries of the
intervals belonging to the partition $\Cal A_0$.
In fact, we want to define all the
powers of $\wt T$ as well, so we may be in trouble at countably many points
(all the preimages of the boundaries of the partition).
We will not worry about such points since they can be consistently ignored
(loosely speaking, the value of $\wt Tg$ at such points can be defined by
taking left and/or right limits--see \cite{HK} or \cite{BK} for details).}
The relevance of the
cones $\Cal C_a$ and the inequality (2.1) are exemplified by the following
result.
\proclaim{Lemma 2.1}
For each $a> \frac A{1-2\lambda^{-1}}$, there exists $\sigma<1$ such that
$$
\wt T\Cal C_a\subset\Cal C_{\sigma a}.
$$
\endproclaim
\demo{Proof}
For each $g\in\Cal C_a$, inequality (2.1) yields
$$
\bigvee_0^1 \wt Tg\leq 2\lambda^{-1}\bigvee_0^1 g+ A\int_0^1 g
\leq (2\lambda^{-1}a+A)\int_0^1 g.
$$
The result follows by choosing $a=A(\sigma-2\lambda^{-1})^{-1}$, and
noticing that $\int g=\int \wt Tg$.
\enddemo
\vskip1cm
\subhead \S 3 DECAY OF CORRELATIONS
\endsubhead
\vskip1cm
In the previous section we have found a cone of functions that is left
invariant by the P-F operator. This it is not quite enough
to obtain a contraction in the corresponding Hilber metric: the diameter of
the image must also be investigated. The nature of the problem is elucidated by
the following Lemma.
\proclaim{Lemma 3.1} Calling $\Theta_a$ the Hilbert metric associated to the
cone $\Cal C_a$, for each $\nu<1$, and $g\in\Cal C_{\nu a}$
$$
\Theta_a(g,\,1)\leq \ln\left[\frac{\max\left\{(1+\nu)\int_0^1 g;\;
\sup\limits_{x\in[0,\,1]} g(x)\right\}}
{\min\left\{(1-\nu)\int_0^1g;\;\inf\limits_{x\in[0,\,1]} g(x)\right\}}\right].
$$
\endproclaim
\demo{Proof}
We have to find the set of $\lambda$ and $\mu$ such that
$\lambda 1\preceq g\preceq \mu 1$. Let us start with the first inequality; it
is satisfied iff $g-\lambda\in\Cal C_a$, i.e.
$$
\aligned
\lambda &\leq g(x) \qquad\forall x\in[0,\,1]\\
&\text{and}\\
\lambda &\leq \int_0^1 g-a^{-1}\bigvee_0^1 g .
\endaligned
$$
Consequently,
$$
\alpha=\sup\lambda=\min\left\{\inf_{x\in[0,\,1]} g(x);\,
\int_0^1 g-a^{-1}\bigvee_0^1 g\right\},
$$
see \S 1 for a definition of $\alpha$ and $\beta$ in the present context.
Since $g\in\Cal C_{\nu a}$, it follows
$$
\int_0^1 g-a^{-1}\bigvee_0^1 g\geq (1-\nu)\int_0^1 g ,
$$
that is $\alpha\geq \min\{(1-\nu)\int g,\,\inf g\}$.
Analogously, one can compute $\beta\leq \max\{(1+\nu)\int g,\,\sup g\}$.
\enddemo
Note that, up to now, we do not have any control on the inf of
a function belonging to our cones, therefore the above Lemma shows not
only that more work is
needed, but also in which direction to concentrate our efforts.
The first step is to notice that, if a function belongs to the cone $\Cal
C_a$, then it cannot be small too often. This is made precise by the
following.
\proclaim{Lemma 3.2} Given a partition, mod-0, $\Cal P$ of $[0,\,1]$, if each
$p\in\Cal P$ is a connected interval with Lebesgue measure less than
$\frac 1{2a}$, then, for each $g\in\Cal C_a$, there
exists $p_0\in\Cal P$ such that
$$
g(x)\geq \frac 12 \int_0^1 g \quad \forall x\in p_0.
$$
\endproclaim
\demo{Proof}
Consider the set
$$
\Cal P_-=\left\{p\in\Cal P\;\bigg|\;\exists x_p\in p\,:\, g(x_p)<\frac
12\int_0^1 g\right\}.
$$
Clearly the Lemma is proven if we show that $\Cal P_-\neq\Cal P$.
Let us suppose the contrary. For
$$
\int_p g\leq m(p) \left(g(x_p)+\bigvee_p g\right)<
\frac {m(p)}2\int_0^1 g+\frac 1{2a}\bigvee_p g ,
$$
and remembering that $g\in\Cal C_a$,
we have
$$
\int_0^1 g< \frac 12\int_0^1 g+ \frac 1{2a}\bigvee_0^1 g\leq
\int_0^1 g
$$
which is a contradiction.
\enddemo
To continue, we define a particular class of partitions:
\proclaim{Definition 3.3} For each $n\in\Bbb N$,
$$
\Cal A_n=\bigvee_{j=0}^{n} T^{-j}\Cal A_0.
$$
\endproclaim
It is immediately clear that $T^{n+1}$ is monotone and regular on each
element of the partition $\Cal A_n$ (in fact, this could be used as an
alternative definition of $\Cal A_n$); moreover, $\Cal A_n$ consists of
intervals with Lebesgue measure smaller than $\lambda^{-n}$.
We can then choose $n_0$
such that all the elements of $\Cal A_{n_0}$ have
measure less than $\frac 1{2a}$ (for example $n_0=\left[\frac{\ln
2a}{\ln\lambda}\right]+1$ would do).
\proclaim{Definition 3.4} We call a map ``covering" if,
for each $n\in\Bbb N$ there exists $N(n)$ such that,
for each $I\in\Cal A_n$,\footnote{We have already remarked that what
happens at the boundaries of the partitions is immaterial. In the same
vein, each time that we write an equality between sets we always mean it
apart a finite number of points.}
$$
T^{N(n)}I=[0,\,1].
$$
\endproclaim
The above property
corresponds to condition $(H2)$ in \cite{Co}. The importance of the notion
of ``covering" is emphasized by the following Lemma.
\proclaim{Lemma 3.5} If the map $T$ is covering, then, for each $a>\frac
A{1-2\lambda^{-1}}$,
$$
\text{diam}(\wt T^{N(n_0)}\Cal C_a)\leq \Delta<\infty .
$$
\endproclaim
\demo{Proof}
Let $g\in\Cal C_a$, then, according to Lemma 3.2, there exists $I_0\in\Cal
A_{n_0}$ such that $g(x)\geq\frac 12\int_0^1 g$ for each $x\in I_0$.
By the covering property, for each $x\in [0,\,1]$ (apart, at most,
finitely many points) there exists $y\in I_0$ such that
$T^{N(n_0)}y=x$; hence
$$
\left(\wt T^{N(n_0)}g\right)(x)=\sum_{y\in T^{-N(n_0)}x}g(y)|D_yT^{N(n_0)}|^{-1}
\geq\frac {\int_0^1 g}{2\|DT\|_\infty^{N(n_0)}}.
$$
According to Lemma 2.1, $\wt T^{N(n_0)}\Cal C_a\subset\Cal C_{\sigma_1 a}$,
with\footnote{To obtain the following formula it is enough to notice that,
iterating (2.1), for each $k>0$, holds $\bigvee_0^1\wt T^k g\leq
\left(2\lambda^{-1}\right)^k\bigvee_0^1g+\frac{1-\left(2\lambda^{-1}\right)^k}
{1-2\lambda^{-1}}A\int_0^1g$; apply then Lemma 2.1 directly to $\wt
T^{N(n_0)}$.}
$$
\sigma_1=(2\lambda^{-1})^{N(n_0)}+\frac{1-(2\lambda^{-1})^{N(n_0)}}
{1-2\lambda^{-1}}Aa^{-1} .
$$
Let $\delta(g)=\frac{\inf\wt T^{N(n_0)} g}{\int g}$, up to now we have seen that
$\delta(g)\geq(2\|DT\|^{N(n_0)})^{-1}$ for all $g\in\Cal C_a$.
Using Lemma 3.1, we can then estimate
$$
\aligned
\text{diam}\left(\wt T^{N(n_0)}\Cal C_a\right)
&\leq 2\sup_{g\in\Cal C_a}\ln\left[\frac{\max\left\{(1+\sigma_1)\int g;\;
\inf \wt T^{N(n_0)}g
+\bigvee_0^1\wt T^{N(n_0)}g\right\}}{\min\left\{(1-\sigma_1)\int g;\;\inf
\wt T^{N(n_0)}g\right\}}\right]\\
&\leq 2\sup_{g\in\Cal C_a}\ln
\left[\frac{\max\left\{(1+\sigma_1);\;\delta(g)
+a\sigma_1\right\}}{\min\left\{(1-\sigma_1);\;\delta(g)\right\}}\right]\\
&\leq 2\ln
\left[\frac{\max\left\{(1+\sigma_1);\;1
+a\sigma_1\right\}}{\min\left\{(1-\sigma_1);\;(2\|DT\|^{N(n_0)})^{-1}
\right\}}\right] .
\endaligned
$$
\enddemo
The above Lemma, together with the results of \S 1, implies exponential decay
of the correlations for covering maps.
\proclaim{Theorem 3.6} If $T$ is covering, then,
for each $f\in L^1([0,\,1])$ and $g\in BV$,
$\int_0^1 g=1$,
$$
\left|\int_0^1gf\circ T-\int_0^1 fd\mu\right|\leq K_n\Lambda^{n}\|f\|_1
\left(1+b\bigvee_0^1 g\right),
$$
with $\Lambda=\tanh(\frac\Delta 4)^{\frac 1{N(n_0)}}$,
$K_n=\e^{\Lambda^{n-N(n_0)}\Delta}
\Lambda^{-N(n_0)}\Delta\|\phi\|_\infty$ and $b=(a-B)^{-1}$ ($B=\frac
A{1-2\lambda^{-1}}$).
\endproclaim
\demo{Proof}
Consider $g\in\Cal C_{a}$, $a>\frac A{1-2\lambda^{-1}}$,
normalized so that $\int_0^1g=1$ (i.e.,
$g$ can be thought as the density of a measure). Then,
$$
\left|\int_0^1 f(\wt T^n g-\phi)dx\right| \leq\|f\|_1
\left\|\frac{\wt T^ng}{\phi}-1\right\|_\infty\|\phi\|_\infty .
$$
\proclaim{Lemma 3.7}If $\Cal C_+=\{g\in BV\;|\;
g(x)\geq 0,\;\;\forall x\in [0,\,1]\}$, and $\Theta_+$ is the corresponding
Hilbert metric, then, for each $g_1,\,g_2\in\Cal C_a$,
$$
\Theta_+(g_1,\,g_2)\leq \Theta(g_1,\,g_2)
$$
\endproclaim
\demo{Proof}
Since $\Cal C_+\supset\Cal C_a$, the identity is a map from
$BV$ to itself that maps $\Cal C_a$ into $\Cal C_+$.
The result follows then from Theorem 1.1.
\enddemo
A simple computation yields
$$
\Theta_+(g_1,\,g_2)=\ln\sup_{x,\,y\in[0,\,1]}\frac{g_1(x)g_2(y)}
{g_1(y)g_2(x)}.
$$
Using the previous facts, and the trivial equality
$$
\frac{(\wt T^n g)(x)}{\phi(x)}=\frac{\wt T^n g(x)\phi(y)}
{\wt T^ng(y)\phi(x)}
\frac{\wt T^n g(y)}{\phi(y)},
$$
we have
$$
\e^{-\Theta_+(\wt T^ng,\,\phi)} \frac{\wt T^ng(y)}{\phi(y)}\leq
\frac{\wt T^ng(x)}{\phi(x)}\leq
\e^{\Theta_+(\wt T^ng,\,\phi)} \frac{\wt T^ng(y)}{\phi(y)}
$$
for each $x,\,y\in[0,\,1]$.
Because $\int_0^1(\wt T^n g-\phi)=0$, there must exist
$y_n^+,\,y_n^-\in[0,\,1]$ such that
$\wt T^n g(y_n^-)\leq\phi(y_n^-)$ and
$\wt T^n g(y_n^+)\geq\phi(y_n^+)$.
Using the previous inequalities, with $y=y_n^-$ and $y=y_n^+$, respectively,
we obtain,
for each $x\in[0,\,1]$,
$$
\e^{-\Theta_+(\wt T^ng,\,\phi)} \leq\frac{\wt T^n g(x)}{\phi(x)}\leq
\e^{\Theta_+(\wt T^ng,\,\phi)}
$$
and
$$
\left\|\frac{\wt T^n g}{\phi}-1\right\|_\infty
\leq\e^{\Theta_+(\wt T^ng,\,\phi)} -1
\leq\e^{\Theta(\wt T^ng,\,\phi)} -1 .
$$
According to Theorem 1.1 and Lemma 3.5
$$
\aligned
\Theta(\wt T^ng,\,\phi)&\leq \Theta([\wt T^{N(n_0)}]^{[\frac
n{N(n_0)}]}g,\,[\wt T^{N(n_0)}]^{[\frac
n{N(n_0)}]}\phi)\\
&\leq\tanh(\frac\Delta 4)^{[\frac n{N(n_0)}]-1} \Theta(\wt
T^{N(n_0)}g,\,\phi).
\endaligned
$$
Hence,
$$
\left\|\frac{\wt T^n g}{\phi}-1\right\|_\infty
\leq \e^{\Lambda^{n-N(n_0)}\Theta(\wt T^{N(n_0)} g,\,\phi)}-1
\leq \e^{\Lambda^{n-N(n_0)}\Delta}\Delta\Lambda^{n-N(n_0)} .
$$
This estimate shows that for each $f\in L^1([0,\,1])$, $g\in\Cal C_{a}$
$$
\aligned
&\left|\int_0^1 f\circ T^n gdx-\int_0^1f\phi dx\right|
\leq K_n\|f\|_1\Lambda^n\\
& K_n=\e^{\Lambda^{n-N(n_0)}\Delta}\Delta\Lambda^{-N(n_0)}\|\phi\|_\infty .
\endaligned
$$
\par
Let us now consider $g\in BV$, $g\geq 0$, and $\int_0^1 g=1$.
If $\bigvee_0^1 g\leq a$, we have the above estimate, otherwise
we define $g_\rho=(g+\rho \phi)(1+\rho)^{-1}$, then $\int_0^1 g_\rho=1$
and we have
$$
\bigvee_0^1 g_\rho=\left[\bigvee_0^1
g+\rho\bigvee_0^1\phi\right](1+\rho)^{-1}.
$$
Iterating (2.1) one obtains $\bigvee_0^1\phi\leq\frac
A{1-2\lambda^{-1}}=B$, then
$$
\bigvee_0^1 g_\rho\leq
\left[\bigvee_0^1
g+\rho B\right](1+\rho)^{-1}
$$
Choosing,
$$
\rho=\frac{\bigvee_0^1g-a}{a-B}
$$
we have $g_\rho\in\Cal C_{a}$; hence, the decay of correlations for $g_\rho$
implies the decay of correlations for $g$.
The result for arbitrary $g\in BV$
follows, since any function can be written as the difference
of two positive functions.
\enddemo
In the next section we will see that, provided $\phi(x)>0$, all
the mixing maps are covering, whereby completing the proof of Theorem 0.1.
\vskip1cm
\subhead \S4 GENERAL PROPERTIES OF EXPANDING MAPS
\endsubhead
\vskip1cm
In this section we address some questions concerning piecewise expanding
maps posed by P. Collet in \cite{Co}. We will see that the property that some
image of any interval covers all $[0,\,1]$, is a quite general feature of mixing
maps. This shows that the results obtained in the paper apply to a wide
class of maps.
We start by giving a checkable criterion for the hypotheses of Theorem 0.1.
\proclaim{Definition 4.1} We call a map ``weakly--covering" if there exists
$N_0\in\Bbb N$ such that, for each $I\in\Cal A_0$,
$$
\bigcup_{j=0}^{N_0} T^jI=[0,\,1].
$$
\endproclaim
This is a weaker version of $(H1)$ in \cite{Co}. The next Lemma shows that
weak--covering is all is needed to insure that the first of the hypotheses
of Theorem 0.1 holds.
\proclaim{Lemma 4.2} If a map is weakly-covering, then there exists
$\gamma>0$ such that $\inf\phi\geq\gamma$.
\endproclaim
\demo{Proof}
A consequence of weak-covering and expansivity is that
the property of being weakly-covering does not
depend substantially on the partition.
\proclaim{Sub--Lemma 4.3} If a map is weakly-covering, then, for each
$n\in\Bbb N$, there exists $N_0(n)\in\Bbb N$ such that, for each
$I\in\Cal A_n$,
$$
\bigcup_{j=0}^{N_0(n)} T^jI=[0,\,1].
$$
\endproclaim
\demo{Proof}
Let $I\in\Cal A_n$, then $T^n$ is smooth on $I$, accordingly
$|T^nI|\geq\lambda^n|I|$. If $T^nI$ covers an element of $\Cal A_0$ then the
Lemma is proven; if not, since $T^nI$ is connected, it can intersect at
most two elements of $\Cal A_0$, so it is naturally broken in, at most,
two pieces, let $I_1$ be the larger of the two, clearly
$|I_1|\geq\lambda^n|I|/2$. Let $b_n=\inf\limits_{J\in\Cal A_n}|J|$; then
$|I_1|\geq\lambda^n\frac{b_n}2$.
We can then carry
out a recursive argument: consider $TI_1$; by construction it is connected,
intersect it with the elements of $\Cal A_0$, either it will cover one
element or it will be divided in, at most, two sub-intervals, call $I_2$
the larger one; consider $TI_2$, and so on. It follows that $|I_{k+1}|\geq
\left(\frac\lambda 2\right)^k|I_1|$, which implies that, eventually, $I_k$
will cover an element of $\Cal A_0$, this is all is needed to prove the
Lemma.
\enddemo
By definition $\wt T\phi=\phi$, in addition, by Lemma 3.2 there exists
$I_0\in\Cal A_{n_0}$ such that $\phi(x)\geq\frac 12$ for each $x\in I_0$.
By Sub-Lemma 4.3, for each $x\in [0,\,1]$ there exists $j\leq N_0(n_0)$ and
$y_*\in I_0$ such that $T^jy_*=x$. Hence,
$$
\phi(x)=\wt T^j\phi(x)=\sum_{y\in T^{-j}x}\phi(y)|D_y T^j|^{-1}
\geq \phi(y_*)\|DT\|_\infty^{-j}\geq \frac 12\|DT\|_\infty^{-N_0(n_0)} .
$$
\enddemo
The main result of this section is contained in the following Theorem.
\proclaim{Theorem 4.4} If an expanding map is mixing and
$\inf \phi\geq\gamma>0$, then it is covering.
\endproclaim
\demo{Proof}
For each interval $I\subset[0,\,1]$, define
$$
\chi_I(x)=\left\{\aligned
\frac1{|I|}\quad\text{for }x\in I\\
0\quad\text{for }x\not\in I
\endaligned\right.
$$
then, $\bigvee_0^1\chi_I\leq\frac 2{|I|}$ and $\int_0^1\chi_I=1$.
Given any two intervals $I,\,I'$, the mixing property implies
$$
\lim_{n\to\infty}\int_0^1\chi_{I'}\chi_I\circ T^n=\int_0^1\chi_I\phi
\geq \gamma.
$$
Consider some $n_1\in\Bbb N$ (to be chosen later), there
exists $N_*\in\Bbb N$ such that, for each $I,\,I'\in\Cal A_{n_1}$
$$
\int_0^1\chi_I\wt T^n\chi_{I'}\geq\frac \gamma 2\qquad \forall n\geq N_* .
$$
Choose $I_0\in\Cal A_{n_1}$, consider $\wt T^n\chi_{I_0}$, from (2.1) follows
$$
\bigvee_0^1\wt T^n\chi_{I_0}\leq (2\lambda^{-1})^n\frac 2{|I_0|}+
A\sum_{i=0}^{n-1}(2\lambda^{-1})^i\int_0^1\chi_{I_0}\leq
(2\lambda^{-1})^n \frac 2{|I_0|}+\frac A{1-2\lambda^{-1}}.
$$
Let $b_n=\inf\limits_{I\in\Cal A_n}|I|$, choose $N_1\geq N_*$ such that
$$
\frac {2(2\lambda^{-1})^{N_1}}{b_{n_1}}\leq \frac A{1-2\lambda^{-1}}.
$$
Then,
$$
\bigvee_0^1\wt T^n\chi_{I_0}\leq \frac {2A}{1-2\lambda^{-1}}
\quad\forall n\geq N_1 .
$$
Consider the set $\Cal B_-=\{I\in\Cal A_{n_1}\;|\;\text{exists }x\in I :
\wt T^{N_1}\chi_{I_0}(x)<\frac \gamma 4\}$, and let $L=\#\Cal
B_-$.\footnote{By $\#$ we mean the cardinality of a set.}
First of all, notice that for each $I\in\Cal A_{n_1}$ there exists $y\in I$
such that $\wt T^{N_1}\chi_{I_0}(y)\geq\frac \gamma 2$, if not
$$
\int_0^1\chi_I\wt T^{N_1}\chi_{I_0}<\frac\gamma 2\int_0^1\chi_I=
\frac\gamma 2
$$
contrary to our assumptions on $N_1$.
Consequently, for each $I\in\Cal B_-$,
$$
\bigvee_I\wt T^{N_1}\chi_{I_0}\geq\frac\gamma 4,
$$
which implies
$$
L\leq\frac{8A}{(1-2\lambda^{-1})\gamma}\equiv L_0 .
$$
But, if $\#\{T^{-n}x\}\leq L_0$ we have
$$
\gamma\leq\phi(x)=\sum_{y\in T^{-n}x}\phi(y)|D_yT^n|^{-1}\leq
L_0\|\phi\|_\infty\lambda^{-n}
$$
which shows that, for $N_2=\left[\frac{\ln L_0\|\phi\|_\infty\gamma^{-1}}
{\ln\lambda}\right]+1$, $\#\{T^{-N_2}x\}> L_0$.
Choose $n_1=N_2$; since $T^{-n_1}x$ has at most one point in each element of
$\Cal A_{n_1}$ and $\#\{T^{-n_1}x\}>L$, it follows that there exists $y\in
T^{-n_1}x$ such that $y\in I\not\in\Cal B_-$; hence
$T^{n_1+N_1}I_0=[0,\,1]$.
The statement is then proven by using the same reasoning employed in
Sub-Lemma 4.3.
\enddemo
Summarizing, if a map is weakly-covering then, in force of Lemma 4.2,
we have $\gamma>0$ and, by Theorem 4.4, the map is covering.
Hence, we can prove
the exponential decay of correlations thanks to Theorem 3.6.
We have proved Theorem 0.1.
\vskip1cm
\subhead APPENDIX (Constants)
\endsubhead
\vskip1cm
This appendix contains explicit formulas for all the constants
involved, therefore substantiating my claim that the present
approach provides explicit bounds for the rate of decay of correlations.
Note that all the estimates, and the best choice of $a$, depend
on the function $N(n)$. In an application $N(n)$ can be computed for the
values one is interested in. Unfortunately, it is not clear how to derive a
general bound for $N(n)$ since it contains global informations on the
dynamics of the map.
To be concrete we will study an example.
\vskip.2cm
\line{\bf Example\hfil}
\vskip.2cm
Let us consider the map
$$
T(x)=\left\{\aligned
\frac 92 \left(\frac 19 -x\right)&\quad x\in\left(0,\,\frac
19\right)\\
\frac 92 \left(x-\frac 19\right) &\quad x\in\left(\frac 19,\,
\frac 39\right)\\
\frac 92 \left(\frac 59-x\right)&\quad x\in\left(\frac 39,\,
\frac 59\right)\\
\frac 92 \left(x-\frac 59\right)&\quad x\in\left(\frac 59,\,
\frac 79\right)\\
\frac 92 \left(1-x\right)&\quad x\in\left(\frac 79,\,1\right)
\endaligned
\right.
$$
The partition on which
$T$ is defined is:
$$
\Cal A_0=\left\{\left(0,\,\frac 1{9}\right);\;\left(\frac 1{9},\,\frac
13\right);\;
\left(\frac 13,\,\frac 59\right);\;\left(\frac 59,\,\frac 79\right);\;
\left(\frac 79,\, 1 \right)\right\}
$$
The map satisfies our assumptions since $|DT|=\frac 92>2$ and it is easy to
check that it is covering (remember that, by Lemma 4.2 and Theorem 4.4,
it suffice to check
that $T$ is weakly-covering).
The smaller element of the partition $\Cal A_0$ has size $\frac 19$,
accordingly $A=4$ (see (2.1)). The choice of $a$ is subject to the constraint
$a>\frac A{1-2\lambda^{-1}}=\frac{36}5$ (see Lemma 2.1), while $n_0$
must satisfy $\sup\limits_{I\in\Cal A_{n_0}}|I|\leq\frac 1{2a}$, it is then
immediately clear that we must choose $n_0=1$.
The partition $\Cal A_1$ is
$$
\Cal A_1=\left\{\left(0,\,\frac 1{27}\right);\;\left(\frac 1{27},\,
\frac 7{81}\right);\;
\left(\frac 7{81},\,\frac 19\right);\;\left(\frac 19,\,\frac {11}{81}\right);\;
\left(\frac {11}{81},\, \frac {15}{81} \right);\;\dots\right\}
$$
Clearly $b_1=\frac 2{81}$.
All the elements of $\Cal A_1$ have as image an element of $\Cal A_0$ that
it is mapped on all $[0,\,1]$ apart for the ones that are mapped on
$(0,\,\frac 19)$ and for $(0,\,\frac 1{27})$. A direct computation shows
that $T^3 (0,\,\frac 1{27})=[0,\,1]$ and $T^2 (0,\,\frac 19)=[0,\,1]$,
so $N(1)=3$.
If we choose $a=7.25$, then $\sigma_1\leq .994$ (see Lemma 3.5 for a
definition of $\sigma_1$). Since $a\sigma_1\geq 1+\sigma_1$ and
$1-\sigma_1\geq \frac 12\left(\frac 29\right)^3$, the formula in the proof
of Lemma 3.5 yields
$$
\frac\Delta 2\leq\ln\left[1+2a\sigma_1\|DT\|^{N(1)}\right]\leq \ln 1314
$$
$$
\Lambda=\tanh(\frac\Delta 4)^{\frac 13}\approx 1-\frac 23 e^{-\frac\Delta
2}\leq 1-\frac 1{1971}
$$
moreover, $\lim\limits_{n\to\infty}K_n\leq 140$ and $b\leq .3$.
The above estimates are probably off by a couple of orders of magnitude, but
they should not be considered too unsatisfactory: using the only other
known rigorous estimates, \cite{Co}, one would have been lead to consider at
least $N(20)$ obtaining an estimate of $(1-\Lambda)^{-1}$ wrong by at least
$20$ orders of magnitude. In addition, the present estimate can
certainly be improved: first of all it is easy to check that for each
$x\in[0,\,1]$ $\#\{T^{-3}x\}\geq 2$, this allows immediately to divide by
two the number inside the log (cf. the proof of Lemma 3.5); second, some
advantages can be obtained by using a different partition (instead of the
dynamical one) and a different number (instead than $\frac 12$) in Lemma
3.2.\footnote{Using these two ideas (i.e., the partition $\{(0,\,\frac
19);\; (\frac 19,\,\frac 29);\;(\frac 29,\,\frac 39);\;\dots\}$ and $\frac
7{36}$ instead than $\frac 12$, which allows to set $n_0=0$) it is already
possible to obtain the improved estimate $\Lambda\leq 1-\frac 1{377}$.}
Moreover, the result can conceivably be improved by
choosing a cone of functions better adapted to the particular example at
hand.
However, the problem of finding optimal bounds it is a business all in
itself and not the one we have been concerned with in this paper.
\Refs
\widestnumber\key{XXXX}
\ref\key{BK}\by V. Baladi, G. Keller\paper Zeta functions and transfer
operators for piecewise monotone transformations
\jour Communications in Mathematical Physics\vol 127\pages 459--477
\yr 1990
\endref
\ref\key{BY}\by M. Benedicks, L-S. Young\paper Absolutely continuous
invariant measures and random perturbations for certain one-dimensional
maps\jour Ergodic Theory and Dynamical Systems\vol 12\pages 13--37\yr 1992
\endref
\ref\key{B1}\by Garret Birkhoff\paper Extension of Jentzsch's Theorem
\jour Transaction American Mathematical Society\vol 85\yr 1957
\pages 219--227
\endref
\ref\key{B2}\by Garret Birkhoff\book Lattice Theory\publ American
Mathematical Society Colloquium Publications, Providence, Rhode Island
\bookinfo {\bf 25} 3rd ed. \yr 1967
\endref
\ref\key{DS}\by P. Diaconis, D. Stroock\paper Geometric bounds for
eigenvalues of Markov chains
\jour Annals of Applied Probability\vol 1\pages 36--61\yr 1991
\endref
\ref\key{Co}\by Collet\paper An estimate of the decay of correlations for
mixing non Markov expanding maps of the interval
\paperinfo preprint
\yr 1984
\endref
\ref\key{F} \by P. Ferrero\paper Contribution a la theorie des etats
d'equilibre en mechanique statistique\paperinfo Theses\yr 1981
\endref
\ref\key{FS1}\by P. Ferrero, B. Schmitt\paper Produits
al\'eatories d'op\'erateurs matrices de transfert
\jour Probability Theory and Related Fields\vol 79 \pages 227--248
\yr 1988
\endref
\ref\key{FS2}\by P. Ferrero, B. Schmitt\paper On the rate of convergence
for some limit ratio theorem related to endomorphisms with a non regular
invariant density\paperinfo preprint
\endref
\ref\key{FS2}\by P. Ferrero, B. Schmitt\paper Ruelle's Perron-Frobenius
Theorem and Projective Metrics\jour Colloquia Mathematica Societas J\'anos
Bolyai\vol 27\paperinfo Random Fields, Esztergom (Hungary)\yr 1979
\endref
\ref\key{HK}\by F. Hofbauer, G. Keller\paper Ergodic properties of
invariant measures for piecewise monotone transformations
\jour Math. Zeit. \vol 180\pages 119--140\yr 1982
\endref
\ref\key{H}\by B.R. Hunt\paper Estimating invariant measures and Lyapunov
exponents\paperinfo preprint.
\endref
\ref\key{Ke}\by G. Keller\paper Un th\'eorem de la limite centrale
pour une classe de transformations monotones par morceaux\jour Comptes
Rendus de l'Acad\'emie des Sciences, S\'erie A\vol 291\pages 155--158
\yr 1980
\endref
\ref\key{ITM}\by C. Ionescu Tulcea et Marinescu
\jour Annals of Mathematics\vol 52\pages 140--147\yr 1950
\endref
\ref\key{L}\by C. Liverani\paper Decay of correlations\paperinfo preprint
\endref
\ref\key{LY}\by A. Lasota, J. Yorke\paper On existence of invariant
measures for piecewise monotonic transformations
\jour Trans. Amer. Math. Soc. \vol 186\pages 481--487\yr 1973
\endref
\ref\key{Ru1}\by D. Ruelle\book Thermodynamic formalism
\publ Addison-Wesley \publaddr New York\yr 1978
\endref
\ref\key{Ru2}\by D. Ruelle\paper The Thermodynamic Formalism for Expanding
Maps\jour Communications in Mathematical Physics\vol 125\pages 239--262
\yr 1989
\endref
\ref\key{Ru3}\by D. Ruelle\paper An extension of the theory of fredholm
determinants\jour IHES\vol 72\pages 175--193 \yr 1990
\endref
\ref\key{Ry}\by Marek Rychlik\paper Regularity of the metric entropy for
expanding maps\jour Transactions of the American Mathematical Society\vol
315, 2\pages 833--847\yr 1989
\endref
\ref\key{Sa} \by H. Samelson\paper On the Perron-Frobenius Theorem
\jour Michigan Mathematical Journal\vol 4\pages 57--59 \yr 1956
\endref
\ref\key{Y}\by L-S. Young\paper Decay of correlations for certain quadratic
maps\jour Communications in Mathematical Physics\vol 146\pages 123--138
\yr 1992
\endref
\ref\key{Z} \by K. Ziemian\paper On the Perron-Frobenius operator and limit
theorems for some maps of an interval\inbook Ergodic Theory and
Related Topics II \bookinfo Proceedings \publ Teubner Textez\"ur Math.
{\bf 94}\pages 206--211 \yr 1987
\endref
\endRefs
\enddocument
ENDBODY