\newtheorem{theorem}{Theorem}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}


\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 $0=a_0<a_1<a_2<\cdots<a_J=1$.
We shall furthermore make the following assumptions:

\noindent (i) There exists a constant $\rho<1$ so that $|T'|\geq\rho^{-1}$ on every
atom $A\in{\cal A}$.

\noindent (ii) At the endpoints $a_j$ of the atoms of $\cal A$ the map $T$
shall assume one or the other value of the continuous extensions of $T$
to the closures of the adjacent intervals.

\noindent (iii) $T$ has the Markov property, that is if 
$T(A)\cap A' \not= \emptyset$ for some $A,A'\in{\cal A}$ then
$A'\subset T(A)$.

\noindent (iv) We require that $T$ is topologically mixing, that is, there
is a positive integer $N$ so that the branches of monotonicity of $T^N$
are onto $[0,1]$. Without loss of generality we shall assume that $N=1$.

We can now define the transfer operator $\cal L$ by:
$$
{\cal L}\phi(x) = \sum_{y\in T^{-1}x} \frac{\phi(y)}{|T'(y)|},
$$
for suitable functions $\phi:[0,1]\rightarrow{\bf C}$.
On the space BV of functions with bounded variation 
${\cal L}:{\rm BV}\rightarrow{\rm BV}$, the spectral radius of $\cal L$ 
is equal to $1$, and $1$ is a simple
eigenvalue with a strictly positive eigenfunction $h$ which we assume to
be normalised.  Moreover the essential spectral
radius $\vartheta<1$, where
$$
\vartheta=\limsup_{n\rightarrow\infty}\frac{1}{|(T^n)'|^{1/n}}.
$$
If we put 
$$
\tau=\max\{|\lambda|:\lambda\not= 1\;\; {\rm in\;the\;spectrum\;of}\;\;
{\cal L}:{\rm BV}\rightarrow{\rm BV}\},
$$
(spectral gap), then the convergence of $\cal L$ to the eigenspace
spanned by $h$ is exponential at rate $\tau<1$, that is
$$
\|{\cal L}^n\phi-h\mu(\phi)\|\leq {\rm const.}\tau'^n\|\phi\|,
$$
for every $\tau'>\tau$, 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 $0=p_0<p_1<p_2<\cdots<p_m=1$, then
we put 
$$
{\rm var}\,(\phi,{\cal P}) = \sum_{j=1}^m |\phi(p_{j-1}-\phi(p_j)|
$$
for the variation of the function $\phi$ (if $\phi$ is continuous on
$[0,1]$) with respect to the partition $\cal P$. If $\phi$ is piecewise
continuous we take the variation over the intervals of piecewise
continuity.
Clearly, ${\rm var}\,\phi = \sup_{\cal P}{\rm var}(\phi,{\cal P})$,
where the supremum is over all partitions $\cal P$ of the unit interval.
(Observe that ${\rm var}\,\phi = \,{\rm var}(\phi,{\cal P})$, for all
partitions $\cal P$, if $\phi$ is monotonically increasing or decreasing.)
Put
$$
{\rm Var}\,(\phi,{\cal P})
=\left|{\rm var}\,\phi-{\rm var}\,(\phi,{\cal P})\right|,
$$
and denote by ${\rm diam}\,{\cal P}=\max_j (p_j-p_{j-1})$ the {\em diameter}
of the partition $\cal P$. 

\begin{definition}
We say a function $\phi$ on the unit interval has {\em $\gamma$-uniform
variation} for some $\gamma>0$ 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 $y_0<y_1<\cdots<y_Q$.
As $|(T^k)'|$ is (uniformly in $k$) $\gamma$-H\"{o}lder continuous,
the variation over the partition $\cal Q$ is then
\begin{eqnarray}
{\rm var}\,\left(\frac{1}{|(T^k)'|},{\cal Q}\right)
 &=&\sum_{j=1}^Q\left|\frac{1}{|(T^k)'(y_{j-1})|}-\frac{1}{|(T^k)'(y_j)|}\right|
\nonumber\\
&=& \sum_{j=1}^Q\frac{1}{|(T^k)'(y_j)|}
\left|1-\frac{|(T^k)'(y_j)|}{|(T^k)'(y_{j-1})|}\right|\nonumber\\
&=& \sum_{j=1}^Q\frac{1+{\cal O}(({\rm diam}\,{\cal Q})^{\gamma})}{|(T^k)'(y_j)|}
\left|\log\frac{|(T^k)'(y_j)|}{|(T^k)'(y_{j-1})|}\right|\label{*}.
\end{eqnarray}
Hence, letting ${\rm diam}\,{\cal Q}$ go to zero, one obtains
\begin{eqnarray}
{\rm var}\,\frac{1}{|(T^k)'|}
&\leq&\sup_A\left|\frac{1}{(T^k)'}\right|\,{\rm var}_A\,\log|(T^k)'|\nonumber\\
&\leq&\sup_A\left|\frac{1}{(T^k)'}\right|
\sum_{\ell=0}^{k-1}\,{\rm var}_{T^{\ell}A}\,\log|T'|\nonumber\\
&\leq&\sup_A\left|\frac{k}{(T^k)'}\right|\,{\rm var}\,\log|T'|.\nonumber
\end{eqnarray}
This proves the statement (II) of the lemma.

To prove the inequality (I), let $\cal P$ be some partition. Equation
\ref{*} yields
$$
{\rm var}\,\frac{1}{|(T^k)'|}
\leq\sum_{P\in{\cal P}}\sup_{P\cap A}\left|\frac{1}{(T^k)'}\right|
{\rm var}_{P\cap A}\,\log|(T^k)'|
$$
and
$$
{\rm var}\,\left(\frac{1}{|(T^k)'|},{\cal P}\right)
\geq
\left(1-{\cal O}(({\rm diam}\,{\cal P})^{\gamma})\right)
\sum_{P\in{\cal P}}\inf_{P\cap A}\left|\frac{1}{(T^k)'}\right|
{\rm var}_{P\cap A}\,\left(\log|(T^k)'|,{\cal P}\right).
$$
Now, since
$$
\inf_{P\cap A}\left|\frac{1}{(T^k)'}\right|
\geq\left(1-{\cal O}(({\rm diam}\,{\cal P})^{\gamma})\right)
\sup_{P\cap A}\left|\frac{1}{(T^k)'}\right|,
$$
for $P\in{\cal P}$, we can finally estimate as follows:
\begin{eqnarray}
{\rm Var}\,\left(\frac{1}{|(T^k)'|},{\cal P}\right)
&\leq&\sum_{P\in{\cal P}}\left(
\sup_{P\cap A}\left|\frac{1}{(T^k)'}\right|{\rm var}_{P\cap A}\,\log|(T^k)'|
\right \nonumber\\
& &\hspace{1in}\left -\inf_{P\cap A}\left|\frac{1}{(T^k)'}\right|
{\rm var}_{P\cap A}\,\left(\log|(T^k)'|,{\cal P}\right)\right)
\nonumber\\
& &+c_1({\rm diam}\,{\cal P})^{\gamma}\sup_A\left|\frac{1}{(T^k)'}\right|
{\rm var}_A\,\left(\log|(T^k)'|,{\cal P}\right)\nonumber\\
&\leq& \sup_A\left|\frac{1}{(T^k)'}\right|
\left({\rm Var}_A\,\left(\log|(T^k)'|,{\cal P}\right)
+c_2({\rm diam}\,T^k{\cal P})^{\gamma}\right)\nonumber\\
&\leq& C_5\sup_A\left|\frac{1}{(T^k)'}\right|
({\rm diam}\,T^k{\cal P})^{\gamma}\nonumber,
\end{eqnarray}
for some constants $c_1,c_2$. In the last two inequalities we used that
with some constant $c_3$ (observe that $\log |T'|\in{\rm UV}_{\gamma}$)
$$
{\rm Var}_A\,\left(\log|(T^k)'|,{\cal P}\right)
\leq\sum_{\ell=1}^{k} 
{\rm Var}_{T^kA}\,\left(\log |T'|, T^{\ell}{\cal P}\right)
\leq c_3({\rm diam}\,T^k{\cal P})^{\gamma}\frac{1}{1-\rho},
$$
since  $T^k$ is one-to-one on $A$ and
${\rm diam}\,T^{\ell}{\cal P}\leq\rho^{k-\ell}{\rm diam}\,T^k{\cal P}$.
Similarly
$
{\rm var}_A\,\left(\log|(T^k)'|,{\cal P}\right)
\leq \frac{c_3}{1-\rho}({\rm diam}\,T^k{\cal P})^{\gamma}.
$

Put $C_5=\max(c_2+c_3/(1-\rho),\,{\rm var}\,\log|T'|)$.\hfill$\Box$

\vspace{3mm} 


\noindent  Let $S_k$ denote the inverse branches of $T^k$. If $\varphi\in S_k$,
then we denote its range by $A_{\varphi}$ which in fact is an atom in 
${\cal A}^k$.
If $k<n$ then every inverse branch $\varphi\in S_k$ 
has a corresponding inverse branch $\tilde\varphi$ of $T_n^k$ whose
range $A_{\tilde\varphi}\in{\cal A}^k$ coincides with $A_{\varphi}$.

\begin{lemma}\label{inequality}
For every $\vartheta'>\vartheta$ there exists a constant $C_6$ so that
for all $k<n$:
$$
\left|1-\frac{|(T_n^k)'\tilde\varphi|}{|(T^k)'\varphi|}\right|_{\infty}
\leq nC_6\vartheta'^{n-k}.
$$
\end{lemma}

\noindent {\bf Proof.} Given $\vartheta'>\vartheta$ 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 $k<n$ and observe that $\varphi(x)=\tilde\varphi(x)$ 
on the endpoints $x$ of atoms in ${\cal A}^{n-k}$. 
Therefore, by the mean value theorem,
\begin{eqnarray}
\left|1-\frac{|(T_n^k)'\tilde\varphi|}{|(T^k)'\varphi|}\right|_{\infty}
&\leq&c_2 
\left|\log\frac{|(T_n^k)'\tilde\varphi|}{|(T^k)'\varphi|}\right|_{\infty}
\nonumber\\
&\leq&c_2 \max_{B\in{\cal A}^{n-k}}\left(\,{\rm var}_{T^{k}B}\,\log|(T^k)'|
+\,{\rm var}_{T_n^{k}B}\,\log|(T_n^k)'|\right)
\nonumber\\
&\leq&c_2 \max\log|(T^k)'|\,2\,{\rm diam}\,{\cal A}^{n-k}
\nonumber\\
&\leq&(n-k)c_3\vartheta''^{n-k}
\nonumber\\
&\leq&nC_6\vartheta'^{n-k},
\nonumber
\end{eqnarray}
for some constants $c_2, c_3$ and $C_6$.\hfill$\Box$


\vspace{3mm} 

\noindent {\bf Proof of theorem \ref{UV-convergence}.} We have to show
that there exists a constant $\sigma<1$ so that the conditional 
variation 
${\rm Var}(h-h_n,{\cal P})\leq	 c_1\sigma^n({\rm diam}\,{\cal P})^{\gamma}$
for all partitions $\cal P$ of $[0,1]$ and some constant $c_1$.
 Assume $k<n$ and given an inverse branch $\varphi\in S_k$ let, as above,
 $\tilde\varphi$ be the associated inverse branch of $T_n^k$ which has the same
range $A_{\varphi}=A_{\tilde\varphi}\in{\cal A}^k$.
Let $\cal P$ be a partition of the unit interval. Then, for any $k$,
\begin{eqnarray}
{\rm Var}(h-h_n,{\cal P}) 
&=& {\rm Var}({\cal L}^kh-{\cal L}_n^kh_n,{\cal P})\nonumber\\
&=& {\rm Var}\left(\sum_{\varphi\in S_k}
\left(\frac{h\varphi}{|(T^k)'\varphi|}
-\frac{h_n\tilde\varphi}{|(T_n^k)'\tilde\varphi|}\right),
{\cal P}\right)\nonumber\\
&\leq& \mbox{\rm Var}\left(\sum_{\varphi\in S_k}
\frac{1}{|(T_n^k)'\varphi|}(h\varphi-h_n\tilde\varphi), {\cal P}\right)
\nonumber\\
&&+{\rm Var}\left(\sum_{\varphi\in S_k} h_n\tilde\varphi
\left(\frac{1}{|(T_n^k)'\tilde\varphi|}-\frac{1}{|(T^k)'\varphi|}\right), {\cal P}\right)
\nonumber\\
&\leq&I+II+III+IV,\nonumber
\end{eqnarray}
where $S_k$ are the inverse branches of $T^k$ and their ranges are
the atoms of ${\cal A}^k$.
Let us now estimate the four parts separately.

The first term can be bounded as follows:
\begin{eqnarray}
I&=&\sum_{\varphi\in S_k}\left|\frac{1}{(T_n^k)'\varphi}\right|_{\infty}
{\rm Var}_{T^kA_{\varphi}}\left(h\varphi-h_n\tilde\varphi, 
{\cal P}\right)\nonumber\\
&\leq&\vartheta'^k\sum_{\varphi\in S_k}
{\rm Var}_{A_{\varphi}}\left(h-h_n\hat\varphi_0, T^{-k}{\cal P}\right)\nonumber\\
&\leq&\vartheta'^k
({\rm Var}\,(h,T^{-k}{\cal P})+{\rm Var}\,(h_n,T_n^{-k}{\cal P}))
\nonumber\\
&\leq&c_2\vartheta'^k\rho^{\gamma k}({\rm diam}\,{\cal P})^{\gamma},\nonumber
\end{eqnarray}
where $c_2\leq {\rm U}_{\gamma}(h)+{\rm U}_{\gamma}(h_n)$,
 $\hat\varphi_0=\tilde\varphi T^k$ is a homeo from $T^kA_{\varphi}$
to itself, and $\vartheta'\in(\vartheta,1)$ ($k$ large enough).
We used that ${\rm diam}\,T^{-k}{\cal P}\leq\rho^k{\rm diam}\,{\cal P}$
and ${\rm diam}\,T_n^{-k}{\cal P}\leq\rho^k{\rm diam}\,{\cal P}$.

For the second term we get with the help of lemma \ref{variation}(I):
\begin{eqnarray}
II&=&\sum_{\varphi\in S_k}|h\varphi-h_n\tilde\varphi|_{\infty}
{\rm Var}_{T^kA_{\varphi}}\,\left(
\frac{1}{|(T_n^k)'\tilde\varphi|},{\cal P}\right)
\nonumber\\
&\leq&c_3\kappa^n\sum_{\varphi\in S_k}
{\rm Var}_{A_{\varphi}}\,\left(
\frac{1}{|(T_n^k)'|},T_n^{-k}{\cal P}\right)\nonumber\\
&\leq&c_3C_5c_4\kappa^n({\rm diam}\,{\cal P})^{\gamma}
\sum_{\varphi\in S_k}\frac{1}{|(T_n^k)'\tilde\varphi x|}
\nonumber\\
&\leq&c_3C_5c_4\kappa^n({\rm diam}\,{\cal P})^{\gamma}h_n(x),
\nonumber
\end{eqnarray}
where we have used that $T^kA_{\varphi}=[0,1]$ (mixing property)
for all inverse branches $\varphi\in S_k$, and 
$\sup_{A_\varphi}|1/(T_n^k)'|\leq c_4/|(T_n^k)'(y)|$ for some constant $c_4$
and any $y$ which we choose to be equal to $\tilde\varphi x$. Hence
$II\leq c_5\kappa^n({\rm diam}\,{\cal P})^{\gamma}$ for some $c_5$.
 

Next we estimate the third term as follows
$$
III=\sum_{\varphi\in S_k}
\left|1-\frac{|(T_n^k)'\tilde\varphi|}{|(T^k)'\varphi|}\right|_{\infty}
{\rm Var}_{T^kA_{\varphi}}\,
\left(\frac{h_n\tilde\varphi}{|(T_n^k)'\tilde\varphi|},{\cal P}\right),
$$
where the first term in the sum was estimated in lemma \ref{inequality}.
We thus  obtain with lemma \ref{variation}(I) (applied to the map $T_n$ 
instead of $T$)
\begin{eqnarray}
III&\leq&nC_6\vartheta'^{n-k}\sum_{\varphi\in S_k}
{\rm Var}_{T^kA_{\varphi}}\,\left(
\frac{h_n\tilde\varphi}{|(T_n^k)'\tilde\varphi|},{\cal P}\right)
\nonumber\\
&\leq&nC_6\vartheta'^{n-k}\sum_{\varphi\in S_k}
{\rm Var}_{A_{\varphi}}\,\left(
\frac{h_n}{|(T_n^k)'|},T_n^{-k}{\cal P}\right)\nonumber\\
&\leq&nC_6\vartheta'^{n-k}\sum_{\varphi\in S_k}\left(
\left|\frac{1}{(T_n^k)'}\right|_{\infty}{\rm Var}_{A_{\varphi}}\,
(h_n,T_n^{-k}{\cal P})\right \nonumber\\
&&\hspace{1in}\left +|h_n|_{\infty}{\rm Var}_{A_{\varphi}}\,
\left(\frac{1}{|(T_n^k)'|},T_n^{-k}{\cal P}\right)\right)
\nonumber\\
&\leq&nC_6\vartheta'^{n-k}\left(
\vartheta'^k{\rm Var}\,
(h_n,T_n^{-k}{\cal P})
+|h_n|_{\infty}c_4C_5|h_n|_{\infty}({\rm diam}\,{\cal P})^{\gamma}\right)
\nonumber\\
&\leq&nc_6\vartheta'^{n-k}({\rm diam}\,{\cal P})^{\gamma}\nonumber,
\end{eqnarray}
for some constant $c_6$.

While in the first three expressions we obtain exponential contraction
even if $\gamma=0$, in the last term the assumption that $|T'|$ has 
uniform variation is essential even in the bounded variation (BV-norm)
case (see theorem \ref{BV}). We have
$$
IV=\sum_{\varphi\in S_k}\left|
\frac{h_n}{(T_n^k)'}\circ\tilde\varphi\right|_{\infty}
{\rm Var}_{T^kA_{\varphi}}\,
\left(1-\frac{|(T_n^k)'\varphi|}{|(T^k)'\tilde\varphi|},{\cal P}\right).
$$
We estimate the term with the variation as follows:
\begin{eqnarray}
{\rm Var}_{T^kA_{\varphi}}\,
\left(1-\frac{|(T_n^k)'\tilde\varphi|}{|(T^k)'\varphi|},{\cal P}\right)
&=&{\rm Var}_{A_{\varphi}}\,
\left(1-\frac{|(T_n^k)'\hat\varphi_0|}{|(T^k)'|},T^{-k}{\cal P}\right)\nonumber\\
&\leq&c_7{\rm Var}_{A_{\varphi}}\,
\left(\log\frac{|(T^k)'\hat\varphi_0|}{|(T_n^k)'|},T^{-k}{\cal P}\right)\nonumber\\
&\leq&c_7\sum_{\ell=0}^{k-1}{\rm Var}_{A_{\varphi}}\,
\left(\log\frac{|T'T^{\ell}|}{|T_n'T_n^{\ell}\hat\varphi_0|},T^{-k}{\cal P}
\right)\nonumber\\
&=&c_7\sum_{\ell=0}^{k-1}{\rm Var}_{T^{\ell}A_{\varphi}}\,
\left(\log\frac{|T'|}{|T_n'\hat\varphi_{\ell}|},
T^{\ell-k}{\cal P}\right)\nonumber,
\end{eqnarray}
for some constant $c_7$, where 
$\hat\varphi_{\ell}=T_n^{\ell}\tilde\varphi T^{k-\ell}$ is a homeomorphism 
of $T^{\ell}A_{\varphi}$ to itself. We estimate the terms inside the sum
individually as follows (for some $c_8, c_9,c_{10}$):
\begin{eqnarray}
\sum_{\varphi\in S_k}\left|
\frac{h_n}{(T_n^k)'}\circ\tilde\varphi\right|_{\infty}
{\rm Var}_{T^{\ell}A_{\varphi}}\,
\left(\log\frac{|T'|}{|T_n'\hat\varphi_{\ell}|},T^{\ell-k}{\cal P}\right)
\nonumber\hspace{-3in}\\
&\leq&\sum_{\varphi\in S_k}\left|
\frac{h_n}{(T_n^k)'}\circ\tilde\varphi\right|_{\infty}
c_8\left({\rm Var}_{T^{\ell}A_{\varphi}}\,
\left(\log|T'|, T^{\ell-k}{\cal P}\right)\right \nonumber\\
&&\hspace{1.5in}\left +{\rm Var}_{T^{\ell}A_{\varphi}}\,
\left(\log|T_n'|, T_n^{\ell-k}{\cal P}\right)\right)
\nonumber\\
&\leq&c_8\vartheta'^{\ell}\sum_{\psi\in S_{k-\ell}}
\left({\rm Var}_{A_{\psi}}\,
\left(\log|T'|, T^{\ell-k}{\cal P}\right)
+{\rm Var}_{A_{\psi}}\,
\left(\log|T_n'|, T_n^{\ell-k}{\cal P}\right)\right)
\nonumber\\
&\leq&c_9\vartheta'^{\ell}\left(
\left({\rm diam}\,T^{\ell-k}{\cal P}\right)^{\gamma}
+\left({\rm diam}\,T_n^{\ell-k}{\cal P}\right)^{\gamma}\right)\nonumber\\
&\leq&c_{10}\vartheta'^{\ell}\rho^{\gamma(k-\ell)}\left({\rm diam}\,
{\cal P}\right)^{\gamma}\nonumber,
\end{eqnarray}
as ${\rm diam}\,T^{\ell-k}{\cal P}\leq\rho^{k-\ell}\rm diam}\,{\cal P}$
and ${\rm diam}\,T_n^{\ell-k}{\cal P}\leq\rho^{k-\ell}\rm diam}\,{\cal P}$.
Thus we obtain with some $c_{11}$:
$$
IV\leq c_{10}\sum_{\ell=0}^{k-1}\vartheta'^{k-\ell}\rho^{\gamma\ell}
\left({\rm diam}\,{\cal P}\right)^{\gamma}
\leq kc_{11}\theta^k\left({\rm diam}\,{\cal P}\right)^{\gamma}.
$$
where $\theta=\max(\vartheta',\rho^{\gamma})$, $\theta<1$.

Finally with a suitable choice for $k$, say $k=n/2$, we get that
$$
{\rm Var}(h-h_n,{\cal P})\leq I+II+III+IV
\leq c_{12}\left(\vartheta'^k+\kappa^k+n\vartheta'^{n-k}+k\theta^k\right)
\leq c_1\sigma^n,
$$
for all $n$ and for some $\sigma<1$ (for which
$\sigma^2\geq\max(\vartheta',\kappa,\theta)$).\hfill$\Box$



\vspace{3mm} 

\noindent {\bf Proof of theorem \ref{BV}.} We have to show
that ${\rm var}(h-h_n)\leq	c_1\sigma^n$ for some $\sigma<1$
and some constant $c_1$. As before $S_k$ denote the inverse branches
of $T^k$, and if $\varphi\in S_k$ then we denote the corresponding inverse 
branch of $T_n^k$ by $\tilde\varphi$ of $T_n^k$. We proceed as in the proof
of theorem \ref{UV-convergence} and obtain for any $k<n$:
$$
{\rm var}(h-h_n) = {\rm var}({\cal L}^kh-{\cal L}_n^kh_n)
\leq I+II+III+IV.
$$
Again we proceed to estimate the terms one by one, where the first 
three terms are essentially estimated by putting $\gamma=0$.

The first term:
\begin{eqnarray}
I&=&\sum_{\varphi\in S_k}\left|\frac{1}{(T_n^k)'\varphi}\right|_{\infty}
{\rm var}_{T^kA_{\varphi}}\left(h\varphi-h_n\tilde\varphi\right)\nonumber\\
&\leq&\vartheta'^k\sum_{\varphi\in S_k}\left(
{\rm var}_{A_{\varphi}}\,h+{\rm var}_{A_{\varphi}}\,h_n\right)\nonumber\\
&\leq&c_2\vartheta'^k.\nonumber
\end{eqnarray}

The second term is estimated as (using lemma \ref{variation}(II)):
\begin{eqnarray}
II&=&\sum_{\varphi\in S_k}|h\varphi-h_n\tilde\varphi|_{\infty}
{\rm var}_{T^kA_{\varphi}}\,\left(
\frac{1}{|(T_n^k)'\tilde\varphi|}\right)
\nonumber\\
&\leq&c_3\kappa^n\sum_{\varphi\in S_k}
{\rm var}_{A_{\varphi}}\,\left(
\frac{1}{|(T_n^k)'|}\right)\nonumber\\
&\leq&c_3C_5\kappa^n
\sum_{\varphi\in S_k}\max_{A_{\varphi}}\frac{1}{|(T_n^k)'\tilde\varphi x|}
\nonumber\\
&\leq&c_4\kappa^n.
\nonumber
\end{eqnarray}
 

Using lemma \ref{inequality}, the third term becomes
\begin{eqnarray}
III&=&\sum_{\varphi\in S_k}
\left|1-\frac{|(T_n^k)'\tilde\varphi|}{|(T^k)'\varphi|}\right|_{\infty}
{\rm var}_{T^kA_{\varphi}}\,
\frac{h_n\tilde\varphi}{|(T_n^k)'\tilde\varphi|}\nonumber\\
&\leq&nC_6\vartheta'^{n-k}\sum_{\varphi\in S_k}
{\rm var}_{A_{\varphi}}\,\frac{h_n}{|(T_n^k)'|}\nonumber\\
&\leq&nC_6\vartheta'^{n-k}\sum_{\varphi\in S_k}\left(
\left|\frac{1}{(T_n^k)'}\right|_{\infty}{\rm var}_{A_{\varphi}}\,
h_n+|h_n|_{\infty}{\rm var}_{A_{\varphi}}\,
\frac{1}{|(T_n^k)'|}\right) \nonumber\\
&\leq&nC_6\vartheta'^{n-k}\left(
\vartheta'^k{\rm var}\,h_n
+|h_n|_{\infty}C_5c_5|h_n|_{\infty}\right)
\nonumber\\
&\leq&nc_6\vartheta'^{n-k}\nonumber,
\end{eqnarray}
for some constants $c_5,c_6$, where we used lemma \ref{variation}(II).

For the fourth term we have to use that $|T'|$ has 
uniform variation. As before
$$
IV=\sum_{\varphi\in S_k}\left|
\frac{h_n}{(T_n^k)'}\circ\tilde\varphi\right|_{\infty}
{\rm var}_{T^kA_{\varphi}}\,
\left(1-\frac{|(T_n^k)'\varphi|}{|(T^k)'\tilde\varphi|}\right),
$$
where
\begin{eqnarray}
{\rm var}_{T^kA_{\varphi}}\,
\left(1-\frac{|(T_n^k)'\tilde\varphi|}{|(T^k)'\varphi|}\right)
&=&{\rm var}_{A_{\varphi}}\,
\left(1-\frac{|(T_n^k)'\hat\varphi_0|}{|(T^k)'|}\right)\nonumber\\
&\leq&c_7\,{\rm var}_{A_{\varphi}}\,
\log\frac{|(T^k)'\hat\varphi_0|}{|(T_n^k)'|}\nonumber\\
&=&c_7\sum_{\ell=0}^{k-1}{\rm var}_{T^{\ell}A_{\varphi}}\,
\log\frac{|T'|}{|T_n'\hat\varphi_{\ell}|}\nonumber
\end{eqnarray}
with some  $c_7$, where 
$\hat\varphi_{\ell}=T_n^{\ell}\tilde\varphi T^{k-\ell}$.
In order to estimate the variation term inside the sum for a given 
value of $\ell$, consider on $T^{\ell}A_{\varphi}$ the partition 
${\cal Q}_{\ell}$
induced by $T^{\ell}{\cal A}^n$. Notice that $|T_n'|$ and 
$|T_n'\hat\varphi_{\ell}|$ are both constant on the atoms of ${\cal Q}_{\ell}$.
In every atom $Q\in{\cal Q}_{\ell}$ there is (by the mean value theorem) 
a point $y_Q$ so that so that $\left|T'(y_Q)\right|=\left|T_n'|_Q\right|$.
The points $\{y_Q:Q\in{\cal Q}_{\ell}\}$ form a partition of 
$T^{\ell}A_{\varphi}$ which we shall call ${\cal R}_{\ell}$.
As $\log |T'|$ has $\gamma$-uniform variation, one obtains:
$$
{\rm var}_{T^{\ell}A_{\varphi}}\,
\log\frac{|T'|}{|T_n'\hat\varphi_{\ell}|}
={\rm Var}_{T^{\ell}A_{\varphi}}\,\left(\log|T'|,{\cal R}_{\ell}\right)
\leq {\rm U}_{\gamma}(\log|T'|)\left({\rm diam}\,{\cal R}_{\ell}\right)^{\gamma}.
$$
Moreover, since 
$$
{\rm diam}\,{\cal R}_{\ell}\leq2\,{\rm diam}\,{\cal Q}_{\ell}
\leq2\,{\rm diam}\,T^{\ell}{\cal A}^n\leq2\,{\rm diam}\,{\cal A}^{n-\ell}
\leq2\rho^{n-\ell},
$$
we deduce that
\begin{eqnarray}
\sum_{\varphi\in S_k}\left|
\frac{h_n}{(T_n^k)'}\circ\tilde\varphi\right|_{\infty}
{\rm var}_{T^{\ell}A_{\varphi}}\,
\log\frac{|T'|}{|T_n'\hat\varphi_{\ell}|}
&\leq&\vartheta'^{k-\ell}\sum_{\psi\in S_{k-\ell}}
{\rm Var}_{A_{\psi}}\,\left(
\log|T'|,{\cal R}_{\ell}\right)\nonumber\\
&\leq&c_8\vartheta'^{k-\ell}\rho^{\gamma(n-\ell)}\nonumber,
\end{eqnarray}
and, with some $c_9$:
$$
IV\leq c_8\sum_{\ell=0}^{k-1}\vartheta'^{k-\ell}\rho^{\gamma(n-\ell)}
\leq kc_9\theta^{n-k},
$$
where $\theta\geq\max(\vartheta',\rho^{\gamma})$, $\theta<1$.

Finally with a suitable choice for $k$, say $k=n/2$, we get that
$$
{\rm var}(h-h_n,{\cal P})\leq I+II+III+IV
\leq c_{11}\left(\vartheta'^k+\kappa^k+n\vartheta'^{n-k}+k\theta^k\right)
\leq c_1\sigma^n,
$$
for all $n$ and for some $\sigma<1$.\hfill$\Box$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%            Example
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Example}

\noindent In this section we present a map $T$ of the unit interval to 
itself which is piecewise $C^{1+L}$ and for which 
${\rm var}\,(h-h_{n_j})$ converges subexponentially fast to zero
as the sequence $n_j$ goes to infinity. As before $h_n$ are the densities
for the $n$-th natural affine approximation $T_n$ of $T$.

As in  the proof of theorem \ref{BV} let us consider (for a suitable
sequence of $n$-values)
$$
{\rm var}\,(h-h_n) = {\rm var}\,({\cal L}^kh-{\cal L}_n^kh_n)
\leq I+II+III+IV,
$$
where as above $I+II+III\leq c_1(\vartheta^k+\kappa^n+n\vartheta^{n-k})$
for $k<n$, where the numbers $\vartheta,\kappa<1$ are as above.
 We shall now construct $T$ so that the fourth term $IV$ can
be estimated from below (polynomially in $\log n_j$). We shall proceed
in four steps. We shall construct the map (i); then estimate the 
conditional variation $\log T'$ from below for a certain partition (ii)
and from above for some finer partitions (iii); and finally we
shall obtain lower bounds on the variation  ${\rm var}\,(h-h_n)$ (iv).

\vspace{2mm}

\noindent (i) We approximate $T$ in the following way in $C^1$
by a sequence of piecewise $C^{1+L}$-maps $V_j$, $j=1,2,\dots$. 
Let $P$ be some large
integer which is to be chosen later and put $V_1$ for the piecewise
linear map $V_1(x)=Px \bmod 1$, for $x\in[0,1]$. Then $V_1'$ is constant 
equal to $P$ on $P$ subintervals of length $1/P$ that form a partition 
$\cal A$ of $[0,1]$. The quantities $\vartheta'=(P-\frac{1}{3P})^{-1}$ and 
$\vartheta=(P+\frac{1}{3P})^{-1}$ will be a lower respectively upper bound 
for the contraction rates for the inverse branches of the maps $V_j$
and $T$. Choose $\beta\geq2\frac{\log \vartheta}{\log \vartheta'}$ ($\beta>1$),
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 $-2$)

\setlength{\unitlength}{1in}
\begin{picture}(5,1.8)

\put(1.75,1){\line(1,1){.5}}
\put(2.25,1.5){\line(1,-2){.5}}
\put(2.75,.5){\line(1,1){.5}}

\put(2,.2){Squiggle}
\end{picture}

\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 $A\in{\cal P}^k$, $k<n$, where ${\cal R}_{k-1}$ is a partition of 
$A$ whose diameter $\vatheta\leq2\vartheta'^{n-k}$ and whose cardinality
$|{\cal R}_{k-1}|$ is bounded from above by $\vartheta^{-(n-k)}$.
Since in our case $T^{k-1}A=[0,1]$ for all
$A\in{\cal A}^k$, we therefore get that (recall that by construction $T'>1$)
$$
{\rm Var}_{T^{k-1}A}\,(\log T',{\cal R}_{k-1})
={\rm Var}\,(\log T',{\cal R}_{k-1}).
$$
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 $\ell<k-1$, and let us find upper estimates for 
${\rm Var}_{T^{\ell}A}\,(\log T',{\cal R}_{\ell})$  
for $A\in{\cal A}^k$, where ${\cal R}_{\ell}$ are 
partitions whose diameters are bounded above by $2\vartheta'^{n-\ell}$.
Clearly, whenever $A\in{\cal A}^k$, then
$$
\vartheta^{l-1-\ell}\leq|T^{\ell}A|\leq\vartheta'^{k-1-\ell}.
$$
Suppose we have a squiggle of some length $s$.
If we approximate by a locally constant function which is constant on intervals
of lengths $<s/3$ say, then the conditional variation can only be produced by 
those intervals which contain corners (kinks) of the squiggle (of which there
are four in each squiggle). Hence if we only consider squiggles of size 
$\geq\alpha_{m-1}$ we get
\begin{equation}\label{first}
{\rm Var}_{T^{\ell}A\cap([0,1]\setminus H_m)}\,(\log T',{\cal R}_{\ell})
\leq {\rm diam}\,({\cal R}_{\ell})4N_m2|T^{\ell}A|(1-|H_m|),
\end{equation}
(since all squiggles of lengths $\leq\alpha_m$ dwell on $H_m$)
where $N_m$ is the total number of squiggles of lengths $\geq\alpha_{m-1}$.
We estimate ($\beta>1$ 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 $\ell<k=[n/2]$) we get
\begin{equation}\label{upper}
\sum_{\ell=0}^{k-2}\mbox{\rm Var}_{T^{\ell}A}\,(\log T',{\cal R}_{\ell})
\leq\frac{6}{m}\sum_{\ell=0}^{k-2}\vartheta'^{k-1-\ell}
\leq\frac{6}{m}\frac{\vartheta'}{1-\vartheta'}.
\end{equation}

\vspace{2mm}

\noindent (iv) Let us now complete the estimate of the variation in our example.
As above, let $A_{\varphi}\in{\cal A}^k$ be the ranges of the inverse 
branches
$\varphi\in S_k$ of $T^k$, and consider on $T^{\ell}A_{\varphi}$ the 
partition ${\cal Q}_{\ell}$ induced by $T^{\ell}{\cal A}^n$. 
(As above, $|T_n'|$ and $|T_n'\hat\varphi_{\ell}|$ are both constant 
on the atoms of ${\cal Q}_{\ell}$.)
In every atom $Q\in{\cal Q}_{\ell}$ there is (by the mean value theorem) 
a point $y_Q$ so that $\left|T'(y_Q)\right|=\left|T_n'|_Q\right|$.
The points $\{y_Q:Q\in{\cal Q}_{\ell}\}$ form a partition of 
$T^{\ell}A_{\varphi}$ which we shall call ${\cal R}_{\ell}$.
One obtains for $\ell=0,\dots,k-1$:
$$
{\rm var}_{T^{\ell}A_{\varphi}}\,
\log\frac{|T'|}{|T_n'\hat\varphi_{\ell}|}
={\rm Var}_{T^{\ell}A_{\varphi}}\,\left(\log|T'|,{\cal R}_{\ell}\right),
$$
and 
$$
{\rm diam}\,{\cal R}_{\ell}\leq2\,{\rm diam}\,{\cal Q}_{\ell}
\leq2\,{\rm diam}\,T^{\ell}{\cal A}^n\leq2\,{\rm diam}\,{\cal A}^{n-\ell}
\leq2\vartheta'^{n-\ell}.
$$
Therefore, if we use the estimates (\ref{lower}) and (\ref{upper}) which we 
deduced above, we obtain:
\begin{eqnarray}
{\rm var}_{A_{\varphi}}\,\log\left|\frac{(T^k)'}{(T_n^k)'}\right|
&=&{\rm var}_{A_{\varphi}}\,\sum_{\ell=0}^{k-1} 
\log\left|\frac{T'T^{\ell}}{T_n'T_n^{\ell}}\right|
\nonumber\\
&\geq&{\rm var}_{A_{\varphi}}\,
\log\left|\frac{T'T^{k-1}}{T_n'T_n^{k-1}}\right|
-{\rm var}_{A_{\varphi}}\,\sum_{\ell=0}^{k-2} 
\log\left|\frac{T'T^{\ell}}{T_n' T_n^{\ell}}\right|
\nonumber\\
&=&{\rm Var}_{T^{k-1}A_{\varphi}}\,
(\log T',{\cal R}_{k-1})
-\sum_{\ell=0}^{k-2} {\rm Var}_{T^{\ell}A_{\varphi}}\,
(\log T',{\cal R}_{\ell})
\nonumber\\
&\geq&\frac1m-\frac{6}{m}\frac{\vartheta'}{1-\vartheta'}.
\nonumber
\end{eqnarray}
If we choose $P$ large enough so that 
$\frac{6\vartheta'}{1-\vartheta'}\leq\frac12$, then
$$
{\rm var}_{A_{\varphi}}\,\log\left|\frac{(T^k)'}{(T_n^k)'}\right|
\geq \frac1{2m}.
$$
This finally allows us to estimate the term IV from below by
$${\rm IV}\geq c_1\frac1{2m}\sum_{\varphi\in S_k}
\frac{h_n\varphi}{|(T_n^k)'\tilde\varphi|}
\geq c_1\frac1{2m}\inf h_n
\geq\frac{c_1}{4m}\inf h,
$$
for large enough $n$ and some positive constant $c_1$. Consequently,
since we chose $k=[n/2]$, we obtain
$$
{\rm var}\,(h-h_n)\geq IV-(I+II+III)
\geq \frac{c_2}{m}-c(\vartheta^k+\kappa^n+n\vartheta^{n-k})
\geq \frac{c_3}{m(n)}\geq\frac{c_4}{\log n}
$$
for some positive constants $c_2,c_3,c_4$.






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\end{thebibliography} 



\end{document}

