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\begin{document}
\bibliographystyle{plain}
\title {Convergence of the natural approximations of piecewise monotone
interval maps}
\date{}
\author{Nicolai Haydn \thanks{Mathematics Department, University of Southern California,
Los Angeles, 90089-1113. Email:$<$nhaydn@mtha.usc.edu$>$.}}
\maketitle
\noindent{\bf Abstract:}
We consider piecewise monotone interval mappings which are topologically mixing
and satisfy the Markov property. It has previously been shown that the invariant
densities of the natural approximations converge exponentially fast in
uniform pointwise topology to the
invariant density of the given map provided it's derivative is piecewise
Lipshitz continuous. In this paper we show that in general one does not
obtain exponential convergence in the bounded variation norm.
Here we prove that if the derivative of the interval map is
H\"{o}lder continuous
and its variation is well approximable ($\gamma$-uniform variation for
$\gamma>0$), then the densities converge exponentially fast in the
norm.
\section{Introduction}
The dynamics of (mixing) expanding piecewise monotone maps has been extensively
studied,
in particular the spectral properties of the associated transfer operators
have been established by notably Hofbauer and Keller \cite{HK} \cite{K}.
On the
Banach space of functions of bounded variations the transfer operator
has a simple largest real eigenvalue whose eigenfunction is the density
of the invariant measure. The remainder of the spectrum has then strictly
smaller radius which in particular implies that the correlation functions
decay exponentially
fast. For a piecewise affine map which satisfies the Markov property the transfer
operator can be written as a matrix, which makes is more practical to
find its invariant density \cite{U}. It has been shown in \cite{BIS} that
the densities of affine approximations of an expanding piecewise monotone map
indeed converge exponetially fast pointwise uniformly to the actual
density of the invariant
measure if the map has (piecewise) Lipshitz continuous derivatives.
In section 3 we provide an example which shows that under these conditions
the convergence cannot in general expected to be exponential in the bounded
variation norm. However, our main result, theorems \ref{UV-convergence}
and \ref{BV},
shows that the approximating densities convergence in the norm exponentially
fast if one assumes that the variation of the map's derivative can sufficiently
well be approximated by variations over partitions (the derivative has
uniform variation---see below).
Let $T$ be a piecewise $C^{1+\gamma}$ continuous map (that is its first derivative
is $\gamma$-H\"{o}lder continuous) of the unit interval
$[0,1]$ to itself such that its restriction to the atoms of a finite
partition ${\cal A}=\{(a_{j-1},a_j): j=1,\dots J\}$ are monotone with
H\"{o}lder continous derivative $T'$, where $0=a_0\tau$, where $\mu$ denotes the Lebesgue measure on
$[0,1]$ and $\|\phi\|=|\phi|_{\infty}+{\rm var}_{[0,1]}\phi$
is the usual BV-norm.
\begin{definition}
The {\em $n$th piecewise affine approximation} of $T$ is the transform
$T_n:[0,1]\rightarrow[0,1]$ such that $T|_A$ is affine for all
$A\in{\cal A}^n=\bigvee_{j=0}^{n-1}T^{-j}{\cal A}$ (the $n$-th
join) and matches with $T$ at the endpoints of the atoms $A$.
\end{definition}
\noindent Denote by ${\cal L}_n$ the transfer operator for the $n$-th
affine approximation:
$$
{\cal L}_n\phi(x) = \sum_{y\in T_n^{-1}x} \frac{\phi(y)}{|T_n'(y)|},
\;\; \phi\in{\rm BV}.
$$
Clearly $1$ is a simple eigenvalue of ${\cal L}_n$ with strictly
positive eigenfunction $h_n$. Assume $h_n$ is normalised.
\begin{theorem} \label{BIS}
Let $T$ be a piecewise monotone interval map which satisfies the conditions
(i)--(iv) (for some $\gamma>0$).
There exists a constant $C_1$ such that
$$
{\rm var}\, h_n \leq C_1,\,\, \forall n.
$$
For every $\tau'>\tau^{1/4}$ there exists a constant $C_2$ such that
$$
|h_n-h|_{\infty}\leq C_2\tau'^n,\;\; n=1,2,3,\dots.
$$
\end{theorem}
\noindent This theorem is proven in \cite{BIS} for $\tau'>\tau^{1/6}$.
However a simplification of \cite{BIS} lemma 2.2 yields the Ionescu
Tulcea-Marinescu type inequality
\begin{equation}\label{Ionescu}
\|({\cal L}^k-{\cal L}_n^k)\phi\|_{\rm BV}
\leq {\rm const.}\,|\phi|_{\infty}+\vartheta^k{\rm var}\,\phi,
\end{equation}
for all $k=0,1,2,\dots,n$ (instead of $k<\frac{2}{3}n$) which then
(in \cite{BIS} lemma 2.3) implies the improved lower bound
$\tau^{1/4}$ for $\tau'$.
We shall prove that the convergence is indeed exponential in the BV-norm
if $T$ satisfies the stronger property of having uniform variation.
In section 3 we shall then produce an example of a piecewise $C^{1+L}$
map (i.e.\ the derivative of the map is piecewise Lipshitz continuous)
for which
$
{\rm var}\,(h-h_{n_j})\geq\frac{{\rm const.}}{\log n_j}
$
for a suitable sequence of integers $n_j$. That is, for that example
the convergence of the densities is very slow in the bounded variation
norm.
If a finite partition $\cal P$ of the unit interval consists of the
intervals between the points $0=p_00$ if there exists a constant $U$
so that
$$
{\rm Var}\,(\phi,{\cal P})\leq U({\rm diam}\, {\cal P})^{\gamma},
$$
for all partitions $\cal P$ of $[0,1]$.
We shall put $U_{\gamma}(\phi)$ for the smallest possible choice of the
constant $U$.
\end{definition}
\noindent If $f$ and $g$ are functions on the unit interval with
finite $U_{\gamma}$ values, then it is easily seen that
$U_{\gamma}(f+g)\leq U_{\gamma}(f)+U_{\gamma}(g)$ and
$U_{\gamma}(cf)=|c|U_{\gamma}(f)$ for constants $c$. For constant functions
$c$ we have $U_{\gamma}(c)=0$. Moreover
$U_{\gamma}(fg)\leq|g|_{\infty}U_{\gamma}(f)+|f|_{\infty}U_{\gamma}(g)$ .
We can therefore define a norm $\|\cdot \|_{\gamma}$ by
$$
\|\phi\|_{\gamma}=|\phi|_{\infty}+U_{\gamma}(\phi),
$$
and introduce for some partition $\cal P$ of the unit interval the function space
$$
{\rm UV}_{\gamma}=
\{\phi\in C^{\gamma}({\cal P}): \;\|\phi\|_{\gamma}<\infty\}
$$
of functions with $\gamma$-uniform variation, where $C^{\gamma}({\cal P})$
consists of all functions on $[0,1]$ which are $\gamma$-H\"{o}lder continuous
on the atoms $\cal P$. The space $({\rm UV}_{\gamma},\|\cdot\|_{\gamma})$
is a Banach space. Clearly ${\rm UV}_{\gamma}\subset{\rm UV}_{\gamma'}$
for ${\gamma'}\leq{\gamma}$, and moreover
${\rm UV}_{\gamma}\subset\mbox{\rm BV}$ for all positive $\gamma$.
(Naturally $U_{\gamma'}(\phi)\leq U_{\gamma}(\phi)$ if $\gamma'\leq\gamma$.)
Continuous functions in ${\rm UV}_{\gamma}$ are not necessarily
$\gamma$-H\"{o}lder continuous as the example $\phi=x^{\gamma/2}$,
$x\in[0,1]$ shows. Since $\phi$ is increasing
${\rm var}\,\phi={\rm var}\,(\phi,{\cal P})$ and therefore
${\rm Var}\,(\phi,{\cal P})=0$ for every partition $\cal P$ of the
unit interval. Thus $\phi\in{\rm UV}_{\gamma}$ although $\phi$ is not
$\gamma$-H\"{o}lder continuous.
Let us denote by $S_k$ the inverse branches of $T^k$ and put $A_{\varphi}$
for the range of the inverse branches $\varphi\in S_k$. Note that
$A_{\varphi}\in{\cal A}^k$.
\begin{lemma}
Let $T$ be a transformation of the unit interval to itself.
Assume $|T'|\in {\rm UV}_{\gamma}$ ($\gamma>0$), then $\cal L$ maps
${\rm UV}_{\gamma}$ into itself.
\end{lemma}
\noindent {\bf Proof.} Clearly $|{\cal L}\phi|_{\infty}\leq|\phi|_{\infty}$.
To estimate the variation we proceed as follows
\begin{eqnarray}
{\rm Var}\,({\cal L}\phi,{\cal P})
&=&\left|{\rm var}\,{\cal L}\phi-{\rm var}\,({\cal L}\phi,{\cal P})\right|
\nonumber\\
&\leq&\sum_{\varphi\in S_1}\left|
{\rm var}\,\frac{\phi\varphi}{|T'\varphi|}-
{\rm var}\,\left(\frac{\phi\varphi}{|T'\varphi|},{\cal P}\right)
\right|\nonumber\\
&\leq&|S_1|\,{\rm Var}\,\left(\frac{\phi}{|T'|},T^{-1}{\cal P}\right)\nonumber.
\end{eqnarray}
This implies that $\|{\cal L}\phi\|_{\gamma}\leq c_1\|\phi\|_{\gamma}$ for some
constant $c_1$.\hfill$\Box$
\vspace{3mm}
\noindent For the transfer operator and its affine approximations one
can now deduce inequality (\ref{Ionescu}) on the space ${\rm UV}_{\gamma}$
for $\gamma>0$ (an explicit proof can be found by picking one's way
through the proof of theorem \ref{BV}). It follows that the approximating
densities $h_n$ in fact lie in ${\rm UV}_{\gamma}$ for all $\gamma>0$.
The following theorem, which is proven in the following section, asserts
that convergence of the approximating densities is in fact exponential
in the spaces of $\gamma$-uniform variation.
\begin{theorem}\label{UV-convergence}
Let $T$ be a transformation of the unit interval to itself so that
$|T'|\in {\rm UV}_{\gamma}$ ($\gamma>0$). Then there exists a number
$\sigma\in (0,1)$ and a constant $C_3$ so that
$$
\|h-h_n\|_{\gamma}\leq C_3\sigma^n,
$$
for all $n=1,2,\dots$.
\end{theorem}
\begin{theorem}\label{BV}
Let $T$ be a transformation of the interval which satisfies the conditions
(i)--(iv) and let $\cal L$ be the associated transfer operator acting
on the Banach space of functions of bounded variation. If we assume that
$T'$ has $\gamma$-uniform variation for some positive $\gamma$, then there
exists a constant $C_4$ and a $\sigma\in(0,1)$ so that
$\|h-h_n\|_{\rm BV}\leq C_4\sigma^n\;\;\;\forall\,n$.
In particular $\,{\rm var}\,(h-h_n)\leq C_4\sigma^n$, for all $n$.
\end{theorem}
\vspace{3mm}
\noindent The proof of these two theorems is the subject of the
following section.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%% Proof of Convergence theorems
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convergence}
\begin{lemma}\label{variation}
There exists a constant $C_5$ so that for every interval $A\subset[0,1]$
on which $T^k$ is one-to-one:
\noindent (I) For every partition $\cal P$ of $[0,1]$ one has
$$
{\rm Var}_A\,\left(\frac{1}{|(T^k)'|}, {\cal P}\right)
\leq C_5 \sup_A \frac{1}{|(T^k)'|}({\rm diam}\,T^k{\cal P})^{\gamma}.
$$
\noindent (II)
$$
{\rm var}_A\,\frac{1}{|(T^k)'|} \leq kC_5 \sup_A \frac{1}{|(T^k)'|}.
$$
\end{lemma}
\noindent {\bf Proof.} Let $\cal Q$ be a partition of $A$.
Assume $\cal Q$ is given by the points $y_0\vartheta$ there exists a constant $C_6$ so that
for all $k\vartheta$ then for any
$\vartheta''\in(\vartheta,\vartheta')$ we can find $c_1$ so that
$\frac{1}{|(T^k)'|}\leq c_1\vartheta''^k$ for all $k$. Since $T^k$
is one-to-one on the atoms of ${\cal A}^k$ we conclude that
${\rm diam}\,{\cal A}^k\leq c_1\vartheta''^k$ for
all $k$. Now let $k1$),
and define a sequence of integers $n(m)=[\beta^m]$ ($[\cdot]$ denotes
integer part). That is $m\sim\log_\beta n(m)$. The sequence of positive numbers
$\alpha_m=\vartheta^{n(m)}$, $m=1,2,\dots$, is at least
exponentially fast decreasing to zero ($\alpha_m\sim\vartheta^{\beta^m}$).
Note that $\vartheta'^{n(m)/2}\leq\alpha_{m-1}$.
Suppose we already constructed the maps $V_1,V_2,\dots,V_{m-1}$, and
let us now proceed to find $V_m$. First, denote by $H_j$ the union of
(open) intervals on which $V_j'$ is constant (in fact equal to $P$).
Then $H_{m-1}$ is the disjoint union of (finitely many) intervals, say
$I_{m-1,1},I_{m-1,2},\dots$. Each of these intervals $I_{m-1,\ell}$
is divided into $[|I_{m-1,\ell}|\frac{1}{m\alpha_m}]+1$ ($|\cdot|$ denotes
the length of the interval) many pieces and on
the middle of each of these pieces we replace the constant value $P$
of $V_{m-1}'$ by a {\em squiggle} (the left and right thirds both have
slope $1$ while the middle third has slope $-2$)
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\noindent of length $\alpha_m$ (the length of a squiggle is the euclidean distance
between the left and right endpoints). In this way we find the derivative
$V_m'$ of the map $V_m$. This completes the construction of $V_m$.
Notice that $V_m(x)=V_{m-1}(x)$ for $x$ outside the squiggles which were
introduced at the step $m$ (i.e.\ for
$x\in[0,1]\setminus(H_m\setminus H_{m-1})$).
For the lebesgue measure (sum of the lengths of subintervals) of the
`linear set' $H_m$ of $V_m$ (i.e.\ $H_m=\{x\in[0,1]: V_m'(x)=P\}$) we obtain
\begin{eqnarray}
|H_m|&=&|H_{m-1}|-\sum_{\ell}
\left(\left[|I_{m-1,\ell}|\frac{1}{m\alpha_m}\right]+1\right)\alpha_m\nonumber\\
&\geq& |H_{m-1}|(1-\frac{1}{m}) \nonumber\\
&=& |H_{m-1}|\frac{m-1}{m}\nonumber\\
&\geq& |H_1|\frac{1}{m}.\nonumber
\end{eqnarray}
Thus $|H_m|\sim 1/m$. Let us furthermore observe that the number $M_m$
of squiggles of length $\alpha_m$ is
\begin{equation}\label{squiggles}
M_m\sim|H_m|\frac1{m\alpha_m}\sim\frac{\vartheta^{-\beta^m}}{m^2}
\sim\frac{\vartheta^{-n}}{m^2},
\end{equation}
and the total length of the $\alpha_m$-squiggles
is $|H_{m-1}\setminus H_m|\sim\frac1{m^2}$.
It follows from the construction that all the maps $V_m$ are $C^{1+L}$
on the atoms of $\cal A$
(in fact $2$ is a Lipshitz constant for all the derivatives $V_m'$),
and their derivatives satisfy $1/\vartheta\leq V_m'\leq1/\vartheta'$.
By Arzela-Ascoli the maps $V_m$ converge in $C^1$ to a limit $T$
which is piecewise of class $C^{1+L}$.
Clearly, $T$ has the Markov property and is topologically mixing (in fact
$N=1$). It thus satisfies the conditions (i)--(iv) (with $\rho=\vartheta'$).
If ${\cal A}^n=\bigvee_{j=0}^{n-1}T^{-j}{\cal A}$ denotes the $n$-th join of
$\cal A$, then $\vartheta'^{-n}\leq|{\cal A}^n|\leq\vartheta^{-n}$ (cardinality)
and
$\vartheta^n\leq|A|\leq\vartheta'^n$ (length) for every atom $A\in{\cal A}^n$.
\vspace{2mm}
\noindent (ii)
Let us next estimate ${\rm Var}_{T^{k-1}A}\,(\log T',{\cal R}_{k-1})$,
for $A\in{\cal P}^k$, $k1$)
$$
{\rm Var}_{T^{k-1}A}\,(\log T',{\cal R}_{k-1})
={\rm Var}\,(\log T',{\cal R}_{k-1}).
$$
If we put $k=[n/2]$ and $n=n(m)$, then $|{\cal A}^{n-k}|\leq M_m$
by (\ref{squiggles}) for large enough $m$.
We can therefore conclude that the total length
of those squiggles of lengths $\leq\alpha_m$ which are subsets of single
atoms of the partition ${\cal R}_{k-1}$ is at least $|H_{m+1}|\sim\frac1{m+1}$.
Every such squiggle contributes $\frac43$-times its length to the variation
${\rm Var}\,(\log T',{\cal R}_{k-1})$. Hence
\begin{equation}\label{lower}
{\rm Var}\,(\log T',{\cal R}_{k-1})\geq\frac43|H_{m+1}|\geq\frac1m.
\end{equation}
\vspace{2mm}
\noindent (iii) Now let $\ell1$ large enough):
$$
N_m=\sum_{k=2}^{m-1}M_k
\sim\sum_{k=2}^{m-1}\frac1{k^2\alpha_k}\leq \frac1{m\alpha_{m-1}}.
$$
On the other hand for squiggles $\leq\alpha_m$ we get as before
\begin{equation}\label{second}
{\rm Var}_{T^{\ell}A\cap H_m}\,(\log T',{\cal R}_{\ell})
\leq\frac43|T^{\ell}A|\cdot|H_m|\sim\frac4{3m}|T^{\ell}A|
\end{equation}
(the variation of a squiggle is $4/3$ times its length).
As $|T^{\ell}A|\leq\vartheta'^{k-1-\ell}$ if $A\in{\cal A}^k$, we
now obtain from equations (\ref{first}) and (\ref{second})
\begin{eqnarray}
{\rm Var}_{T^{\ell}A}\,(\log T',{\cal R}_{\ell})
&\leq&\frac4{3m}|T^{\ell}A|+8|T^{\ell}A|N_m\vartheta'^{n-\ell}
\nonumber\\
&\leq&\vartheta'^{k-1-\ell}\frac1m
\left(\frac43+\frac{4\vartheta'^{n-\ell}}{\alpha_{m-1}}\right),\nonumber
\end{eqnarray}
as $N_m\leq1/m\alpha_{m-1}$. Hence, since by assumption
$\vartheta'^{n-\ell}\alpha_{m-1}^{-1}\leq1$, (as $\ell