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\topmatter

\title
A PROBABILISTIC APPROACH TO INTERMITTENCY
\endtitle
\author
Carlangelo Liverani, Beno\^it Saussol, Sandro Vaienti
\endauthor
\affil University of Rome {\sl Tor Vergata},\\
Centre de Physique Th\'eorique, Marseilles,\\
PhyMat, University of Toulon.
\endaffil
\address
Liverani Carlangelo,
Mathematics Department,
University of Rome II, Tor Vergata,
00133 Rome, Italy.
\endaddress
\address
Saussol Beno\^\i t and Vaienti Sandro,
Centre de Physique Th\'eorique,
CNRS, Luminy case 907,
13288 Marseille Cedex 9, France.
PhyMat, Mathematics Department,
University of Toulon,
BP 132, 83957 La Garde Cedex, France.
\endaddress
\email
liverani@mat.utovrm.it, saussol@cptsg3.univ-mrs.fr, vaienti@cptsg2.univ-mrs.fr
\endemail
\date
\today
\enddate

\abstract
We present an original approach which allows to investigate the statistical
properties of a non-uniform hyperbolic map of the interval. Based on a
stochastic
approximation of the deterministic map, this method gives essentially the
optimal
polynomial bound for the decay of correlations, the degree depending on the
order of
the tangency at the neutral fixed point.
\endabstract
\thanks
\bf We gratefully acknowledge many interesting discussions with V. Baladi, X.
Bressaud,
V. Maume, B. Schmitt and G. Keller. One of us (B.S.) acknowledges partial
support of CEE grant CHRX-CT94-0460.
Finally, it is a pleasure to thank the ESI, Vien, and the
Institute for Advanced Studies, Jerusalem, where part
of this work was done.
\endthanks
\endtopmatter

\vskip -.5cm
\centerline{\bf CONTENT}

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\roster
\item"0."  Introduction\dotfill p. \ \ 1
\item"1."  The Model\dotfill p. \ \ 2
\item"2."  An Invariant Cone\dotfill p. \ \ 3
\item"3."  A Random Perturbation\dotfill p. \ \ 5
\item"4."  Decay of Correlations\dotfill p. 10
\item"5."  General considerations\dotfill p. 11
\item" "   References\dotfill p. 14

\endroster

\baselineskip=\riga
\lineskip=\rigat
\vskip .5cm

\document
\subhead
\S 0 Introduction
\endsubhead
Recently the study of the convergence to the equilibrium in hyperbolic systems
has witnessed several new results ranging from new methods to treat systems
with
singularities \cite{Li1}, \cite{Yo} or with partially hyperbolic behavior
\cite{BY}, to methods for studying Anosov flows \cite{Ch2}, \cite{Do},
\cite{Li2}.

Thanks to such results we can now regard the study of the decay of correlations
for uniformly hyperbolic systems as reasonably understood (albeit there is much
room for improvements, especially as flows and dependence on smoothness of
observable
are concerned).
On the contrary the available results on the convergence to the equilibrium in
non-uniform hyperbolic systems are extremely unsatisfactory. The study of such
systems stands as a challenge. In particular, it is evident the need to develop
new strategies to investigate such problems.

This is the focus of the present paper where we study a one parameter
family of intermittent maps. These applications are expanding, except at a
neutral
fixed point, where hyperbolicity is lost. The local behavior of the map at this
point
is responsible for various phenomenon. Let us denote by $1+\a$ the order of the
tangency at the critical point.
For $\a=0$ we have a purely expanding map, which has a unique equilibrium
state for the potential $\varphi=-\log DT$, with exponential decay of
correlations.
For $0<\a<1$, the map possess an absolutely continuous probability measure
(SRB measure), which is still an equilibrium state (it is no more unique,
since the Dirac mass at the origin is invariant and $\delta_0(\varphi)=0$).
For $\a\geq 1$, there are no absolutely continuous invariant probability
measure, whereas one still has a $\sigma$-finite absolutely invariant measures
\cite{PS}.

We focus here on the second region of the parameter, and propose to find
the density of the invariant measure, and the rate of decay of the
correlation functions.
In this domain, one cannot expect a spectral gap for the Perron-Frobenius
operator (see the end of section 4),
therefore none of the usual strategies in this setting can be followed.
Our approach is based on the following philosophy :
sure, the map is not hyperbolic, but it is the case nearly everywhere;
thus, if we perform a random perturbation of the map, the neutral fixed point
should be lost in a cloud of hyperbolic points and the intermittent effect
could
be suppressed.
This naive argument, rather surprisingly, works.

An interesting property of such a method is the following.
For smooth expanding maps, the same idea can be carried out, but yields a
sub-exponential rate of decay, while the decay is well known to be exponential.
On the contrary, in the present case, the power
law found appears to be near optimal, as remarked in section 4.

This is an indication that our crude approach performs better in the
non-uniform case than in the uniform one. These consideration are at the base
of
our belief that this type of strategy could yield relevant results in more
general situations.

The plan of the paper is as follows: In section one we present our model and
discuss
some related literature. Section two is devoted to the study of the invariant
measure.
The section may have an interest in  itself since it gives a very direct
approach to
obtaining the invariant measure and its properties for such a map (for a
comparison
with other techniques see \cite{CF,Th}).
Section three introduces the key idea of the paper, that is the random
perturbation
and its instrumental properties. In section four we harvest the facts from the
previous sections and obtain the announced result. In addition, we point out
that our results suffice to establish the CLT for $C^{(1)}$ observable,
provided that $\alpha < 1/2$ (Remark 4.2-(3)).
The last section contains few considerations on
how to treat the general problem of expanding maps with neutral fixed points.
Since the focus of the present paper is on the method and not on the class of
one-dimensional maps to which it can be applied, we content ourselves with few
pointed, but sketchy, considerations.

\subhead
\S 1 The model
\endsubhead
Let us consider for $0<\a<1$ the map $T:[0,\,1]\to[0,\,1]$
$$
T(x)=\left\{\aligned
            &x(1+2^\alpha x^\alpha)\quad \forall x\in\left[0,\,1/2\right)\\
            &2x-1\quad\quad\forall x\in\left[1/2,\,1\right]
             \endaligned
             \right.
$$
The importance of this kind of intermittent maps was addressed by Prellberg
and Slawny in \cite{PS}, where the relationship with a statistical model
introduced by Fisher \cite{FF} and successively studied by Gallavotti \cite{Ga}
was emphasized. In the papers \cite{GW}, \cite{W}, the dynamical behavior of
these
maps  was taken as a model for the intermittency of turbulent flows \cite{PM}.
In the paper \cite{PS} several mathematical results were announced, concerning
the ergodic and statistical properties of such maps, notably the presence of
phase transitions for the topological pressure (see also \cite{Lo}).
The paper \cite{W} deals with a piecewise linear version of the map and focus
especially on the recurrence properties of the orbits. These latter properties
have been put on a solid mathematical basis by Collet, Galves and Schmitt in
\cite{CGS} in the piecewise linear case, and by Campanino and Isola for the
non-linear ($\sigma$-finite) case \cite{CI1,CI2,CI3}.
The problem of the decay of correlations was considered in the piecewise
linear case and for the finite absolutely continuous (w.r.t. Lebesgue)
measure in \cite{LSV} and \cite{Mo}. Both papers obtain an algebraic
($n^{-\gamma}$) upper bound for the decay of correlations, the first by using
Markov approximations [some results were successively improved by Chernov
in \cite{Ch1}], the second by exhibiting the absence of spectral gap for the
Perron-Frobenius operator using the induction procedure, already invoked in
\cite{PS} (it is interesting to remark that Mori's work follows the analysis
of these maps carried out by Takahashi in a series of papers \cite{Ta1,Ta2}).

The decay of correlations for the non-linear case is a more difficult problem.
In our knowledge, the following methods have been proposed.

In the paper \cite{Yu}, Yuri applied Markov approximations, generalizing
the works of \cite{LSV} and \cite{Ch1};
the paper \cite{Is} aims to extend to the non-linear case the approach
of Mori, still inducing, and gives a description of the {\it zeta}
function, and finally in \cite{FS} the authors propose an interesting technique
based on Hilbert metrics, yet the implementation of such an idea is still
incomplete.

Let us go back to our particular model.
In the next section, we will prove that there exists\footnote{Such a result
can also be obtained by inducing \cite{Th}. The method used here is
more direct and provides additional informations on the properties of the
invariant
density, although such extra informations will not be essential in the
following.
See section five for a more detailed discussion.} a locally Lipschitz function
$h\in C^{(0)}(]0,1])\cap L^1([0,1])$ such that $Ph=h$.
\subhead
\S 2 An invariant Cone
\endsubhead

If we define the cone
$\Cal C_0=\{f\in C^{(0)}(]0,\,1])\;|\; f\geq 0,\,f \hbox{ is decreasing} \}$
it is immediate to see that $\Cal C_0$ is left invariant by the P-F operator.
To see a bit more let us call $X$ the identity, $X(x)=x$.
\proclaim{Lemma 2.1} The cone
$\Cal C_1 = \{ f\in \Cal C_0 \;|\; X^{\a+1}f \hbox{ \rm is increasing} \}$,
is left invariant by the operator $P$.
\endproclaim
\demo{Proof}
Let $f\in \Cal C_1$, then
$$
x^{\a+1} Pf(x)= \sum_{y \in T^{-1}x} \left( \frac{Ty}y \right )^{\a+1}
\frac{y^{\a+1}f(y)}{D_yT} .
$$
Setting $T^{-1}x=\{y_1,\,y_2\}$, $y_1\leq y_2$, and $\xi=2^\a y_1^\a$ we can
write
$$
x^{\a+1}Pf(x)= \frac{(1+\xi)^{\a+1}}{1+(\a+1)\xi} \; y_1^{\a+1}f(y_1)
+ \frac 12 \left( \frac{2y_2-1}{y_2} \right)^{\a+1}  y_2^{\a+1}f(y_2).
$$
Whence the result,
since $X^{\a+1}f$, $x \mapsto y_1$, $x \mapsto \xi$, $x \mapsto y_2$ are
increasing.
\qed
\enddemo
Let us define
$$
m(f)=\int_0^1 f(x)dx.
$$
Obviously $m(Pf)=m(f)$.

The last interesting property is contained in the following.

\proclaim{Lemma 2.2} The cone
$\Cal C_*=\{f\in \Cal C_1\cap L^1([0,\,1])\;|\; f(x)\leq a x^{-\a}m (f) \}$
is invariant with respect to the operator $P$, provided $a$ is chosen large
enough.
\endproclaim
\demo{Proof}
For each $f\in\Cal C_*$ holds both
$$
f(x) \leq a x^{-\a} m(f),
$$
and
$$
x^{\a+1}f(x)\leq f(1)\leq \int_0^1 f=m(f).
$$
Let us suppose for simplicity that $m(f)=1$.
One has to find a constant $a$, independent of $f$, such that
$Pf(x) \leq a x^{-\a}$ since $m (P f)= m (f) =1$.
$$
\aligned
P f(x)=&
\frac {f(y_1)}{D_{y_1}T} + \frac {f(y_2)}{D_{y_2}T} \\
\leq &
\frac {a y_1^{-\a}}{D_{y_1}T} + \frac {y_2^{-\a-1} }{D_{y_2}T} \\
\leq &
\left \{ \left ( \frac x{y_1} \right )^\a \frac 1{D_{y_1}T} +
\frac 1a \frac {x^\a}{y_2^{\a+1} D_{y_2}T} \right \} a x^{-\a} .
\endaligned
$$
The term in curly bracket is bounded by
$$
\frac {(1+\xi)^\a}{1+(\a+1)\xi} + \frac {2^\a}a \xi  \leq
\frac {1+\a \xi + \frac {2^\a}a \xi (1+(\a+1)\xi)}{1+(\a+1)\xi} \leq
\frac {1+\left (\a+\frac {2^\a(\a+2)}a \right ) \xi}{1+(\a+1)\xi} \leq 1.
$$
Whenever $a \geq 2^\a (\a+2)$, from which the Lemma follows.
\qed\enddemo
Putting together all the previous estimates yields

\proclaim{Lemma 2.3} There exists a locally Lipschitz function $h$ such that
$Ph=h$ and $h(x)\leq a x^{-\a}$.
\endproclaim
\demo{Proof}
The operator $P$ leaves invariant the set $K=\{f\in \Cal C_*\;|\;m(f)=1\}$.
But $X^{\a+1}K$ consists of equibounded equicontinuous functions,\footnote{Let
$f\in K$ and define $\phi(x)=x^{1+\a}f(x)$, then, for $x\geq y$, holds
$$
\aligned
0\leq\phi(x)-\phi(y)&\leq (x^{1+\a}-y^{1+\a})f(x)\leq a(1+\a)x^{-\a}\int_y^x
\xi^\a d\xi\\
&\leq a(1+\a)|x-y| .
\endaligned
$$
}
hence it is compact in $C^{(0)}$. Accordingly, for each $f\in K$ the sequence
$X^{\a+1} \frac 1n\sum_{i=0}^{n-1} P^if$ has accumulations points in $C^{(0)}$.
Let
$h_* \in K$ be such an accumulation point.
Clearly, $h=X^{-1-\a}h_*$ is a fixed point of $P$, hence the result.
The regularity of $h$ is easily obtained by checking that $h\in\Cal C_*$.
\qed \enddemo

Before introducing the random approximation to our dynamics let us remark a
property of the functions in $\Cal C_*$ that will be
instrumental in the following.

\proclaim{Lemma 2.4} For each function $f\in\Cal C_*$
$$
\inf_{x\in[0,\,1]}f(x)=f(1)\geq \min\left\{a,\,
\left[\frac{\a(1+\a)}{a^\a}\right]^{\frac 1{1-\a}}\right\}\int_0^1 f.
$$
\endproclaim
\demo{Proof}
It clearly suffices to consider the case $\int_0^1 f=1$. We have already seen
that
$$
\aligned
f(x)&\leq a x^{-\a},\\
f(x)&\leq x^{-1-\a}f(1) .
\endaligned
$$
We introduce the point $x_*=a^{-1}f(1)$. On its left the first
inequality is
stricter and the opposite holds on its right. If $x_*>1$, then $f(1)>a$,
otherwise
$$
1=\int_0^1f=\int_0^{x_*}f+\int_{x_*}^1 f\leq \int_0^{x_*}a\xi^{-\a}
+\int_{x_*}^1 \xi^{-1-\a}f(1)\leq \frac{a^\a}{\a(1-\a)}f(1)^{1-\a} ,
$$
from which the lemma follows.
\qed\enddemo


\subhead
\S 3 A Random Perturbation
\endsubhead


For simplicity let us identify $[0,1]$ with the circle $S^1$, on $S^1$
the map is not smooth but it is continuous (this is not essential but it will
make our life a bit easier).
Let us define the ``ball" $B_\ve(x)=\{y\in S^1\;|\; |x-y|\leq \ve\}$ and the
averaging operator\footnote{Let us remark that this particular choice of $\Bbb
A_\ve$ has nothing special, any other ``reasonable" choice would do as well.}

$$
\Bbb A_\ve f(x)=\frac 1{2\ve}\int_{B_\ve(x)}f(y) dy.
$$

It is now possible to define the perturbed operator
$$
\Bbb P_\ve =P^{n_\ve} \Bbb A_\ve ,
$$
where $n_\ve\in\Bbb N$ will be specified later.

The following Lemma shows that the perturbed operator is not too different
from the original one, provided we consider observables in $\Cal C_*$.

\proclaim{Lemma 3.1} For each $f\in \Cal C_*$
$$
\|P^{n_\ve} f- \Bbb P_\ve f\|_1\leq c_1 \|f\|_1 \ve^{1-\a} .
$$
Where $c_1=\frac{10 a}{\a(1-\a)}$.
\endproclaim
\demo{Proof}
We assume that $f \in \Cal C_*$ and $\int f=1$.
First, observe that
$$
\|P^{n_\ve} f- \Bbb P_\ve f\|_1\leq
\| f - \Bbb A_\ve f \|_1 .
$$
Next, we recall the estimates
$$
\aligned
f(x) \leq & a x^{-\a} \\
 \frac {f(y)}{f(x)}\leq & \left(\frac xy\right)^{1+\a}\quad\forall x\geq y.
\endaligned
$$
This allows us to bound the $L^1$ norm of the difference between the function
$f$ and its average.
$$
\aligned
\| f - \Bbb A_\ve f \|_1 \leq &
\frac 1{2\ve} \int_\ve^{1-\ve}\gluu dx \int_{B_\ve(x)} \gluu dy \; \left|
f(y)-f(x) \right |
+ \int_{B_\ve(0)} \gluu dx
\left | f(x) - \frac 1{2\ve} \int_{B_\ve(x)} \gluu dy \; f(y) \right | \\
\leq &
\frac 1{2\ve} \int_\ve^{1-\ve} \gluu dx\left\{
\int_{x-\ve}^{x}\gluu dy [f(y)-f(x)]+\int_x^{x+\ve}\gluu dy[f(x)-f(y)]\right
\}\\
&\ +\int_{B_\ve(0)} \gluu f(x) dx + \int_{B_{2\ve}(0)} \gluu f(y) dy \\
\leq &
\frac 1{2\ve} \int_\ve^{1-\ve} \gluu dx \int_{x-\ve}^{x}\gluu dy
f(y)\left[1-\frac{f(y+\ve)}{f(y)}\right]
+ 4a \int_0^{2\ve} \gluu x^{-\a} dx \\
\leq &
\frac {a}{2(1-\a)\ve} \int_\ve^{1-\ve} \gluu dx
\left( 2x^{1-\a}-(x+\ve)^{1-\a}-(x-\ve)^{1-\a} \right )\\
&\ +\frac {a}{2\a} \int_\ve^{1-\ve} \gluu dx
\left(x^{-\a}-(x+\ve)^{-\a} \right )
 + \frac{4a}{1-\a}(2\ve)^{1-\a} \\
\leq & \frac{a}{2(1-\a)(2-\a)\ve}\left\{(2\ve)^{2-\a}-2\ve^{2-\a}
-(1-2\ve)^{2-\a}-1+2(1-\ve)^{2-\a}\right\}\\
&\ +\frac{a}{2\a(1-\a)}\{2^{1-\a}\ve^{1-\a}-1-\ve^{1-\a}+(1-\ve)^{1-\a}\}
+\frac {8a}{1-\a}\ve^{1-\a}\\
\leq & \frac {10}{\a(1-\a)}\ve^{1-\a}.
\endaligned
$$
This proves Lemma 3.1.
\qed\enddemo
Next,
$$
\aligned
\Bbb P_\ve f(x)=&\sum_{y\in T^{-n_\ve}x}\frac 1{2 \ve D_yT^n_\ve}\int_0^1dz
\chi_{B_\ve(y)}(z) f(z)\\
=&\sum_{y\in T^{-n_\ve}x}\frac 1{2 \ve D_yT^n_\ve}\int_0^1dz
\chi_{B_\ve(z)}(y) f(z)\\
=&\frac 1{2\ve}\int_0^1 dz P^{n_\ve}\chi_{B_\ve(z)}(x)f(z)\\
:=&\int_0^1  \Cal K_\ve(x,\,z)f(z)dz.
\endaligned
$$
Our task is to find a lower bound for the kernel $\Cal K_\ve(x,\,z)$.
For this purpose, let us define $T_1$ to be the map $T$ restricted to the
interval $[0,\,1/2]$ and $a_n=T^{-n}_11$. We have the following asymptotic
bound for the sequence $a_n$.

\proclaim{Lemma 3.2} For all integer $n>0$ the following holds
$$
a_n \leq 2^{\frac 1{\a^2}+\frac 1\a } n^{-\frac{1}\a} .
$$
\endproclaim
\demo{Proof}
The Lemma is proven by induction. First it is clearly satisfied for $n=1$.
Next, let us suppose that
$a_n < c n^{-\frac {1}\a }$, and let us prove that $a_{n+1} < c (n+1)^{-\frac
{1}\a
}$. If it is false, then
$$
a_n = a_{n+1} (1+2^\a a_{n+1}^\a)
\geq
c (n+1)^{-\frac {1}\a } (1+2^\a c^\a (n+1)^{-1}).
$$
By the assumption on $a_n$ we obtain
$$
n^{-\frac {1}\a } \geq (n+1)^{-\frac {1}\a } (1+\frac {2^\a c^\a}{n+1})
$$
or equivalently
$$
(1+\frac 1n)^{\frac {1}\a} \geq 1+\frac {2^\a c^\a}{n+1}.
$$
By convexity it follows
$$
(2^{\frac1\a}-1)\frac 1n\geq \frac{2^\a c^\a}{n+1},
$$
that is
$$
c^\a\leq 2^{-\a}(2^{\frac 1\a}-1)\frac {n+1}n\leq  2^{-\a+1}(2^{\frac 1\a}-1),
$$
which is contradictory if we choose $c=2^{\frac 1{\a^2}+\frac 1\a}$.
\qed
\enddemo
We define $\Delta_k=[a_k,\,a_{k-1}]$ for each $k>0$.
We are now able to prove

\proclaim{Proposition 3.3} There exists $\gamma>0$ such that for each
$\ve>0,\,x,\,z\in S^1$
$$
\Cal K_\ve(x,\,z)\geq \gamma,
$$
provided we choose $n_\ve=2^{2+\frac 1\a} \ve^{-\a}$.
\endproclaim
\demo{Proof}
First of all, we choose $k_0=3$.
Next, notice that for each interval $J$ and integer $m$
$$
(P^m\chi_J)(x)\geq \chi_{T^mJ}(x)\inf_{y\in J}(D_yT^m)^{-1}.
$$
Let $\delta_0=a_{k_0}-a_{k_0+1}$.
By Lemma 2.4 it is obvious that there exists $n_0$ and $c_0$
such that for all intervals $I$ of size larger than $\delta_0$ holds
$$
P^n \chi_I \geq c_0
$$
provided $n \geq n_0$.
Thus the task is to control the $\inf_{y\in J}(D_yT^m)^{-1}$,
where $m$ is the time needed for the interval $J$ to become
an interval of size $\delta_0$. Let $I_0=[0,a_{k_0}]$.
Let $J$ be an interval, three possibilities can occur :

(1) $J\cap I_0= \emptyset$.

(2) $J\cap I_0\neq \emptyset$ and $J$ contains, at most, one $a_k$ for $k>k_0$.

(3) $J$ contains more than one $a_k$ for $k>k_0$.

\noindent
We can associate to each $J$ a sequence $n_1,k_1,\ldots,k_{p-1},n_p$ of
integers
($n_1$ may be null), retracing the trajectory of $J$ in the following
way : For a time $n_1$ (1) holds, then the image of $J$ enters the intermittent
region
$I_0$ and (2) holds with $k=k_1+k_0$, so after $k_1$ iterations it exits from
$I_0$.
Then the image of $J$ stays in the hyperbolic region for $n_2$ iterations, and
so
on... Finally we end when the size of the interval becomes larger than
$\delta_0$
or if case (3) happens. Let us see what happens in this regimes.

\roster
\item  Let $\ds D=\sup_{y \in [a_{k_0+1},1]} \frac{D^2_yT}{D_yT^2}$ and
$r=(D_{a_{k_0+1}}T)^{-1}<1$.
For $n\leq n_1$, the usual distortion estimates yield, for each $y\in J$, with
$c_2=D/(1-r)$,
$$
\frac{D_yT^n |J|}{|T^n J|}\leq \e[c_2|T^n J|].
$$
\item
Let $J_1=T^{n_1}J$. Let us see what happens in the intermittent region.
Suppose that $J_1 \subset ]a_{k+1},a_{k-1}[$, i.e. (2) occurs with $k=k_0+k_1$.
In this case a direct computation for $j\leq k_1$ and $y_1,y_2 \in J_1$ implies
$$
\aligned
\frac{D_{y_1}T^j}{D_{y_2}T^j}\leq&\e\left[\sum_{i=1}^{k_1}\frac{\sup_{\xi\in
[a_{k-i+2},a_{k-i}]} D_\xi^2 T}{\inf_{\xi\in [a_{k-i+2},a_{k-i}]}D_\xi
T}|T^iy_2-T^i
y_1|\right]\\
\leq &\e\left[\sum_{i=1}^{k_1}\a(1+\a)a_{k-i+2}^{\a-1}
     (D_{a_{k+2-i}}T^{k_1-i})^{-1}|T^{k_1}J_1|\right] \\
\leq &\e\left[\a (\a+1)
\sum_{q=k_0+2}^{k_0+k_1+1}a_q^{\a-1}(D_{a_q}T^{q-(k_0+2)})^{-1}|T^{k_1}J_1|\right].
\endaligned
$$
Since the first branch of the map is convex, we have
$$
D_{a_q}T^{q-(k_0+2)} \geq r \frac{a_{k_0+1}-a_{k_0+2}}{a_{q-1}-a_q} = r
a_{k_0+2}^{1+\a} a_q^{-(1+\a)}.
$$
Moreover, Lemma 3.2 gives $a_q^{2\a} \leq 2^{2+2/\a} q^{-2}$. This yields,
setting $\ds c_3 = \frac{\a(1+\a)2^{2+2/\a}}{r a_{k_0+2}^{1+\a}}$,
$$
\frac{D_{y_1}T^j}{D_{y_2}T^j}\leq
\e\left[c_3 |T^{k_1}J_1|\right] .
$$
\item
Finally, let us see what happens if (3) holds for some iterate $K=T^jJ$.
If more than one third of the size of $K$ is in $[a_1,1]$, then we consider
$K \cap [a_1,1]$ and case (1) will hold for ever, loosing just factor $1/3$.
Else we cut $K$ into pieces $\Delta_{k_-},\cdots,\Delta_{k_+}$ such that
the union of them is of size bigger than $|K|/3$.
For these $\Delta_k$, the preceding computation yields
$$
P^{k-k_0} \chi_{\Delta_k} \geq \chi_{\Delta_{k_0}}
\frac{\e[-c_3 |\Delta_{k_0}|]}{|\Delta_{k_0}|} |\Delta_k| .
$$
Therefore, with $l=n_0+k_+-k_0$,
$$
P^l \chi_K
\geq
\sum_{k=k_-}^{k_+} P^{l+k_0-k} P^{k-k_0} \chi_{\Delta_k}
\geq
\sum_{k=k_-}^{k_+} c_0 \frac{\e[-c_3 \delta_0]}{\delta_0}|\Delta_k|
\geq
c_0\frac{\e[-c_3 \delta_0]}{\delta_0} \frac{|K|}3 .
$$
\endroster
Since we control what happens on each region, it is possible to estimate
the total distortion after $m=n_1+k_1+\cdots+n_p+l$ iterations, where $l=n_0$
if case (3) never happens ($l=n_0+k_+-k_0$ if case (3) occurs).
$$
\aligned
P^m \chi_J
\geq&
P^l P^{n_p} P^{k_{p-1}} \cdots P^{n_2} P^{k_1} P^{n_1} \chi_J \\
\geq&
|J|\frac{c_0}{3\delta_0}
\e\left[-c_3\delta_0-c_2|T^{n_p+\cdots+k_1+n_1}J|-\cdots-c_3|T^{k_1+n_1}J|-c_2|T^{n_1}J|)\right]\\
\geq&
|J|\frac{c_0}{3\delta_0}
\e\left[-(c_2+c_3)\delta_0(1+r^{n_p}+r^{n_p+n_{p-1}}+\cdots+r^{n_p+n_{p-1}+\cdots+n_2})\right]\\
\geq&
|J|\frac{c_0}{3\delta_0}
\e\left[\frac{-(c_2+c_3)\delta_0}{1-r}\right] :=\gamma |J|.
\endaligned
$$
To conclude, we need to fix $n_\ve$. We choose the supremum over all possible
values of $m=n_1+k_1+\cdots+n_p+l$, associated to intervals $J$ of size $2\ve$.
It is immediate to see that the worst case scenario is when case (3)
happens at the beginning, and $J=]-2\ve/3,\,4\ve/3[$.
In this case, $m$ is such that $a_{k_0+m} \leq 2\ve/3$.
Clearly, $n_\ve=2^{2+\frac 1\a} \ve^{-\a}$ is large enough and the
Lemma is proven.
\qed\enddemo

\subhead
\S 4 Decay of Correlations
\endsubhead

Proposition 3.3 allows immediately to conclude that $\Bbb P_\ve$ has an
invariant
density $h_\ve$ to which it converges exponentially fast\footnote{
Let us briefly recall the argument: set $\Omega=[0,\,1]$ and
consider  $f\in L^1(\Omega)$ with $\int_{\Omega}f=0$. Remember that $\Bbb P_\ve
1=\Bbb P_\ve^*1=1$ and let $\Omega^+_\ve=\{x\in\Omega\;|\;\Bbb P_\ve f\geq
0\}$;
$\Omega_+=\{x\in\Omega\;|\; f\geq 0\}$, then
$$
\aligned
\|\Bbb P_\ve f\|_1&=2\int_{\Omega^+_\ve}dx\int_{\Omega}dy \Cal K_\ve(x,\,y)
f(y)=2\int_{\Omega}dy f(y)\int_{\Omega^+_\ve}dx[ \Cal K_\ve(x,\,y)-\gamma]\\
&\leq 2\int_{\Omega_+}dyf(y)\int_{\Omega}dx [\Cal K_\ve(x,\,y)-\gamma]
=(1-\gamma)2\int_{\Omega_+}dyf(y)\\
&=(1-\gamma)\|f\|_1.
\endaligned
$$
Accordingly, for each $f\in L^1(\Omega)$ and defining $\Pi f=\int_\Omega f$,
holds
$$
\|(\Bbb P_\ve-\Pi)^nf\|_1=\|\Bbb P_\ve^n(\Idfoot-\Pi)f\|_1\leq
(1-\gamma)^n\|f\|_1 .
$$
}
in $L^1$. In the following we will call $\mu$ the invariant measure $d\mu=hdx$.
Using all the above facts, we are able to prove our main result.

\proclaim{Theorem 4.1}
For all $g \in L^\infty$, $f\in \Cal C^{(1)}([0,1])$ such that $\int f
d\mu=0$ the following holds
$$
\left |\int g\circ T^n f d\mu \right | \leq
c_4C\left(\|f\|_{C^{(1)}}\right)\|g\|_\infty n^{-\frac 1\a+1}(\log n)^{\frac
1\a}.
$$
Where $C:\RR \to \RR$ is affine.
\endproclaim
\demo{Proof}
Let $f\in\Cal C_*+\RR$,
$\int_0^1f=0$ and $g\in L^\infty$, $\|g\|_\infty=1$ we have
$$
\aligned
\left|\int_0^1gP^nf\right|\leq& \|P^nf-\Bbb P_\ve^{\frac n{n_\ve}}f\|_1+
\|\Bbb P_\ve^{\frac n{n_\ve}}f\|_1\\
\leq& \sum_{i=1}^{\frac n{n_\ve}}\|P^{(i+1)n_\ve}f-\Bbb P_\ve P^{i n_\ve}f\|_1+
\e\left[-(1-\gamma) \frac n{n_\ve}\right] \|f\|_1 \\
\leq& 2c_1\|f\|_1\frac n{n_\ve} \ve^{1-\a} + \e\left[-(1-\gamma) \frac
n{n_\ve}\right] \|f\|_1 \\
\leq& c_4 \|f\|_1 n^{-\frac 1\a+1}(\log n)^{\frac 1\a} .
\endaligned
$$
by choosing $\ve=n^{-\frac 1\a}(\log n^{(1-\gamma)(\frac 1\a-1)})^{\frac 1\a}$.

This is not yet the decay of correlation with respect to the absolutely
continuous
invariant measure $d\mu=h dx$ of our dynamical system. To get such a result, we
need
to notice that if $f \in C^{(1)}$, then we can choose $\lambda, \nu, \delta \in
\RR$
such that
$f_{\lambda,\nu,\delta}(x):=(f(x)+\lambda x+\nu)h(x)+\delta \in \Cal C_*$,
and $(\lambda x+\nu)h(x)+\delta \in \Cal C_*$, the dependence of the
parameters with respect to the $C^{(1)}$ norm of $f$ being affine.
Finally, the decay of correlations with respect to $\mu$,
for each $f \in C^{(1)}, \int fd\mu=0$ and $g\in L^\infty$ can be estimated
as follows
$$
\left |\int g\circ T^n f d\mu \right | \leq
c_4C\left(\|f\|_{C^{(1)}}\right) n^{-\frac 1\a+1}(\log n)^{\frac 1\a}.
$$
Where $C:\RR \to \RR$ is affine.
\qed\enddemo

\proclaim{Remark 4.2}
\roster
\item
The proof of Theorem 4.1 allows to estimate the difference between
$h_\ve$ and $h$; namely, to prove the estimate:\footnote{\rm It suffices to
write
$$
\aligned
\|h-h_\ve\|_1&\leq \|P^n1-h\|_1+\|\Bbb P_\ve^{\frac n{n_\ve}}1-h_\ve\|_1+\|\Bbb
P_\ve^{\frac n{n_\ve}}1- P^n1\|_1 \\
&\leq\hbox{const.} \{n^{1-\frac 1\a}(\log n)^{\frac 1\a}+(1-\gamma)^{n\ve^\a}+
n\ve\}
\endaligned
$$
and to choose $n\sim \ve^{-\a}\log\ve^{-1}$.}
$\|h_\ve-h\|_1\leq \hbox{const.} \ve^{1-\a}(\log\ve^{-1})^{\frac 1\a}$.
\item
Theorem 4.1 allows to get an estimate for the rate of decay for
H\"older continuous observables. More precisely, given a $\beta $-H\"older
continuous function $f$, we can bound the correlations by $n^{-\beta(\frac
1\a-1)}$,
up to some logarithmic correction.
\item
We have obtained a polynomial decay,
with a bound comparable to the one found in \cite{Mo} for the piecewise linear
case and the one stated in \cite{Is} for the general case. Moreover, compared
with
numerical simulations \cite{LSV}, our bound appears to be extremely close to
optimal:
the expected one is the same apart from the logarithmic correction.
\item
As a side consequence of our work we have that $\sum_{n=0}^\infty P^n f$
 converges
in $L^1$ provided $\a<\frac 12$. According to \cite{Li3} this estimate suffices
to
prove the Central Limit Theorem. That is, given $f\in \Cal C^{(1)}([0,\,1])$
such that
$\int_0^1 fd\mu=0$
$$
\frac1{\sqrt n}\sum_{i=0}^n f\circ T^i
$$
converges, in distribution, to a Gaussian variable with variance $\int_0^1f^2
d\mu+2\sum_{i=1}^\infty\int_0^1f f\circ T^i d\mu$.
\endroster
\endproclaim

\subhead
\S 5 General Considerations
\endsubhead

The reader may be under the impression that the proposed approach is specific
to the special maps studied here. In particular, section two seems quite model
dependent. In fact, while the estimates done there would apply certainly to
similar maps, it is true that some more work is needed to present them in a
completely general fashion\footnote{Yet, the results extend immediately
to any map of the interval which is $\Cal C^{(1)}$--conjugate to our model.
More precisely, suppose that $\widetilde T:[0,1] \to [0,1]$ and
$\Phi \in \hbox{Diff}^{(1)}([0,1])$ satisfies $\widetilde T \circ \Phi = \Phi
\circ
T$. Then, for the absolutely continuous $\widetilde T$--invariant
measure $\widetilde \mu$  defined by $\widetilde \mu (f)=\mu (f \circ \Phi)$,
it is straightforward to see that the power law decay is the same for $T$ and
$\widetilde T$ for $\Cal C^{(1)}$ observable.}.

Nevertheless, section two has been introduced more for the purpose of making
the paper self contained and to emphasize the existence  of a very direct
method of proving the existence of an invariant measure, than for real
necessity.

Here we discuss briefly what is really essential in order to apply the
present method.

Let us consider a map $T:[0,\,1]\to[0,\,1]$ expanding but for the fixed point
$0$.
where $D_0T=1$.
Assume $D^2T$ continuous but in the point $0$ and
$$
\aligned
|D^2_xT| &\leq C x^{\a-1} \\
|D_xT|   &=   1 + c x^\a + o(x^\a) .
\endaligned
$$
Assume further that there exists an invariant probability measure absolutely
continuous with respect to Lebesgue (see the discussion at the end of this
section).
Calling $h$ its density we have $Ph=h$.
Then, in analogy with Lemma 2.1 holds\footnote{
Such a result should extends easily to maps expanding but for some fixed points
$\{p_j\}$ where $D_{p_j}T=1$. We can define $\theta(x)=\min_j |x-p_j|$ and
assume $D^2T$ continuous but in the points $\{p_j\}$ and
$$
\aligned
|D^2_xT| &\leq C \theta(x)^{\a-1} \\
|D_xT|   &=   1 + c_j \theta(x)^\a + o(\theta(x)^\a) .
\endaligned
$$
Then the same cone with $\frac {a+b\theta(x)}{\theta(x)}$ instead of
$\frac {a+bx}x$ should be invariant.}

\proclaim{Lemma 5.1}There exist $a>\alpha$ and $b>0$ such that the cone
$$
\Cal C=\left \{
f\in C^{(1)}(]0,\,1])\;\Bigm | \; 0 \leq f(x) \leq 2 h(x) \! \int_0^1 \! f ;\;
|f'(x)| \leq \frac {a+bx}x f(x)\right \},
$$
is invariant with respect to the Perron-Frobenius operator.
\endproclaim
\demo{Proof}
Obviously, the first condition is invariant by $P$.
If $f$ belongs to the cone
$$
\aligned
|(Pf)'(x)|
& =
\left | \sum_{y \in T^{-1}x} \frac {D^2_yT}{|D_yT|^3} f(y) +
 \frac 1{|D_yT|^2} f'(y) \right | \\
&\leq
\sum_{y \in T^{-1}x} \left ( \frac {2 C y^{\a-1}}{|D_yT|^2} +
\frac {a+by}{y|D_yT|} \right ) \frac{f(y)}{|D_yT|} \\
&\leq
\frac {a+bx}x Pf(x) \sup_{y \in [0,1]} \left [
\frac {2C}{|D_yT|^2} \frac {y^{\a-1}T(y)}{a+bT(y)} +
\frac {T(y)}{y|D_yT|} \frac {a+by}{a+bT(y)} \right ]
\endaligned
$$
Let $\Omega(y)$ being the term in brackets.
We have
$$
T(y) = \int_0^y D_tT dt = \int_0^y (1+c t^\a + o(t^\a)) dt =
 y + \frac c{\a+1} y^{\a+1} + o(y^{\a+1}) .
$$
Which shows that $\Omega(y) \leq
1+\Bigl ( \frac {2C}a + c\bigl ( \frac 1{\a+1}-1 \bigr ) \Bigr ) y^\a +
o(y^\a)$.
Therefore there exists $\delta>0$ such that $\Omega(y)\leq 1$ on $[0,\delta]$
provided $a$ is big enough.
Next, outside this neighborhood, we have $y>\delta$ and $|D_yT|>\gamma>1$, so
$$
\Omega(y) \leq \frac 1a \frac {2 C \delta^{\a-1}}{\gamma^2} +
 \frac ab \frac 1{\delta \gamma} + \frac 1\gamma < 1
$$
provided $a$ and $b/a$ are big enough.
\qed\enddemo
In addition, we assume that for some $0<\beta<1$, $0<\gamma_0<1$ and $c_*>0$,
$$
\gamma_0 \leq h(x)\leq c_*x^{-\beta} .
\tag 5.1
$$
It follows that a sharper cone is invariant. Note that in the process of
proving
the next lemma we will establish the analogous of Lemma 2.4.

\proclaim{Lemma 5.2} There exists $\delta>0$ and $0<\gamma < \gamma_0$ such
that
$$
\Cal C_*=\{f\in \Cal C\;|\;  f(x)\geq \gamma\int_0^1 f\hbox{ {\rm for}
}x\leq\delta\},
$$
is invariant with respect to the Perron-Frobenius operator.
\endproclaim
\demo{Proof}
The proof starts by establishing the analogous of Lemma 2.4
\proclaim{Sub-Lemma 5.3} If $f\in\Cal C$, then
$$
\min_{[\delta,1]} f\geq \frac{\delta^a}{2e^b} \int_0^1 f
$$
\endproclaim
\demo{Proof}
We have the bounds
$$
f(x)\leq 2c_*x^{-\beta}\int_0^1 f
$$
and
$$
|f'(x)|\leq (a x^{-1}+b)f(x)
$$
coming from the cone and the assumed estimate on the invariant measure.
For $x\geq y>\delta$ the second bound yields
$$
\left(\frac yx\right)^a e^{-b(x-y)}\leq \frac {f(x)}{f(y)}\leq
\left(\frac xy\right)^a e^{b(x-y)}.
$$
Accordingly, normalizing $\int f=1$,
$$
1=\int f=\int_0^\delta f+\int_\delta^1 f\leq \frac
{2c_*}{1-\beta}\delta^{1-\beta}+
\delta^{-a}e^b\min_{x\in[\delta,\,1]}f.
$$
Next, by choosing $\delta$ sufficiently small, we have $ \frac
{2c_*}{1-\beta}\delta^{1-\beta}\leq \frac 12$ from which the result follows.
\qed\enddemo

We choose $\delta $ so small that for all $y\leq\delta$ hold
$|D_yT|^{-1} \geq 1-2c\delta^\a$ and Sub-Lemma 5.3 together with
$1-2c\delta^\a+\mu^{-1} > 1$. Let $x\leq \delta$, then $T^{-1}x$ will consist
of, at least, two points $y_1\leq\delta$ and $y_2\geq\delta$. Let
$\mu=\|DT\|_\infty$, then, by choosing $\gamma$ small, holds
$$
\aligned
Pf(x)\geq& |D_{y_1}T|^{-1}f(y_1)+\mu^{-1}f(y_2)\geq
\left[(1-2c\delta^\alpha)\gamma+\mu^{-1}\min\left\{\gamma,\,\frac{\delta^a}{2e^b}\right\}
\right]\int_0^1 Pf\\
&\geq \gamma\int_0^1 Pf.
\endaligned
$$
\qed\enddemo
This is enough to show that $\inf f \geq \gamma \int f$ whenever
$f \in \Cal C_*$, which implies that (since the constant function
$1$ belongs to the cone $\Cal C_*$)
$$
\inf_{n\geq 0} \inf_{[0,1]} P^n1 \geq \gamma > 0 .
\tag 5.2
$$
The just mentioned facts are all is really needed to make our approach work.
Section three remains the same since the
distortion estimates depend only on the behavior of the neutral fixed point
which is ensured by our assumption on $D^2T$.
Accordingly section four follows exactly in the same way yielding a polynomial
decay depending on $\alpha$ and $\beta$.

In conclusion, one can obtain a polynomial decay of correlations for a large
class of maps provided (5.1) holds. Here we do not address how to obtain such
an estimate on the invariant density, although several approaches (besides the
ones in the style of what we have done in section two) are possible (see, for
example, \cite{Th} where a result of the type (5.1) is obtained by inducing).

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