\documentclass[research]{raex}
\FirstPageNumber{449}
\Volume{36} % publication information for your paper
\IssueNumber{2} % publication information for your paper
\Year{2010/2011} % publication information for your paper
\Editor{Zbigniew Nitecki} % for your paper
\Received{January 22, 2011} % publication information for your paper
%\CoverAuthor{Jozef Bobok and Martin Soukenka\\}
%\CoverTitle{Irreducibility, Infinite Level Sets and Small Entropy }
\begin{Author}
% First and last name of author
\FirstName{Jozef}\LastName{Bobok}
% full postal address including postal code and country
\PostalAddress{KM FSv \v CVUT, Th\'akurova 7, 166 29 Praha 6, Czech Republic}
% email address for author
\Email{bobok@mat.fsv.cvut.cz}
% grant support is indicated by giving \Thanks
\Thanks{The first author was partly supported by the Grant Agency of the Czech Republic contract number
201/09/0854. He also gratefully acknowledges the support of the MYES
of the Czech Republic via contract MSM 6840770010.}
\end{Author}
\begin{Author}
% First and last name of author
\FirstName{Martin}\LastName{Soukenka}
% full postal address including postal code and country
\PostalAddress{KM FSv \v CVUT, Th\'akurova 7, 166 29 Praha 6, Czech Republic or \'{U}T AV \v{C}R, Dolej\v{s}kova 1402/5, 182 00 Praha 8, Czech Republic}
% email address for author
\Email{soukenkam@mat.fsv.cvut.cz}
% grant support is indicated by giving \Thanks
%\Thanks{}
\end{Author}
\begin{MathReviews}
\primary{37B40}
\secondary{26A30}
\end{MathReviews}
\begin{KeyWords}
\keyword{interval map}
\keyword{knot point}
\keyword{Lebesgue measure}
\keyword{topological entropy}
\end{KeyWords}
\title{Irreducibility, infinite level sets, and small entropy}
\markboth{J. Bobok and M. Soukenka}{Irreducibility, infinite level sets, and small entropy}
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\begin{document}
\maketitle
\begin{abstract}
We investigate continuous piecewise affine interval maps with
countably many laps that preserve the Lebesgue measure. In
particular, we construct such maps having knot points (a point $x$
where Dini's derivatives satisfy $D^{+}f(x)=D^{-}f(x)= \infty$ and $D_{+}f(x)=D_{-}f(x)=
-\infty$) and estimate their topological entropy. Our main result is: for any $\eps>0$ we
construct a continuous interval map $g=g_{\eps}$ such that (i) $g$ preserves the Lebesgue measure; (ii)
knot points of $g$ are dense in $[0,1]$ and
for a $G_{\delta}$ dense
set of $z$'s, the set $g^{-1}(\{z\})$ is infinite; (iii) $\ent(g)\le\log2+\eps$.
\end{abstract}
\section{Introduction}\label{s1}
A map $f\colon~X\to X$ is called $m$-fold on $Y\subset X$, if for every $y\in Y$
a set $f^{-1}(y)$ contains at least $m$ points.
For a set $X$, we call a subset $Y\subset X$ cocountable if its
complement $X\setminus Y$ is (at most) countable, and say that a map
$f\colon~X\to X$ is cocountably $m$-fold if it is globally $2$-fold
and $m$-fold on some cocountable subset $Y\subset X$.
In \cite{Bo05} the author proved the following estimate on
topological entropy:
\begin{theorem}\label{t4} The topological entropy of any continuous
cocountably $m$-fold map $f\colon~[0,1]\to [0, 1]$ satisfies
$\ent(f)\ge \log m$.\end{theorem}
This result is rather delicate, as there is a simple Raith's example
of a continuous map $f\colon~[0,1]\to [0,1]$ that is $m$-fold (for an
arbitrarily chosen $m\in\bbn$) except at $y=1$, which has a single
preimage point, but its non-wandering set consists of the fixed
endpoints, so that the entropy is zero (see \cite{BoNi05} for more
detailed information). It is folklore knowledge that analogous
examples can be constructed on any $n$-dimensional manifold
(orientable or non-orientable, also with boundary).
Moreover, in \cite{BoNi07} the authors showed that the set of
points, where the $m$-fold conditions fail in the hypotheses of
Theorem \ref{t4}, cannot be allowed to be uncountable, even if it is
nowhere dense. Namely, for each integer $m>0$ there exists a
continuous map $f\colon~[0,1]\to [0,1]$ such that $f$ is globally
$2$-fold, $f$ is $m$-fold on a set $Y=[0,1]\setminus K$, where $K$
is a nowhere dense, closed (uncountable) set and at the same time
$\ent(f)=\log2$.
Despite Theorem \ref{t4} and related examples, the problem of
understanding of relationship of two characteristics of an interval
(or a tree) map - its topological entropy and cardinalities of level
sets - is not completely solved. On the one hand the proofs used in
\cite{Bo05}, \cite{BoNi07} are rather difficult with many
technicalities, on the other hand all known (counter)examples work
with a ``poor'' set of non-wandering points. Thus, one could expect
some strengthened version of Theorem \ref{t4} stated for a class of
irreducible interval maps (transitive, with a dense set of periodic
points) proved by essentially simplified methods.
As a canonical expression of mentioned insufficient grasp of the subject
we can introduce the following conjectures:
\begin{conjecture}\label{co1}Any continuous nowhere differentiable interval map preserving the Lebesgue measure
has infinite topological entropy.\end{conjecture}
We recall that by a knot point of function $f$ we mean a point $x$
where Dini's derivatives satisfy $D^{+}f(x)=D^{-}f(x)= \infty$ and $D_{+}f(x)=D_{-}f(x)=
-\infty$.
\begin{conjecture}\label{co2}Any continuous interval map preserving the Lebesgue measure $\lambda$ and with a knot point $\lambda$-a.e.
has infinite topological entropy.\end{conjecture}
Note that the existence of continuous interval maps used in the
hypotheses has been proved in \cite{Bo91}.
The goal of this paper is to provide more sophisticated examples
related to Conjectures \ref{co1}, \ref{co2}. To this goal we
investigate continuous piecewise affine interval maps with countably
many laps and preserving the Lebesgue measure. We construct such
maps having finitely many knot points and estimate their topological
entropy. As the main result of this paper stated in Theorem \ref{t5} we obtain the following: for any $\eps>0$ we
construct a continuous interval map $g=g_{\eps}$ such that (i) $g$
is nowhere monotone and preserves the Lebesgue measure
(irreducibility); (ii) knot points of $g$ are dense in $[0,1]$ and for a $G_{\delta}$ dense
set of $z$'s, the set $g^{-1}(\{z\})$ is infinite (infinite level
sets); (iii) $\ent(g)\le\log2+\eps$ (small entropy). Two applications are presented in Corollary \ref{c2} and Theorem \ref{t6}.
The paper is organized as follows. In Section \ref{s2} we give some
basic notation, definitions and known results (Theorems \ref{t3},
\ref{t2}, \ref{t1}). Section \ref{s3} is devoted to the both local
and global perturbations and the map $g$ cited above is
constructed.
Finally, in Section \ref{s4} we prove the main results - Theorem
\ref{t5} and its Corollary \ref{c2}. We also present one application to the
$n$-dimensional case - Theorem \ref{t6}.
\section{Definitions and known results}\label{s2}
As general references one can use \cite{Wa82} or \cite{KaHa95}.
Let $X$ be a compact metric space and $f\colon~X\to X$ be a continuous map. By $\mathfrak M(X)$ we denote the set of all Borel normalized measures on $X$. The weak$^*$ topology on $\mathfrak M(X)$ is defined by taking the sets
\begin{equation*}V_{\mu}(f_1,\dots,f_k;\eps_1,\dots,\eps_k)=\Big\{\nu\colon~\Big\vert\int f_j\,{\rm d}\mu -\int f_j\, {\rm d}\nu\Big\vert<\eps_j,~j=1,\dots,k\Big\}\end{equation*}
as a basis of open neighborhood for $\mu\in\mathfrak M(X)$ with $\eps_j>0$ and $f_j$ being a continuous function defined on $X$. The map $f$ transports every measure $\mu\in\mathfrak M(X)$ into another measure $f_*\mu\in\mathfrak M(X)$. In what follows if we say ``measure''
we in fact mean Borel normalized measure and if we measure some
set then we assume that it is measurable. The support of $\mu$ is the smallest closed set $S\equiv \supp\mu$ such that $\mu(S)=1$.
If $\mu=f_*\mu$ then $\mu$ is said to be invariant ($\mu$ is preserved by $f$). It is equivalent to the condition $\mu(f^{-1}(S))=\mu(S)$ for any measurable
$S\subset X$. Let $\mathfrak M(f)$ be the set of measures
preserved by $f$. A point $p\in X$ is said to be periodic if for some positive integer $n$, $f^n(p)=p$. The set of all periodic points of $f$ is denoted by $\Per(f)$. A measure $\mu\in\mathfrak M(f)$ the $\supp\mu$ of which coincides with one periodic orbit (cycle) is said to be a $CO$-measure and the set of all $CO$-measures which are concentrated on cycles is denoted by $\mathfrak P(f)$.
We say that $S\subset X$ is $f$-invariant if $f(S)\subset S$.
A measure $\mu\in\mathfrak M(f)$ is called ergodic if for any
$f$-invariant set $S\subset X$ either $\mu(S)=0$ or $\mu(S)=1$.
We denote the set of all $f$-invariant ergodic measures by $\mathcal
E(f)$. If $\mu$
is ergodic then either $\supp\mu=\orb(p)$ for some periodic point $p\in\Per(f)$ or
$\supp\mu$ is a perfect set.
For $\mu\in\mathfrak M(f)$, the measure-theoretic entropy of $f$ is a quantity
$$h_{\mu}(f)=\sup_{\zeta}\lim_{n\rightarrow \infty} H_{\mu}(\zeta_n),$$
where the supremum is taken over all finite measurable partition $\zeta$ of $X$, $$H_{\mu}(\zeta_n)=-\sum_{A\in\zeta_n}\mu(A)\log\mu(A)$$
and $\zeta_n=\{A_{i_0}\cap f^{-1}A_{i_1}\cap\cdots\cap f^{-(n-1)}A_{i_{n-1}}\colon~A_{i_j}\in\zeta\}$. The topological entropy $\ent(f)$ of $f$ can be defined as \cite{Wa82}
$$\ent(f)=\sup_{\mu}h_{\mu}(f),$$ where the supremum is taken over all $\mu$ from $\mathfrak M(f)$.
In particular, when $X=[0,1]$ and $f\colon~[0,1]\to [0,1]$ is continuous the map $f$ will be called an interval map.
\begin{theorem}\cite{BlCo92}\label{t7} Let $f\colon~[0,1]\to [0,1]$ be an interval map preserving the Lebesgue measure. The set $\mathfrak P(f)$ is dense in $\mathfrak M(f)$ {\rm (}in the weak$^*$ topology{\rm )}. In particular, the set of all periodic points of $f$ is dense in $[0,1]$.
\rightline{$\Box$}
\end{theorem}
\begin{proposition}\label{p2}Let $f\colon~[0,1]\to [0,1]$ have a dense set of periodic points and let $f$ be $2$-fold on $Y\subset X$, where $[0,1]\setminus Y$ is finite. Then the set $\{x\in [0,1]\colon~x\notin\Per(f)~\&~f(x)\in\Per(f)\}$ is dense in $[0,1]$.\end{proposition}
\begin{proof}Choose an interval $J\subset [0,1]$. By our assumption there are closed intervals $K,J_1,J_2$ such that
\begin{equation*}J_1\subset J~, J_1\cap J_2=\emptyset,~f(J_1)=f(J_2)=K\subset f(J).\end{equation*}
Since the set $\Per(f)$ is dense, there is a periodic point $p\in J_2$ and also a non-periodic point $x\in J_1\subset J$ for which $f(x)=f(p)\in\Per(f)$.\end{proof}
We will need following ergodic decomposition.
\begin{theorem}\cite{Phe66}\label{t3} Let $\mu\in\mathfrak M(f)$. Then
there is a measure $m$ on $\mathcal E(f)$ such that
$\mu(S)=\int_{\mathcal E(f)}\lambda(S)\,{\rm d}m$ for any measurable set
$S$.
\rightline{$\Box$}
\end{theorem}
Fix $f\colon~[0,1]\to [0,1]$ and $x\in [0,1]$. The Lyapunov
exponent, $\lambda_f(x)$, is given by
\begin{equation*}\lambda_f(x)=\lim_{n\to\infty}\frac{1}{n}\log\vert
(f^n)'(x)\vert\end{equation*} if the limit exists. The Lyapunov characteristic $\chi\colon~[0,1]\to [0,\infty]$ is defined as
\begin{equation}\label{e9}
\chi_f(x) = \left\lbrace
\begin{array}{c l}
\lambda_f(x),\quad \lambda_f(x)>0,\\
0,\quad\ \ \ \ \text{otherwise}.
\end{array}
\right.
\end{equation}
The following known theorem (its one-dimensional version) will be one of the key tools when proving Theorem
\ref{t5}.
\begin{theorem}{\rm (}the Margulis-Ruelle inequality{\rm )} {\rm (see \cite[pp.
281-285]{Ma87})}. \label{t2}Let $f\colon~[0,1]\to [0,1]$ be a
piecewise Lipschitz map, let $\mu$ be an invariant measure for $f$, and assume that $f$ is differentiable $\mu$-a.e.
Then
\begin{equation*}h_{\mu}(f)\le\int_{\supp\mu}\chi_f\,{\rm d}\mu.\end{equation*}
\rightline{$\Box$}
\end{theorem}
For a
pair $(T,g)$ with $T\subset \bbr$ closed and continuous
$g\colon~T\to T$, $g_T\colon~\conv T\to \conv T$ (by $\conv T$ we mean the convex hull of $T$) is a piecewise
affine ``connect-the-dots'' interval map given by $(T,g)$.
An interval map $f\colon~[0,1]\to [0,1]$ has a subsystem $(T,g)$ if
$T\subset [0,1]$ is closed, $g=f\vert{T}$ and $g(T)\subset T$.
A subsystem
$(T,g)$ of $f$ is piecewise monotone, respectively strictly ergodic if $g_T$ is
piecewise monotone, respectively if there is exactly one measure $\mu\in\mathfrak M(f)$ such that
$\supp\mu=T$ and no other measure has its support as a subset of $T$.
\begin{proposition}\label{p1}Let $f\colon~[0,1]\to [0,1]$ be piecewise affine possibly with countably many laps and having a
piecewise monotone
strictly ergodic
subsystem $(T,g)$ supporting an invariant measure $\mu$ with
$h_{\mu}(f)>0$. Then for each $x\in T$,
\begin{equation*}\lambda_f(x)=\int_{[0,1]}\log\vert f'\vert\,{\rm d}\mu\in (0,\infty). \end{equation*}
\end{proposition}
\begin{proof}We have
\begin{equation*}\frac{1}{n}\log\vert(f^n)'\vert=\frac{1}{n}\log\Big(\prod_{j=0}^{n-1}\vert f'(f^j)\vert\Big)
=\frac{1}{n}\sum_{j=0}^{n-1}\log\vert f'(f^j)\vert
\end{equation*}
and the right-hand sums converge on the set $T$ uniformly to a
constant $\lambda_{\mu}=\int_{[0,1]}\log\vert f'\vert\,{\rm d}\mu$ - see
\cite[Theorem 6.19, p.160]{Wa82}. The value $\lambda_{\mu}$ is positive by (\ref{e9}),
our assumption $h_{\mu}(f)>0$ and Theorem \ref{t2}. Since $(T,g)$ is
piecewise monotone, the number $\lambda_{\mu}$ is less than
$\infty$.
\end{proof}
The Variational principle represents a basic relationship between
measure-theoretic and topological
entropy. In the context of interval maps one can restrict attention to the subset of strictly ergodic piecewise monotone
pairs and corresponding invariant measures.
\begin{theorem}\cite{Bo03}\label{t1}. Let f be an interval map. Then
\begin{equation*}\ent(f)=\sup_{(T,g)}h_{\mu}(f), \end{equation*}
where the supremum is taken over all strictly ergodic piecewise
monotone subsystems $(T,g)$ of $f$ and corresponding invariant
measures $\mu$.
\rightline{$\Box$}
\end{theorem}
\section{Constructions}
\label{s3}
\subsection{Local perturbation.}
In the first subsection of this section we describe a specific local perturbation of an interval map, i.e. a change of definition of a map on a ``small'' subset of its domain. All is summarized in Definition \ref{d1}.
For $n\ge 1$, the maps $\alpha_5$ are ``connect-the-dots'' maps
with the dots (see Figure 1(a))
\begin{equation*}\{(0,0),(1/5,1),(2/5,0),(3/5,1),(4/5,0),(1,1)\}.\end{equation*}
In order to describe how we will perturb maps we start with a map $\kappa\colon~[0,1]\to [0,1]$
defined as the uniform limit of a sequence $\{\kappa_n\}_{n\ge 1}$:
fix a sequence $\{\delta_n\}_{n\ge 1}$ of positive real numbers with $\delta_1=1/2$ and such that
$10\delta_{n+1}<\delta_n$; then the intervals $K_n=[1/2-\delta_n,1/2+\delta_n]$
satisfy \begin{equation}\label{e12}[0,1]=K_1\supset K_2\supset
K_3\cdots~,~10\lambda(K_{n+1})<\lambda(K_n).\end{equation}
\begin{figure}[!ht]
\resizebox{11cm}{!}{\includegraphics{s1.pdf}}
\caption{Figure 1. (a) The map $\alpha_5$; (b) The map $\kappa_3$.}
\end{figure}
We construct maps $\kappa_n\colon~[0,1]\to [0,1]$ inductively:
\noindent ($n=1$): $\kappa_1=\alpha_5$.
\noindent ($n>1$): If the map $\kappa_{n-1}$ is already defined, we put (see Figure 1(b) for $n=3$)
$\kappa_n=\kappa_{n-1}$ on $[0,1]\setminus K_n$ and $\kappa_n=h\circ\alpha_5\circ h^{-1}_n$ on $K_n$,
where $h_n$, respectively $h$ is affine, preserves orientation and maps the unit interval onto $K_n$, respectively $\kappa_{n-1}(K_n)$.
Clearly, each $\kappa_n$ is continuous and it preserves the Lebesgue
measure. Moreover, by our construction and (\ref{e12})
$$\sup_{x\in [0,1]}\vert \kappa_n(x)-\kappa_{n-1}(x)\vert\le 5^n\lambda(K_n)<\frac{5^n}{10^{n-1}}=\frac{5}{2^{n-1}},$$
hence the map $\kappa=\lim_n\kappa_n$ exists, it is continuous and the
Lebesgue measure preserving again. Since the map $\kappa$ depends on the sequence $\Delta=\{\delta_n\}_{n\ge 1}$, we will
sometimes use the notation $\kappa=\kappa[\Delta]$.
Let $f\colon~[0,1]\to [0,1]$ be an interval map, consider a point $x\in (0,1)$ and a $\beta>0$ such that $0\le x-\betaf(x+\beta)$, a decreasing $(x,\beta,\Delta)$-perturbation of $f$ is defined analogously by using the map $1-\kappa[\Delta]$ instead of $\kappa[\Delta]$.
\end{definition}
\subsection{Global perturbation.}
In the second subsection we apply above local perturbation repeatedly to obtain a global change of definition of a map on a dense subset of its domain.
For a piecewise affine map $f$ (possibly with countably many laps) let $W(f)$ be the set consisting of all points in which $f$ is not differentiable and endpoints $0,1$.
Let $\{J_m\}_{m\geq 1}$ be the sequence of all rational subintervals of $(0,1)$. Consider the full tent map
$f:[0,1]\rightarrow [0,1]$ given by $f(x)=1-|1-2x|,\,x\in [0,1]$.
Fix an $\eps>0$. We inductively define maps
$g_m$:
\noindent ($m=0$): $g_0=f$, $x_0=1$, $p_0=0$.
\noindent ($m>0$): Since by Theorem \ref{t7} the map $g_{m-1}$ has a dense set of
periodic points and each point from $[0,1)$ has at least two
$g_{m-1}$-preimages, by Proposition \ref{p2} there is a point $x_m$ such that
\begin{eqnarray}\label{e11}x_m\in J_m,~x_m\notin \Per(g_{m-1}),~g_{m-1}(x_m)=p_m\in\Per(g_{m-1}),\\ \label{e8}~p_m\notin\bigcup_{j=1}^{m-1}\orb
(p_j),~x_m\notin W(g_{m-1})\cup\{x_0,\dots,x_{m-1}\};\end{eqnarray}
for a sequence $\{k^m_n\}_{n\ge 1}$ of positive integers fulfilling
\begin{equation}\label{e5}
\sum_{n=1}^{\infty}\frac{\log(\vert 5^ng'_{m-1}(x_m)\vert)}{k^m_n+1}<\frac{\eps}{2^m},\end{equation}
there is a sequence $\Delta_m=\{\delta^m_n\}_{n\ge 1}$ (of
sufficiently small delta's, shortly, sufficiently small $\Delta_m$)
and a corresponding (increasing or decreasing)
$(x_m,\beta_m,\Delta_m)$-perturbation $g_m$ of $g_{m-1}$ such that
for each $j\in\{1,\dots,m\}$ and $n\ge 1$
($K^j_n=[1/2-\delta^j_n,1/2+\delta^j_n]$),
\begin{eqnarray}\label{e4}x\in
d_{x_j,\beta_j}(K^j_n)~\implies~\{g_{m}^i(x)\}_{i=1}^{k^j_n}\cap
d_{x_j,\beta_j}(K^j_n)=\emptyset,\\ \label{e7}
\max\{\lambda(g^i_{m}(d_{x_j,\beta_j}(K^j_n))\colon~i=0,\dots,k^j_n\}<1/n\end{eqnarray}
and, in particular, for
$[x_m-\beta_m,x_m+\beta_m]=d_{x_{m},\beta_m}([0,1])$,
\begin{equation}\label{e17}\lambda(g_{m}([x_m-\beta_m,x_m+\beta_m]))<1/m.\end{equation}
\begin{figure}[!ht]
\resizebox{11cm}{!}{
\includegraphics{StansUzly.pdf}
}
\caption{Figure 2. (a) The map $g_{2}$; (b) The map $g_{3}$.}
\end{figure}
We will argue the properties (\ref{e4}), (\ref{e7}) in more details.
\begin{claim}If {\rm (\ref{e4}), (\ref{e7})} is true for $j\in\{1,\dots,m-1\}$ and $g_{m-1}$ then the sequence $\Delta_m=\{\delta^m_n\}_{n\ge 1}$
fulfilling {\rm (\ref{e4}), (\ref{e7})} for $j\in\{1,\dots,m\}$ and
corresponding $g_m$ also exists.\end{claim}
\begin{proof}Since by (\ref{e8})
\begin{equation*}\orb(p_m)\cap\bigcup_{j=1}^{m-1}\orb (p_j)=\emptyset,\end{equation*}
the (\ref{e7}) applied on $g_{m-1}$ means that for a sufficiently small $\tilde\Delta_m$ and corresponding $\tilde g_m$ the properties (\ref{e4}), (\ref{e7}) remain true for $\tilde g_m$ up to finitely many
$n$'s. Taking appropriately $\Delta_m$ smaller than $\tilde\Delta_m$ (if necessary), we
obtain the map $g_m$ fulfilling (\ref{e4}), (\ref{e7}) for
$j\in\{1,\dots,m\}$ and every $n$.
\end{proof}
\begin{claim}\label{c1}For any $m\in\bbn$ and any invariant measure $\mu\in\mathfrak M(g_m)$,
\begin{equation*}\int_{[x_m-\beta_m,x_m+\beta_m]}\log\vert g_m'\vert\,{\rm d}\mu\le
\sum_{n=1}^{\infty}\frac{\log(\vert 5^ng'_{m-1}(x_m)\vert)}{k^m_n+1}.\end{equation*}
\end{claim}
\begin{proof}By the representation Theorem \ref{t3} it is sufficient to assume that $\mu$ is ergodic. Let $x\in\supp\mu$ be a generic point for $\mu$ (see \cite{Wa82}).
Putting $L_n=d_{x_m,\beta_m}(K^m_n)$, from (\ref{e4}) we get
\begin{equation}\label{e14}\mu(L_n)\le\frac{1}{k^m_n+1};\end{equation}
by our definition of $(x_m,\beta_m,\Delta_m)$-perturbation ($g_m$ of
$g_{m-1}$)
\begin{equation}\label{e15}\vert g'_m\vert=\vert 5^ng'_{m-1}(x_m)\vert\text{ on }L_n\setminus L_{n+1}.\end{equation}
Since $[x_m-\beta_m,x_m+\beta_m]=\bigcup_{n=1}^{\infty}(L_n\setminus
L_{n+1})$, from (\ref{e14}) and (\ref{e15}) we obtain
\begin{eqnarray*}\label{e16}\int_{[x_m-\beta_m,x_m+\beta_m]}\log\vert g_m'\vert\,{\rm d}\mu
= \sum_{n=1}^{\infty}\int_{L_n\setminus L_{n+1}}\log\vert g_m'\vert\,{\rm d}\mu \le\\
\le \sum_{n=1}^{\infty}\int_{L_n}\log\vert g_m'\vert\,{\rm d}\mu
\le\sum_{n=1}^{\infty}\frac{\log(\vert
5^ng'_{m-1}(x_m)\vert)}{k^m_n+1}.\end{eqnarray*} \end{proof}
Notice that each $g_m$ preserves the Lebesgue measure and by (\ref{e17})
$$\sup_{x\in [0,1]}\vert g_m(x)-g_{m-1}(x)\vert<1/m;$$
the reader can easily see that \begin{equation}\label{e6}g=\lim_mg_m\end{equation} is defined well and it preserves the Lebesgue measure again.
\section{The main result}
\label{s4}
\begin{theorem}\label{t5}The continuous interval map $g$ defined by {\rm (\ref{e6})} has the following properties:
\begin{itemize}\item[(i)] $g$ is nowhere monotone and preserves the Lebesgue
measure;
\item[(ii)] knot points of $g$ are dense in $[0,1]$ and for a $G_{\delta}$ dense set $Z$ of $z$'s, the set
$g^{-1}(\{z\})$ is infinite;
\item[(iii)] $\ent(g)\le\log2+\eps$.
\end{itemize}\end{theorem}
\begin{proof} The property (i) directly follows from our construction of $g$.
Let us prove (ii). It follows from (\ref{e11}) and our choice of the intervals $J_m$ that the sequence $\{x_m\}$ is dense in $[0,1]$. We will show that $g$ has a knot point at every $x_m$. By the property (\ref{e8}) of our construction, for every $k\ge m$ hold true equalities
\begin{equation}\label{e3}g(x)=g_k(x)=g_m(x)\text{ for every }x\in \{x_m\}\cup d_{x_m,\beta_m}(W(\kappa[\Delta_m])).\end{equation}
Since the map $\kappa[\Delta_m]$ has a knot point at $1/2$ and the
maps $r_{x_m,\beta_m}$, $d_{x_m,\beta_m}$ are affine, Definition
\ref{d1} and (\ref{e3}) give us that also each of the maps $g_k,g$,
$k\ge m$ has a knot point at $x_m=d_{x_m,\beta_m}(1/2)$. It means
that each of the sets
\begin{equation*}S_m:=\{z\in [0,1]\colon~\#
g^{-1}(\{z\})>m\}^{\circ}\end{equation*} is open and dense in $[0,1]$ hence $Z=\bigcap_mS_m$ is $G_{\delta}$ dense.
(iii) Let us fix $g_m$.
Using Theorem \ref{t1} let us fix a continuous strictly ergodic invariant measure $\mu\in\mathfrak M(g_m)$ with $h_{\mu}(g_m)>0$, denote $S=\supp\mu$. Then $(S,\iota=g_m\vert S)$ is an infinite minimal subsystem of $g_m$ and each point of $S$ is (uniformly) recurrent. The map $g_m$ is piecewise affine with
countably many laps accumulated exactly in points $x_1,\dots,x_m$.
By (\ref{e11}), $S\cap \{x_1,\dots,x_m\}=\emptyset$ hence the set
$S$ is a subset of finitely many laps of $g_m$. It implies that the
map $\iota_S$ is Lipschitz and since the measure with respect to $\mu$ of any countable set is zero, both the piecewise affine maps $g_m,\iota_S$ are differentiable
$\mu$-a.e. Applying Theorem \ref{t2}, Proposition \ref{p1} and
(\ref{e9}) we get
\begin{equation*}0