# Exponential mixing for
heterochaos baker maps and the Dyck system

Hiroki Takahasi  Keio Institute of Pure and Applied Sciences (KiPAS),
Department of Mathematics, Keio University, Yokohama, 223-8522, JAPAN
<EMAIL_ADDRESS>
###### Abstract.

We investigate mixing properties of piecewise affine non-Markovian maps acting
on $[0,1]^{2}$ or $[0,1]^{3}$ and preserving the Lebesgue measure, which are
natural generalizations of the heterochaos baker maps introduced in [Y. Saiki,
H. Takahasi, J. A. Yorke. Nonlinearity 34 (2021) 5744–5761]. These maps are
skew products over uniformly expanding or hyperbolic bases, and the fiber
direction is a center in which both contracting and expanding behaviors
coexist. We prove that these maps are mixing of all orders. For maps with a
mostly expanding or contracting center, we establish exponential mixing for
Hölder continuous functions. Using this result, for the Dyck system
originating in the theory of formal languages, we establish exponential mixing
with respect to its two coexisting ergodic measures of maximal entropy.

###### 2020 Mathematics Subject Classification:

Primary 37A25, 37A40; Secondary 37A55

Keywords: piecewise affine map; mixing of all orders; decay of correlations;
the Dyck shift

##  1\. introduction

The baker map (see Figure 1)

$(x,y)\in[0,1)^{2}\mapsto\begin{cases}\vspace{1mm}\displaystyle{\left(2x,\frac{y}{2}\right)}&\text{
on }\displaystyle{\left[0,\frac{1}{2}\right)\times[0,1)},\\\
\displaystyle{\left(2x-1,\frac{y+1}{2}\right)}&\text{ on
}\displaystyle{\left[\frac{1}{2},1\right)\times[0,1)},\end{cases}$

is one of the simplest uniformly hyperbolic dynamical systems. The name
‘baker’ is used since the action of the map is reminiscent of the kneading
dough [14]. The baker map preserves the Lebesgue measure on $[0,1)^{2}$, and
mixing properties with respect to this measure are well-known: It is $K$ and
hence mixing of all orders [26]; has exponential decay of correlations for
Hölder continuous functions [7, 27].

Is there any analogue of the baker map that provides a hands-on, elementary
understanding of complicated phenomena in non-hyperbolic systems? In response
to this question, two piecewise affine maps on $[0,1]^{2}$ or $[0,1]^{3}$ were
introduced in [28], called heterochaos baker maps. The name ‘heterochaos’
comes from their distinctive property that periodic points with different
unstable dimensions coexist densely in an invariant set [28, Theorem 1.1], a
phenomenon aka the unstable dimension variability [12, 19], which was
recognized earlier [1, 6, 22, 30] as a $C^{1}$ robust phenomenon for
diffeomorphisms of closed manifolds of dimension at least three.

Figure 1. The baker map: The $x$-direction is expanding by factor $2$ and the
$y$-direction is contracting by factor $1/2$.

The analysis of the heterochaos baker maps is also motivated by the theory of
symbolic dynamics, which is closely related to automata theory and formal
language theory. There is a universal formal language due to W. Dyck. The Dyck
system [2, 20] is the symbolic dynamics generated by that language.
Interestingly, the heterochaos baker maps are natural geometric models of the
Dyck system [32, Theorem 1.1]. This connection has opened the door for a
complementary analysis of the heterochaos baker maps and the Dyck system,
which consists in

verifying certain dynamical properties of the one side, with the aid of the
other.

For a prototypical result on the heterochaos baker maps with the aid of the
Dyck system, see [32, Theorem 1.2].

In this paper we proceed to a complementary analysis of mixing properties of
the heterochaos baker maps and the Dyck system. We will prove exponential
mixing with respect to the Lebesgue measure for the heterochaos baker maps,
and with the aid of this, prove exponential mixing for the Dyck system with
respect to the two coexisting ergodic measures of maximal entropy. Below we
introduce these two dynamical systems, and state main results.

###  1.1. The heterochaos baker maps

Let $m\geq 2$ be an integer. For $a\in(0,\frac{1}{m})$ define
$\tau_{a}\colon[0,1]\to[0,1]$ by

(1.1)
$\tau_{a}(x)=\begin{cases}\vspace{1mm}\displaystyle{\frac{x-(i-1)a}{a}}&\text{
on }[(i-1)a,ia),\ i\in\\{1,\ldots,m\\},\\\
\displaystyle{\frac{x-ma}{1-ma}}&\text{ on }[ma,1].\end{cases}$

Consider the set of $2m$ symbols

(1.2)
$D=\\{\alpha_{1},\ldots,\alpha_{m}\\}\cup\\{\beta_{1},\ldots,\beta_{m}\\},$

and define

$\Omega_{\alpha_{i}}^{+}=\left[(i-1)a,ia\right)\times\left[0,1\right]\text{
for }i\in\\{1,\ldots,m\\},$

and

$\Omega_{\beta_{i}}^{+}=\begin{cases}\vspace{1mm}\displaystyle{\left[ma,1\right]\times\left[\frac{i-1}{m},\frac{i}{m}\right)}&\text{
for }i\in\\{1,\ldots,m-1\\},\\\
\displaystyle{\left[ma,1\right]\times\left[\frac{i-1}{m},1\right]}&\text{ for
}i=m.\end{cases}$

The sets $\Omega_{\alpha_{i}}^{+}$, $\Omega_{\beta_{i}}^{+}$,
$i\in\\{1,\ldots,m\\}$ are pairwise disjoint and their union equals
$[0,1]^{2}$. Define a map $f_{a}\colon[0,1]^{2}\to[0,1]^{2}$ by

(1.3)
$\begin{split}f_{a}(x,y)=\begin{cases}\displaystyle{\left(\tau_{a}(x),\frac{y}{m}+\frac{i-1}{m}\right)}&\text{
on }\Omega_{\alpha_{i}}^{+},\ i\in\\{1,\ldots,m\\},\\\
\displaystyle{\left(\tau_{a}(x),my-i+m+1\right)}&\text{ on
}\Omega_{\beta_{i}}^{+},\ i\in\\{1,\ldots,m\\}.\end{cases}\end{split}$

See Figure 2 for $m=2$.

Figure 2. The map $f_{a}$ for $m=2$: The domain $\Omega_{\alpha_{i}}^{+}$ and
its image are labeled with $\alpha_{i}$ and $\alpha_{i}^{\prime}$
respectively, the same for $\Omega_{\beta_{i}}^{+}$;
$f_{a}(\Omega_{\beta_{1}}^{+})=[0,1]\times[0,1)$ and
$f_{a}(\Omega_{\beta_{2}}^{+})=[0,1]^{2}$. Figure 3. The map $f_{a,b}$ for
$m=2$: The domain $\Omega_{\alpha_{i}}^{+}$ and its image are labeled with
$\alpha_{i}$ and $\alpha_{i}^{\prime}$ respectively, the same for
$\Omega_{\beta_{i}}^{+}$.

Next, put $\Omega_{\alpha_{i}}=\Omega_{\alpha_{i}}^{+}\times\left[0,1\right]$
and $\Omega_{\beta_{i}}=\Omega_{\beta_{i}}^{+}\times\left[0,1\right]$ for
$i\in\\{1,\ldots,m\\}$. For $a,b\in(0,\frac{1}{m})$ define a map
$f_{a,b}\colon[0,1]^{3}\to[0,1]^{3}$ by

(1.4)
$\begin{split}f_{a,b}(x,y,z)=\begin{cases}\displaystyle{\left(f_{a}(x,y),(1-mb)z\right)}&\text{
on }\Omega_{\alpha_{i}},\ i\in\\{1,\ldots,m\\},\\\
\displaystyle{\left(f_{a}(x,y),bz+1+b(i-m-1)\right)}&\text{ on
}\Omega_{\beta_{i}},\ i\in\\{1,\ldots,m\\}.\end{cases}\end{split}$

See Figure 3. In other words, $f_{a}$ is the projection of $f_{a,b}$ to the
$xy$-plane. Note that $f_{a,b}$ is one-to-one except on the points in the
boundaries of $\Omega_{\alpha_{i}}$, $\Omega_{\beta_{i}}$ where it may be
discontinuous and at most three-to-one. We call $f_{a}$ and $f_{a,b}$
heterochaos baker maps111The baker map is defined on $[0,1)^{2}$, whereas
$f_{a}$ is defined on $[0,1]^{2}$. This minor discrepancy has stemmed from the
intention of the authors of [28] to visualize the ‘heteroclinic set’ in [28]..

The heterochaos baker maps introduced in [28] were precisely $f_{\frac{1}{3}}$
and $f_{\frac{1}{3},\frac{1}{6}}$ with $m=2$. Symbolic dynamics for $f_{a}$,
$f_{a,b}$ and their measures of maximal entropy were investigated in [32].
Saiki et al. [29] considered some variants of $f_{a,b}$ in the context of
fractal geometry and homoclinic bifurcations of diffeomorphisms. Homburg and
Kalle [17] considered iterated function systems intimately related to
$f_{a,b}$.

For any $a\in(0,\frac{1}{m})$, $f_{a}$ preserves the Lebesgue measure on
$[0,1]^{2}$. The map $f_{a,b}$ preserves the Lebesgue measure on $[0,1]^{3}$
if and only if $a+b=\frac{1}{m}$. Put

$g_{a}=f_{a,\frac{1}{m}-a}.$

We will drop $a$ from notation and write $f$, $g$ and so on when the context
is clear.

###  1.2. The Dyck system

Krieger [21] introduced a certain class of subshifts having some algebraic
property, called property A subshifts. It is a fundamental shift space in this
class. The Dyck system is a subshift on the alphabet $D$ in (1.2) consisting
of $m$ brackets, $\alpha_{i}$ left and $\beta_{i}$ right in pair, whose
admissible words are sequences of legally aligned brackets. To be more
precise, let $D^{*}$ denote the set of finite words in $D$. Consider the
monoid with zero, with $2m$ generators in $D$ and the unit element $1$ with
relations

(1.5) $\begin{split}\alpha_{i}\cdot\beta_{j}&=\delta_{i,j},\ 0\cdot 0=0\text{
and }\\\ \gamma\cdot 1&=1\cdot\gamma=\gamma,\ \gamma\cdot
0=0\cdot\gamma=0\text{ for }\gamma\in D^{*}\cup\\{1\\},\end{split}$

where $\delta_{i,j}$ denotes Kronecker’s delta. For $n\geq 1$ and
$\gamma_{1}\gamma_{2}\cdots\gamma_{n}\in D^{*}$ let

${\rm red}(\gamma_{1}\cdots\gamma_{n})=\prod_{i=1}^{n}\gamma_{i}.$

The one and two-sided Dyck shifts on $2m$ symbols are respectively defined by

$\begin{split}\Sigma_{D}^{+}&=\\{\omega\in D^{\mathbb{Z}_{+}}\colon{\rm
red}(\omega_{i}\cdots\omega_{j})\neq 0\text{ for }i,j\in\mathbb{Z}_{+}\text{
with }i<j\\},\\\ \Sigma_{D}&=\\{\omega\in D^{\mathbb{Z}}\colon{\rm
red}(\omega_{i}\cdots\omega_{j})\neq 0\text{ for }i,j\in\mathbb{Z}\text{ with
}i<j\\},\end{split}$

where $\mathbb{Z}_{+}$ denotes the set of non-negative integers. Let
$\sigma_{+}$ and $\sigma$ denote the left shifts acting on $\Sigma_{D}^{+}$
and $\Sigma_{D}$ respectively.

As a counterexample to the conjecture of Weiss [34], Krieger [20] showed that
the Dyck system has exactly two ergodic measures of maximal entropy, and they
are fully supported and Bernoulli. Meyerovitch [25] proved the existence of
tail invariant measures for the Dyck system. To our knowledge, there is no
result on further statistical properties of the measures of maximal entropy
for the Dyck system.

###  1.3. Statements of the results

Under the iteration of $f_{a}$, the $x$-direction is expanding by factor
$\frac{1}{a}$ or $\frac{1}{1-ma}$, while the $y$-direction is a center:
contracting by factor $\frac{1}{m}$ on $\Omega^{+}_{\alpha}$ and expanding by
factor $m$ on $\Omega^{+}_{\beta}$, where

$\Omega^{+}_{\alpha}=\bigcup_{i=1}^{m}\Omega_{\alpha_{i}}^{+}\ \text{ and }\
\Omega^{+}_{\beta}=\bigcup_{i=1}^{m}\Omega_{\beta_{i}}^{+}.$

The local stability in the $y$-direction along each orbit is determined by the
asymptotic time average of the function

(1.6) $\phi^{c}(p)=\begin{cases}-\log m&\text{ on }\Omega^{+}_{\alpha},\\\
\log m&\text{ on }\Omega^{+}_{\beta},\end{cases}$

called the central Jacobian. Write $S_{n}\phi^{c}$ for the sum
$\sum_{k=0}^{n-1}\phi^{c}\circ f_{a}^{k}$. Since $f_{a}$ is a skew product map
over $\tau_{a}$ and $\phi^{c}$ is constant on sets of the form
$\\{x\\}\times[0,1]$, $x\in[0,1]$, the ergodicity of $\tau_{a}$ with respect
to the Lebesgue measure on $[0,1]$ implies that

(1.7) $\lim_{n\to\infty}\frac{1}{n}S_{n}\phi^{c}(p)=(1-2ma)\log m\ \text{ for
Lebesgue a.e. $p\in[0,1]^{2}$.}$

We classify $f_{a}$, $g_{a}$ into three types according to the sign of this
limit value:

  * (i)

$a\in(0,\frac{1}{2m})$ (mostly expanding center),

  * (ii)

$a\in(\frac{1}{2m},\frac{1}{m})$ (mostly contracting center),

  * (iii)

$a=\frac{1}{2m}$ (mostly neutral center).

In [28], $f_{\frac{1}{3}}$ and $g_{\frac{1}{3}}$ with $m=2$ were shown to be
weak mixing with respect to the Lebesgue measure. The proof there can be
slightly modified to show the weak mixingness of $f_{a}$ and $g_{a}$ with
respect to the Lebesgue measure for any
$a\in(0,\frac{1}{m})\setminus\\{\frac{1}{2m}\\}$. Our first result
considerably strengthens this. Let $(X,\mathscr{B},\nu)$ be a probability
space, $T\colon X\to X$ be a measurable map preserving $\nu$ and let $k\geq 2$
be an integer. We say $(T,\nu)$ is $k$-mixing if for all
$A_{0},A_{1}\ldots,A_{k-1}\in\mathscr{B}$,

$\lim_{n_{1},\ldots,n_{k-1}\to\infty}\nu(A_{0}\cap
T^{-n_{1}}(A_{1})\cap\cdots\cap T^{-n_{1}-n_{2}-\cdots-
n_{k-1}}(A_{k-1}))=\prod_{j=0}^{k-1}\nu(A_{j}).$

$2$-mixing is usually called mixing or strong mixing. Clearly $(k+1)$-mixing
implies $k$-mixing, but the converse is unknown. We say $(T,\nu)$ is mixing of
all orders if it is $k$-mixing for any $k\geq 2$. Let ${\rm Leb}$ denote the
Lebesgue measure on $[0,1]^{2}$ or $[0,1]^{3}$.

######  Theorem A.

For any $a\in(0,\frac{1}{m})$, $(f_{a},{\rm Leb})$ and $(g_{a},{\rm Leb})$ are
mixing of all orders.

In fact, we will show that $(f_{a},{\rm Leb})$ is exact and the restriction of
$(g_{a},{\rm Leb})$ to a set of full Lebesgue measure has $K$-property
(Proposition 3.2). These properties imply mixing of all orders [26, Section
2.6].

Concerning rates of $2$-mixing we obtain the following result. All functions
appearing in this paper are real-valued. For a pair $\varphi$, $\psi$ of
functions in $L^{2}(\nu)$, define their correlations by

${\rm Cor}_{n}(T;\varphi,\psi;\nu)=\left|\int\varphi(\psi\circ T^{n}){\rm
d}\nu-\int\varphi{\rm d}\nu\int\psi{\rm d}\nu\right|\text{ for }n\geq 1.$

For a metric space $X$ and $\eta\in(0,1]$ let $\mathscr{H}_{\eta}(X)$ denote
the set of Hölder continuous functions on $X$ with a Hölder exponent $\eta$.
We say $(T,\nu)$ is exponentially mixing if for each $\eta\in(0,1]$ there
exists $\lambda=\lambda(\eta)\in(0,1)$ such that for any pair $\varphi$,
$\psi$ of functions in $\mathscr{H}_{\eta}(X)$ there exists
$C=C(\varphi,\psi)>0$ such that ${\rm Cor}_{n}(T;\varphi,\psi;\nu)\leq
C\lambda^{n}$ for all $n\geq 1$.

######  Theorem B.

For any $a\in(0,\frac{1}{m})\setminus\\{\frac{1}{2m}\\}$, $(f_{a},{\rm Leb})$
and $(g_{a},{\rm Leb})$ are exponentially mixing.

As we recall in Section 2.9, the heterochaos baker maps are geometric models
of the Dyck system [32, Theorem 1.1]: Following the orbits of $f_{a}$ and
$f_{a,b}$ over the partitions $\\{\Omega_{\gamma}^{+}\\}_{\gamma\in D}$ and
$\\{\Omega_{\gamma}\\}_{\gamma\in D}$, one obtains $\Sigma_{D}^{+}$ and
$\Sigma_{D}$ respectively. We exploit this connection to establish exponential
mixing for the Dyck system.

For the one-sided (resp. two-sided) Dyck shift, there exist exactly two shift
invariant ergodic measures of maximal entropy [20], which we denote by
$\nu_{\alpha}^{+}$ and $\nu_{\beta}^{+}$ (resp. $\nu_{\alpha}$ and
$\nu_{\beta}$), leaving the details to Section 5.1. As usual, the metrics $d$
on the shift spaces are the Hamming metrics: For distinct points $\omega$,
$\omega^{\prime}\in\Sigma_{D}^{+}$,

$d(\omega,\omega^{\prime})=\exp(-\min\\{i\geq
0\colon\omega_{i}\neq\omega_{i}^{\prime}\\}),$

and for distinct points $\omega$, $\omega^{\prime}\in\Sigma_{D}$,

$d(\omega,\omega^{\prime})=\exp(-\min\\{i\geq
0\colon\omega_{i}\neq\omega_{i}^{\prime}\text{ or
}\omega_{-i}\neq\omega_{-i}^{\prime}\\}).$

######  Theorem C.

All $(\sigma_{+},\nu_{\alpha}^{+})$, $(\sigma_{+},\nu_{\beta}^{+})$,
$(\sigma,\nu_{\alpha})$, $(\sigma,\nu_{\beta})$ are exponentially mixing.

###  1.4. Outline of proofs and the structure of the paper

Uniformly hyperbolic systems have Markov partitions, and their statistical
properties are well-understood on a symbolic level [7, 27, 31]. The difficulty
in analyzing statistical properties of the heterochaos baker maps consists in
the lack of a Markov partition, which is precisely due to the dynamics in the
$y$-direction.

To overcome this difficulty, we combine inducing and large deviations. A main
tool in our construction is a Markov diagram, originally introduced by
Hofbauer [15, 16] and developed by Keller [18], Buzzi [5] and so on. In
Section 2, for $f_{a}$, $a\in(0,\frac{1}{m})$ we define a stopping time
$R\colon[0,1]^{2}\to\mathbb{Z}_{+}\cup\\{\infty\\}$, and construct a uniformly
expanding induced Markov map $f^{R}$ with infinitely many branches. We show
that $\\{R=\infty\\}$ is a null set for $a\in(0,\frac{1}{2m}]$ (Proposition
2.6), and the Lebesgue measure of the tail $\\{R>n\\}$ decays exponentially in
$n$ for $a\in(0,\frac{1}{2m})$ (Proposition 2.9). Since the heterochaos baker
maps are piecewise affine, all these construction and estimates are purely
combinatorial and distortion estimates are not needed. In Section 3 we show
that both $(f_{a},{\rm Leb})$ and $(g_{a},{\rm Leb})$ are mixing of all orders
for all $a\in(0,\frac{1}{2m}]$. Using the ‘invariance’ of correlations under
the replacement of $a$ by $\frac{1}{m}-a$ (Proposition 3.1), we complete the
proof of Theorem A. In Section 4 we construct towers, and apply Young’s result
[35] to deduce the exponential mixing for the heterochaos baker maps. In order
to estimate error bounds, we perform a large deviation argument for the map
$\tau_{a}$. Using the invariance of correlations again completes the proof of
Theorem B.

A proof of Theorem C in Section 5 relies on a surprising connection between
one of the two ergodic measures of maximal entropy for the Dyck system and the
Lebesgue measure (Proposition 5.7). We emphasize that Theorem C is not an
immediate consequence of Theoerem B, since the coding maps (see Section 2.9
for the definition) from the heterochaos baker maps into the Dyck system do
not preserve Hölder continuous functions. Our idea is to transfer the towers
for the heterochaos baker maps to the Dyck shift spaces, and apply Young’s
result [35] again to obtain the exponential mixing with respect to one of the
two ergodic measures of maximal entropy. Exponential mixing with respect to
the other one follows from the symmetry in the Dyck system (Proposition 5.3).

###  1.5. Comparison with results on partially hyperbolic systems

Like the heterochaos baker maps, skew product maps over uniformly hyperbolic
or expanding bases are considered to be simple models of partially hyperbolic
systems222The splitting of the tangent bundle of $[0,1]^{2}$ into $x$\- and
$y$-directions does not always give rise to a partially hyperbolic splitting
for $f_{a}$: If $0<a<\frac{m-1}{m^{2}}$ then the minimum expansion rate in the
$x$-direction is $\min\\{\frac{1}{a},\frac{1}{1-ma}\\}$, which does not
dominate the expansion rate $m$ in the $y$-direction.. Hence, it is relevant
to compare our results with others on exponential decay of correlations for
partially hyperbolic systems.

Young’s method [35, 36] of deducing nice statistical properties using towers
with fast decaying tails has been successfully applied to some partially
hyperbolic systems, e.g., [3, 8, 9, 13] to deduce exponential decay of
correlations. With more functional analytic methods, exponential decay of
correlations was proved for certain partially hyperbolic diffeomorphisms [10],
and for certain piecewise partially hyperbolic endomorphisms [4]. Clearly the
heterochaos baker maps are covered by none of these existing results.

###  1.6. Comparison with results on subshifts

For a large class of subshifts with ‘non-uniform specification’, Climenhaga
[11] constructed towers and applied Young’s method to obtain nice statistical
properties including exponential decay of correlations. Clearly the Dyck
system does not satisfy the ‘non-uniform specification’, since it would imply
the uniqueness of measure of maximal entropy. One can also directly check the
breakdown of ’non-uniform specification’ using the definition of the Dyck
system.

We believe it would be difficult to construct towers for the Dyck system to
which one can apply Young’s result, purely within the symbolic setting with no
recourse to the connection with the heterochaos baker maps proved in [32]. The
geometric approach to the Dyck system in the proof of Theorem C based on this
connection seems promising, in order to establish many other dynamical
properties of the measures of maximal entropy for the Dyck system. Moreover,
this approach may be extended to some other subshifts with a high symmetry,
such as the Motzkin system [23].

##  2\. Dynamics of the heterochaos baker maps

In this section we analyze the dynamics of the heterochaos baker map $f_{a}$,
$a\in(0,\frac{1}{m})$. In Section 2.1 we introduce a Markov diagram for
$f_{a}$ and describe its combinatorial structure. In Section 2.2 we develop
path counting arguments in the diagram, and in Section 2.3 provide useful
formulas for the Birkhoff sums of the function $\phi^{c}$ in (1.6). In Section
2.4 we introduce a stopping time $R$, and express it as ‘a twice jump time
under a constraint’. In Section 2.5 we give another formula for the stopping
time, and show in Section 2.6 the almost sure finiteness of the stopping time
for $a\in(0,\frac{1}{2m}]$. In Section 2.7 we construct an induced Markov map,
and in Section 2.8 estimate the Lebesgue measure of the tail of the stopping
time. In Section 2.9 we recall the connection between the heterochaos baker
maps and the Dyck system established in [32].

###  2.1. A Markov diagram

Let ${\rm int}$ denote the interior operation in $\mathbb{R}^{2}$ or
$\mathbb{R}^{3}$. Let $A\subset[0,1]^{2}$ be a non-empty set with
$A\subset{\rm int}(\Omega_{\gamma}^{+})$ for some $\gamma\in D$. A set
$B\subset[0,1]^{2}$ is a successor of $A$ if $B=f(A)\cap{\rm
int}(\Omega_{\gamma^{\prime}}^{+})$ holds for some $\gamma^{\prime}\in D$. If
$B$ is a successor of $A$, we write $A\to B$. We set

$\mathcal{V}_{\alpha}=\\{{\rm int}(\Omega_{\alpha_{i}}^{+})\colon
i\in\\{1,\ldots,m\\}\\}\ \text{ and }\ \mathcal{V}_{\beta}=\\{{\rm
int}(\Omega_{\beta_{i}}^{+})\colon i\in\\{1,\ldots,m\\}\\},$

and define $\mathcal{V}_{n}$, $n=0,1,\ldots$ inductively by

$\mathcal{V}_{n}=\begin{cases}\mathcal{V}_{\alpha}\cup\mathcal{V}_{\beta}&\text{
for }n=0,\\\ \mathcal{V}_{n-1}\cup\\{B\colon\text{$B$ is a successor of an
element of $\mathcal{V}_{n-1}$}\\}&\text{ for }n\geq 1.\end{cases}$

We set $\mathcal{V}=\bigcup_{n=0}^{\infty}\mathcal{V}_{n}$. The oriented graph
$(\mathcal{V},\to)$ is called a Markov diagram for $f$. We decompose the set
$\mathcal{V}$ of vertices into infinitely many levels $\mathcal{L}_{r}$,
$r=-1,0,1,\ldots$ as follows:

$\mathcal{L}_{r}=\begin{cases}\mathcal{V}_{\alpha}&\text{ for }r=-1,\\\
\mathcal{V}_{1}\setminus\mathcal{V}_{\alpha}&\text{ for }r=0,\\\
\mathcal{V}_{r+1}\setminus\mathcal{V}_{r}&\text{ for }r\geq 1.\end{cases}$

See Figure 4. Define $l\colon\mathcal{V}\to\mathbb{Z}$ by

$v\in\mathcal{L}_{l(v)}\text{ for each }v\in\mathcal{V}.$

Figure 4. Part of the Markov diagram $(\mathcal{V},\to)$ for $m=2$. We put
$\alpha_{1}=1$, $\alpha_{2}=2$, $\beta_{1}=3$, $\beta_{1}=4$ for simplicity.
The vertices ${\rm int}(\Omega_{\gamma_{1}}^{+})$, $f({\rm
int}(\Omega_{\gamma_{1}}^{+}))\cap{\rm int}(\Omega_{\gamma_{2}}^{+})$,
$f(f({\rm int}(\Omega_{\gamma_{1}}^{+}))\cap{\rm
int}(\Omega_{\gamma_{2}}^{+}))\cap{\rm int}(\Omega_{\gamma_{3}}^{+})$ are
labeled with $\gamma_{1}$, $\gamma_{1}\gamma_{2}$,
$\gamma_{1}\gamma_{2}\gamma_{3}$ respectively.

It is possible to describe all outgoing edges and incoming edges for each
vertex in $\mathcal{V}$. However, we only have to estimate from above the
number of paths of a given length connecting two given vertices. For this
purpose, it suffices to describe the cardinalities of all outgoing edges from
each vertex $v\in\mathcal{V}$ which are either upward, downward or parallel
(shown in black, red, blue arrows respectively in Figure 4), in terms of the
function $l$. The cardinalities of the three sets

$\begin{split}\mathcal{V}_{\uparrow}(v)&=\\{v^{\prime}\in\mathcal{V}\colon
v\to v^{\prime},\ l(v^{\prime})=l(v)+1\\},\\\
\mathcal{V}_{\downarrow}(v)&=\\{v^{\prime}\in\mathcal{V}\colon v\to
v^{\prime},\ l(v^{\prime})=l(v)-1\\},\\\
\mathcal{V}_{\to}(v)&=\\{v^{\prime}\in\mathcal{V}\colon v\to v^{\prime},\
l(v^{\prime})=l(v)\\},\end{split}$

are given as follows:

  * (i)

For each $v\in\mathcal{V}_{\alpha}$,

$\\#\mathcal{V}_{\uparrow}(v)=m+1,\ \\#\mathcal{V}_{\downarrow}(v)=0,\
\\#\mathcal{V}_{\to}(v)=0.$

  * (ii)

For each $v\in\mathcal{V}_{\beta}$,

$\\#\mathcal{V}_{\uparrow}(v)=0,\ \\#\mathcal{V}_{\downarrow}(v)=m,\
\\#\mathcal{V}_{\to}(v)=m.$

  * (iii)

For each $v\in\mathcal{V}\setminus\mathcal{V}_{0}$ with
$v\subset\Omega^{+}_{\alpha}$,

$\\#\\{v^{\prime}\in\mathcal{V}_{\uparrow}(v)\\}=m+1,\
\\#\mathcal{V}_{\downarrow}(v)=0,\ \\#\mathcal{V}_{\to}(v)=0.$

  * (iv)

For each $v\in\mathcal{V}\setminus\mathcal{V}_{0}$ with
$v\subset\Omega^{+}_{\beta}$,

$\\#\\{v^{\prime}\in\mathcal{V}_{\uparrow}(v)\\}=0,\
\\#\mathcal{V}_{\downarrow}(v)=m+1,\ \\#\mathcal{V}_{\to}(v)=0.$

###  2.2. Counting paths in the Markov diagram

Let $n\geq 1$. A path of length $n$ is a word $v_{0}\cdots v_{n}$ of elements
of $\mathcal{V}$ of word length $n+1$ satisfying $v_{k}\to v_{k+1}$ for every
$k\in\\{0,\ldots,n-1\\}$. We say $k\in\\{0,\ldots,n-1\\}$ is a hold time of a
path $v_{0}\cdots v_{n}$ if $\\{v_{k},v_{k+1}\\}\subset\mathcal{V}_{\beta}$.
Let $P_{n}$ denote the set of paths of length $n$ which have no hold time.

In order to estimate the cardinalities of $P_{n}$ and its various subsets, we
consider projections to paths of the symmetric random walk on $\mathbb{Z}$.
Let

(2.1) $Z_{n}=\left\\{l_{0}\cdots
l_{n}\in\mathbb{Z}^{n+1}\colon|l_{k}-l_{k+1}|=1\text{ for
}k\in\\{0,\ldots,n-1\\}\right\\}.$

Define a projection $\Phi_{n}\colon P_{n}\to Z_{n}$ by

$\Phi_{n}(v_{0}v_{1}\cdots v_{n})=l(v_{0})l(v_{1})\cdots l(v_{n}).$

######  Lemma 2.1.

Let $n\geq 1$ and let $l_{0}l_{1}\cdots l_{n}\in Z_{n}$. We have

$\\#\Phi_{n}^{-1}(l_{0}l_{1}\cdots
l_{n})\leq(m+1)m^{\frac{1}{2}(n+l_{n}-l_{0})}.$

###### Proof.

A path $u_{0}\cdots u_{j}$ in the Markov diagram is upward if
$l(u_{k})<l(u_{k+1})$ for all $k\in\\{0,\ldots,j-1\\}$, and downward if
$l(u_{k+1})>l(u_{k})$ for all $k\in\\{0,\ldots,j-1\\}$. Let $r$, $s$ be
integers with $-1\leq r<s$. From the description of the Markov diagram in
Section 2.1, the number of upward paths from one vertex in $\mathcal{L}_{r}$
to another in $\mathcal{L}_{s}$ does not exceed $m^{s-r-1}(m+1)$, and the
number of upward paths from one vertex in $\mathcal{L}_{r}$ to another in
$\mathcal{L}_{s}$ which can be concatenated to a downward path does not exceed
$m^{s-r-1}$. Reciprocally, the number of downward paths from one vertex in
$\mathcal{L}_{s}$ to another in $\mathcal{L}_{r}$ does not exceed $m+1$, and
the number of downward paths from one vertex in $\mathcal{L}_{s}$ to another
in $\mathcal{L}_{r}$ which can be concatenated to an upward path does not
exceed $m$.

Let $l_{0}\cdots l_{n}\in Z_{n}$. We may assume $l_{k}\geq-1$ for
$k\in\\{0,\ldots,n\\}$ for otherwise the desired inequality is trivial. Any
path in $\Phi_{n}^{-1}(l_{0}\cdots l_{n})$ is uniquely written as the
alternate concatenation of upward and downward paths, see Figure 5 for
example. The sum of the lengths of all the upward paths in this concatenation
equals $\frac{1}{2}(n+l_{n}-l_{0})$. Hence we obtain the desired inequality. ∎

Figure 5. Alternate concatenations of upward and downward paths for $m=2$ with
$\alpha_{1}=1$, $\alpha_{2}=2$, $\beta_{1}=3$, $\beta_{1}=4$ for simplicity:
(a) $3\to 1\to 2\to 4$; (b) $1\to 11\to\cdots\to 3\to 1$.

For a vertex $v\in\mathcal{L}_{0}\setminus\mathcal{V}_{\beta}$ and positive
integers $n$, $j$, let $P_{n}(j;v)$ denote the set of paths $v_{0}\cdots
v_{n}$ of length $n$ such that $v_{0}=v$, $l(v_{k})\geq 1$ for all
$k\in\\{1,\ldots,n\\}$ and $l(v_{n})=j$. Note that $P_{n}(j;v)\neq\emptyset$
if and only if $n\equiv j$ mod $2$.

######  Lemma 2.2.

For any $v\in\mathcal{L}_{0}\setminus\mathcal{V}_{\beta}$ and all positive
integers $n$, $j$ with $n\equiv j$ mod $2$, we have

$\\#P_{n}(j;v)\leq\frac{j}{n}\begin{pmatrix}n\\\
\frac{n+j}{2}\end{pmatrix}\frac{m+1}{m}m^{\frac{n+j}{2}}.$

###### Proof.

Since each path in $P_{n}(j;v)$ has no hold time, each element of
$\Phi_{n}(P_{n}(j;v))$ may be viewed a path of the symmetric random walk on
$\mathbb{Z}$ starting at the position $j$ which hits the origin $0$ at step
$n$ for the first time. The reflection principle for the random walk gives

(2.2) $\\#\Phi_{n}(P_{n}(j;v))=\frac{j}{n}\begin{pmatrix}n\\\
\frac{n+j}{2}\end{pmatrix}.$

By Lemma 2.1, for each $v_{0}\cdots v_{n}\in P_{n}(j;v)$ we have

(2.3) $\\#\Phi_{n}^{-1}(l(v_{0})\cdots
l(v_{n}))\leq\frac{m+1}{m}m^{\frac{n+j}{2}}.$

Combining (2.2) and (2.3) we obtain the desired inequality.∎

###  2.3. Formulas for Birkhoff sums of the central Jacobian

Let $n\geq 1$ and let $v_{0}\cdots v_{n}$ be a path in the Markov diagram
$(\mathcal{V},\to)$. The Birkhoff sum $S_{n}\phi^{c}$ on the rectangle
$\bigcap_{k=0}^{n}f^{-k}(v_{k})$ is a constant, which we denote by
$S_{n}\phi^{c}(v_{0}\cdots v_{n})$ with a slight abuse of notation. From the
structure of the Markov diagram described in Section 2.1, we derive two useful
formulas for this constant.

  * (i)

If $v_{0}\cdots v_{n}\in P_{n}$, then

(2.4) $S_{n}\phi^{c}(v_{0}\cdots v_{n})=(l(v_{0})-l(v_{n}))\log m.$

  * (ii)

If $v_{0}\in\mathcal{V}_{\beta}$ and $v_{n}\in\mathcal{V}_{\beta}$ then

(2.5) $S_{n}\phi^{c}(v_{0}\cdots
v_{n})=\\#\\{k\in\\{0,\ldots,n-1\\}\colon\text{$k$ is a hold time of
$v_{0}\cdots v_{n}$}\\}\log m.$

Since any path in $(\mathcal{V},\to)$ can be written as a concatenation of
paths of the above two kinds, the Birkhoff sum of the central Jacobian along
any path is kept in track by a combination of the formulas (2.4) and (2.5).

###  2.4. Stopping time

We denote by $|\cdot|$ the Lebesgue measure on $[0,1]$, $[0,1]^{2}$,
$[0,1]^{3}$. A rectangle (resp. block) is a product of two (resp. three) non-
degenerate intervals in $[0,1]$. For a rectangle $B=B_{x}\times B_{y}$ we
write

$|B|_{x}=|B_{x}|\ \text{ and }\ |B|_{y}=|B_{y}|.$

Similarly, for a block $B=B_{x}\times B_{y}\times B_{z}$ we write

$|B|_{x}=|B_{x}|,\ |B|_{y}=|B_{y}|\ \text{ and }\ |B|_{z}=|B_{z}|.$

For $p\in[0,1]^{2}$ and $n\geq 1$, let $K_{n}(p)$ denote the maximal rectangle
containing $p$ on which $f^{n}$ is affine. Clearly $|K_{1}(p)|_{y}=1$ for
$p\in\Omega^{+}_{\alpha}$, $|K_{1}(p)|_{y}=\frac{1}{m}$ for
$p\in\Omega^{+}_{\beta}$ and
$|K_{n}(p)|_{y}/|K_{n-1}(p)|_{y}\in\left\\{1,\frac{1}{m}\right\\}$ for $n\geq
2$. An integer $n\geq 2$ is a cutting time of $p$ if
$|K_{n}(p)|_{y}/|K_{n-1}(p)|_{y}=\frac{1}{m}$. We define a stopping time

(2.6) $R(p)=\inf\\{n\geq 2\colon\text{$n$ is a cutting time of $p$}\\}.$

Clearly we have $R(p)=2$ if and only if $p$ is contained in the set

$E=\Omega^{+}_{\beta}\cap f^{-1}(\Omega^{+}_{\beta}).$

The stopping time is expressed as follows. Put $S_{0}\phi^{c}\equiv 0$ for
convenience.

######  Proposition 2.3.

For all $p\in[0,1]^{2}$ we have

$R(p)=\begin{cases}\inf\\{n\geq 0\colon S_{n}\phi^{c}(p)=-\log m\text{ and
}f^{n}(p)\in E\\}+2&\text{ on }\Omega^{+}_{\alpha},\\\ \inf\\{n\geq 0\colon
S_{n}\phi^{c}(p)=0\text{ and }f^{n}(p)\in E\\}+2&\text{ on
}\Omega^{+}_{\beta}.\end{cases}$

Before proceeding let us record two equalities. First,

(2.7) $f(\Omega^{+}_{\beta}\setminus E)=\Omega^{+}_{\alpha}.$

The definition of the stopping time (2.6) implies

(2.8) $R\circ f=R-1\ \text{ on }\Omega^{+}_{\beta}\setminus E.$

###### Proof of Proposition 2.3.

By (2.7) and (2.8) it suffices to consider the case
$p\in\Omega^{+}_{\beta}\setminus E$. Clearly we have $R(p)\geq 3$. We claim
$S_{R(p)-2}\phi^{c}(p)=0$, for otherwise (2.5) implies
$S_{R(p)-2}\phi^{c}(p)>0$, and there is $k\in\\{2,\ldots,R(p)-1\\}$ such that
$f^{k-2}(p)\in E$ and $S_{k-2}\phi^{c}(p)=0$. From Lemma 2.4 below, it follows
that $k$ is a cutting time of $p$, a contradiction to the minimality in the
definition of $R(p)$. This claim and (2.4) together imply $S_{n}\phi^{c}(p)<0$
for all $n\geq 1$ with $n\leq R(p)-3$, and so the desired equality holds. ∎

######  Lemma 2.4.

Let $p\in\Omega^{+}_{\beta}$ and $n\geq 2$. Then $n$ is a cutting time of $p$
if and only if $f^{n-2}(p)\in E$ and $S_{n-2}\phi^{c}(p)\geq 0$.

###### Proof.

To show the ‘if’ part, suppose $n\geq 2$, $f^{n-2}(p)\in E$ and
$S_{n-2}\phi^{c}(p)\geq 0$. Take $i\in\\{1,\ldots,m\\}$ with
$f^{n-2}(p)\in\Omega_{\beta_{i}}^{+}$. Let $B$ denote the connected component
of $f^{-n+2}(\Omega_{\beta_{i}}^{+})$ that contains $p$. Then $B$ is a
rectangle, $f^{n-2}|_{B}$ is affine and $|f^{n-2}(B)|_{y}=\frac{1}{m}$. Hence
$n$ is a cutting time of $p$. To show the ‘only if’ part, let $n\geq 2$ be a
cutting time of $p$. The definition of cutting time implies $f^{n-2}(p)\in E$.
Since $p\in\Omega^{+}_{\beta}$, by (2.5) we obtain $S_{n-2}\phi^{c}(p)\geq 0$.
∎

###  2.5. Formula for the stopping time in terms of pullbacks

Let $A$ be a non-empty subset of $[0,1]^{2}$ and let $n\geq 1$. A connected
component of $f^{-n}(A)$ is called a pullback of $A$ by $f^{n}$. If $B$ is a
pullback of $A$ by $f^{n}$ and $f^{n}|_{B}$ is affine, we call $B$ an affine
pullback of $A$ by $f^{n}$, or simply an affine pullback. If $A$ is connected
and $B$ is an affine pullback of $A$ by $f^{n}$, then $f^{n}(B)=A$.

######  Proposition 2.5.

For any $p\in(0,1)^{2}$ such that $R(p)$ is finite and
$f^{R(p)}(p)\in(0,1)^{2}$, we have

$R(p)=\min\left\\{\begin{split}&n\geq 2\colon\text{there exists an affine
pullback $B$ of $(0,1)^{2}$ by $f^{n}$}\\\ &\quad\quad\quad\ \text{such that
}p\in B\text{ and }|B|_{y}=\frac{1}{m}|K_{1}(p)|_{y}\end{split}\right\\}.$

###### Proof.

In the case $R(p)=2$, we have $p\in E$ and the desired equality is obvious.
Suppose $R(p)\geq 3$. Then $p\notin E$ holds. By Proposition 2.3, there exists
$i\in\\{1,\ldots,m\\}$ such that $f^{R(p)-2}(p)\in\Omega^{+}_{\beta_{i}}\cap
E$. The pullback of the rectangle $\Omega_{\beta_{i}}^{+}\cap E$ by
$f^{R(p)-2}$ that contains $p$, denoted by $B^{\prime}$, is an affine pullback
and satisfies $|B^{\prime}|_{y}=|K_{1}(p)|_{y}$. Proposition 2.3 implies ${\rm
int}(\Omega_{\beta_{i}}^{+}\cap E)\subset f^{R(p)-2}(B^{\prime})$. In
particular, $f^{R(p)-1}|_{B^{\prime}}$ is affine and ${\rm
int}(\Omega^{+}_{\beta})\subset f^{R(p)-1}(B^{\prime})$. From the assumption
$f^{R(p)}(p)\in(0,1)^{2}$, there exists an affine pullback $B$ of $(0,1)^{2}$
by $f^{R(p)}$ satisfying $p\in B\subset B^{\prime}$ and
$|B|_{y}=\frac{1}{m}|B^{\prime}|_{y}=\frac{1}{m}|K_{1}(p)|_{y}$. Hence, $R(p)$
does not exceed the minimum in the right-hand side of the desired equality in
the proposition. The reverse inequality is obvious from the property of $B$. ∎

###  2.6. Almost sure finiteness of the stopping time

Clearly, if $a\in(\frac{1}{2m},\frac{1}{m})$ then the set $\\{R=\infty\\}$ has
positive Lebesgue measure. Otherwise this is a null set.

######  Proposition 2.6.

If $a\in(0,\frac{1}{2m}]$ then the stopping time is finite Lebesgue a.e. on
$[0,1]^{2}$.

###### Proof.

Proposition 2.3 and (2.4), (2.5) together imply
$\limsup_{n\to\infty}\frac{1}{n}S_{n}\phi^{c}(p)\leq 0$ for all
$p\in\\{R=\infty\\}$. Recall (1.7) that $\frac{1}{n}S_{n}\phi^{c}$ converges
almost surely to the positive constant $(1-2ma)\log m$. Hence, if
$a\in(0,\frac{1}{2m})$ then $R$ is finite Lebesgue a.e. on $[0,1]^{2}$.

For the rest of the proof of Proposition 2.6 we assume $a=\frac{1}{2m}$. We
need two lemmas.

######  Lemma 2.7.

For Lebesgue a.e. $x\in[0,1]$, there exists a pair $(M,N)$ of positive
integers such that

$\sup_{n\geq N}\frac{1}{\sqrt{n}}|S_{n}\phi^{c}(x,0)|\leq M.$

###### Proof.

Put $v(\phi^{c})=\int_{[0,1]}|\phi^{c}(x,0)|^{2}{\rm d}x$. Let
$(a_{k})_{k=2}^{\infty}$ be the increasing sequence of positive reals
satisfying $(1/\sqrt{2\pi
v(\phi^{c})})\int_{\mathbb{R}\setminus[-a_{k},a_{k}]}\exp\left(-\frac{x^{2}}{2v(\phi^{c})}\right){\rm
d}x=1/k^{2}$ for all $k\geq 2$. For $k\geq 2$ and $n\geq 1$, define

$G_{k,n}=\left\\{x\in[0,1]\colon\frac{1}{\sqrt{n}}S_{n}\phi^{c}(x,0)\in[-a_{k},a_{k}]\right\\}.$

The central limit theorem applied to the sequence of independently identically
distributed random variables $x\in[0,1]\mapsto\phi^{c}(f^{j}(x,0))$
$(j=0,1,\ldots)$ with mean $0$ and variance $v(\phi^{c})$ yields

$\lim_{n\to\infty}|G_{k,n}|=\frac{1}{\sqrt{2\pi
v(\phi^{c})}}\int_{[-a_{k},a_{k}]}\exp\left(-\frac{x^{2}}{2v(\phi^{c})}\right){\rm
d}x=1-\frac{1}{k^{2}}.$

Choose a sequence $(n_{k})_{k=2}^{\infty}$ of positive integers such that
$\sum_{k=2}^{\infty}|[0,1]\setminus\bigcup_{n=n_{k}}^{\infty}G_{k,n}|<\infty$.
By Borel-Cantelli’s lemma,

(2.9)
$\left|\limsup_{k\to\infty}\left([0,1]\setminus\bigcup_{n=n_{k}}^{\infty}G_{k,n}\right)\right|=0.$

Let $G$ denote the set of $x\in[0,1]$ for which there is no positive integer
pair $(M,N)$ satisfying $\sup_{n\geq N}|S_{n}\phi^{c}(x,0)|/\sqrt{n}\leq M$.
For any $x\in G$ we have $|S_{n}\phi^{c}(x,0)|/\sqrt{n}\to\infty$ as
$n\to\infty$. Hence, for any $k\geq 2$ there is $n\geq n_{k}$ such that
$|S_{n}\phi^{c}(x,0)|/\sqrt{n}>a_{k}$, and so
$x\in[0,1]\setminus\bigcup_{n=n_{k}}^{\infty}G_{k,n}$. Since $k\geq 2$ is
arbitrary, (2.9) yields $|G|=0$, which verifies Lemma 2.7. ∎

Define $Q\colon[0,1]^{2}\setminus E\to\mathbb{Z}_{+}\cup\\{\infty\\}$ by

$Q(p)=\begin{cases}\inf\\{n\geq 1\colon S_{n}\phi^{c}(p)=-\log m\text{ and
}f^{n}(p)\in\Omega^{+}_{\beta}\\}&\text{ on }\Omega^{+}_{\alpha},\\\
\inf\\{n\geq 1\colon S_{n}\phi^{c}(p)=0\text{ and
}f^{n}(p)\in\Omega^{+}_{\beta}\\}&\text{ on }\Omega^{+}_{\beta}\setminus
E.\end{cases}$

######  Lemma 2.8.

$Q$ is finite Lebesgue a.e. on $[0,1]^{2}\setminus E$.

###### Proof.

By (2.7) and $Q\circ f=Q-1$ on $\Omega^{+}_{\beta}\setminus E$, it suffices to
show that $Q$ is finite Lebesgue a.e. on $\Omega^{+}_{\alpha}$. Recall that
$f_{\frac{1}{2m}}$ is a skew product over the piecewise affine fully branched
map $\tau_{\frac{1}{2m}}$ in (1.1) that preserves the Lebesgue measure on
$[0,1]$, and $\phi^{c}$ is constant on the sets $\\{x\\}\times[0,1]$,
$x\in[0,1]$. By Lemma 2.7, for Lebesgue a.e. $p\in[0,1]^{2}$ there exists a
positive integer pair $(M,N)$ satisfying $\sup_{n\geq
N}|S_{n}\phi^{c}(p)|/\sqrt{n}\leq M$. Moreover, the definition of $Q$ and the
formulas (2.4), (2.5) together imply $\sup_{k\geq 1}S_{k}\phi^{c}(p)\leq-\log
m$ for all $p\in\Omega^{+}_{\alpha}\cap\\{Q=\infty\\}$. Hence, Lebesgue a.e.
point in $\Omega^{+}_{\alpha}\cap\\{Q=\infty\\}$ is contained in the set

(2.10)
$W=\bigcup_{M=1}^{\infty}\bigcup_{N=1}^{\infty}\bigcap_{n=N}^{\infty}W_{M,n},$

where $M$, $n$ are positive integers and

$W_{M,n}=\left\\{p\in\Omega^{+}_{\alpha}\colon\sup_{k\geq
1}S_{k}\phi^{c}(p)\leq-\log m\ \text{ and }\
S_{n}\phi^{c}(p)\geq-M\sqrt{n}\right\\}.$

It suffices to show that $W$ is a null set. For each $M\geq 1$ we have

$W_{M,n}\subset\bigcup_{v\in\mathcal{L}_{0}\setminus\mathcal{V}_{\beta}}\bigcup_{j=1}^{\lfloor
M\sqrt{n}\rfloor}\bigcup_{v_{0}\cdots v_{n}\in
P_{n}(j;v)}\bigcap_{k=0}^{n}f^{-k}(v_{k}),$

where $\lfloor\cdot\rfloor$ denotes the floor function. Using Lemma 2.2 to
bound $\\#P_{n}(j;v)$ and the identity
$|\bigcap_{k=0}^{n}f^{-k}(v_{k})|_{x}=2^{-n}m^{-\frac{n+j}{2}}$ for each path
$v_{0}\cdots v_{n}$ in $P_{n}(j;v)$, we have

$\begin{split}|W_{M,n}|&\leq\sum_{v\in\mathcal{L}_{0}\setminus\mathcal{V}_{\beta}}\sum_{v_{0}\cdots
v_{n}\in P_{n}(j;v)}\left|\bigcap_{k=0}^{n}f^{-k}(v_{k})\right|\\\
&\leq\sum_{j=1}^{\lfloor M\sqrt{n}\rfloor}\frac{j}{n}\begin{pmatrix}n\\\
\frac{n+j}{2}\end{pmatrix}\frac{m+1}{m}m^{\frac{n+j}{2}}2^{-n}m^{-\frac{n+j}{2}}\leq
n^{-\frac{3}{2}}\sum_{j=1}^{\lfloor
M\sqrt{n}\rfloor}j\leq\frac{M^{2}}{\sqrt{n}},\end{split}$

for all sufficiently large $n$. For the third inequality we have used
Stirling’s formula for factorials to evaluate the binomial coefficient. This
yields $|\bigcap_{n=N}^{\infty}W_{M,n}|\leq\inf_{n\geq N}|W_{M,n}|=0$ for all
$M$, $N\geq 1$, and therefore

$\begin{split}\left|W\right|&\leq\sum_{M=1}^{\infty}\sum_{N=1}^{\infty}\left|\bigcap_{n=N}^{\infty}W_{M,n}\right|=0,\end{split}$

as required. This completes the proof of Lemma 2.8. ∎

By virtue of (2.7), (2.8) and $R=2$ on $E$, it suffices to show that $R$ is
finite Lebesgue a.e. on $\Omega^{+}_{\beta}\setminus E$. Let $2\mathbb{Z}_{+}$
denote the set of positive even integers. If $p\in\Omega^{+}_{\beta}\setminus
E$ and $Q(p)$ is finite then $Q(p)\in 2\mathbb{Z}_{+}$. Put $M_{1}=E$, and
$M_{k}=\Omega^{+}_{\beta}\cap\\{Q=k\\}$ for $k\in 2\mathbb{Z}_{+}$. For each
$k\in\\{1\\}\cup 2\mathbb{Z}_{+}$, $M_{k}$ is written as a finite union
$M_{k}=\bigcup_{\ell}M_{k,\ell}$ of pairwise disjoint rectangles $M_{k,\ell}$
with the properties that $M_{k,\ell}\subset\Omega^{+}_{\beta_{i}}$ for some
$i\in\\{1,\ldots,m\\}$, $|M_{k,\ell}|_{y}=\frac{1}{m}$, $f^{k}|_{M_{k,\ell}}$
is affine and ${\rm int}(\Omega^{+}_{\beta_{j}})\subset
f^{k}(M_{k,\ell})\subset\Omega^{+}_{\beta_{j}}$ for some
$j\in\\{1,\ldots,m\\}$. Lemma 2.8 gives
$|\Omega^{+}_{\beta}\setminus\bigcup_{k\in\\{1\\}\cup
2\mathbb{Z}_{+}}M_{k}|=0$.

The map $H\colon\bigcup_{k\in\\{1\\}\cup
2\mathbb{Z}_{+}}M_{k}\to\Omega^{+}_{\beta}$ given by $H(x,y)=f^{k}(x,y)$ for
$(x,y)\in M_{k}$ has the skew product form

$H(x,y)=(H_{1}(x),H_{2}(x,y)).$

The map $H_{1}$ is a piecewise affine map on the interval $[ma,1]$ with
infinitely many full branches, leaving the normalized Lebesgue measure on the
interval invariant and ergodic. From Birkhoff’s ergodic theorem for $H_{1}$,
$n_{1}(p)=\inf\\{n\geq 1\colon H^{n}(p)\in E\\}$ is finite for Lebesgue a.e.
$p\in\Omega^{+}_{\beta}\setminus E$. Moreover, (2.4) implies
$S_{n_{1}(p)}\phi^{c}(p)=0$ for $p\in\Omega^{+}_{\beta}\setminus E$. This
together with Proposition 2.3 implies $R(p)=n_{1}(p)+2<\infty$ for Lebesgue
a.e. $p\in\Omega^{+}_{\beta}\setminus E$. The proof of Proposition 2.6 is
complete. ∎

###  2.7. An induced Markov map

For each $n\geq 1$ let $\mathscr{P}_{n}$ denote the collection of affine
pullbacks of $(0,1)^{2}$ by $f^{n}$ which are contained in $\\{R=n\\}$. We set

$\mathscr{P}=\bigcup_{n=1}^{\infty}\mathscr{P}_{n}.$

Elements of $\mathscr{P}$ are pairwise disjoint, and the stopping time is
constant on each element. We now define an induced map
$f^{R}\colon\bigcup_{\omega\in\mathscr{P}}\omega\to[0,1]^{2}$ by
$f^{R}|_{\omega}=f^{R|_{\omega}}|_{\omega}$ on each $\omega\in\mathscr{P}$,
and put

$\Delta_{0}^{+}=\bigcap_{n=0}^{\infty}(f^{R})^{-n}\left(\bigcup_{\omega\in\mathscr{P}}\omega\right).$

Partition $\Delta^{+}_{0}$ into
$\\{\omega\cap\Delta^{+}_{0}\\}_{\omega\in\mathscr{P}}$, label the partition
elements with an integer $i\geq 1$, and write
$\\{\Delta^{+}_{0,i}\\}_{i=1}^{\infty}=\\{\omega\cap\Delta^{+}_{0}\\}_{\omega\in\mathscr{P}}$
and $R_{i}=R|_{\Delta^{+}_{0,i}}.$ We have
$\Delta^{+}_{0}=\bigcup_{i=1}^{\infty}\Delta^{+}_{0,i}$, and $f^{R}$ maps each
$\Delta_{0,i}^{+}$ affinely onto $\Delta_{0}^{+}$. Proposition 2.6 implies
$|\Delta_{0}^{+}|=1$ for $a\in(0,\frac{1}{2m}]$.

###  2.8. Stopping time estimates

For $a\in(0,\frac{1}{m})$ put

$\chi(a)=-\log\sqrt{a(1-ma)}.$

Note that $\chi(a)\geq\log\sqrt{4m}>0$. Moreover we have
$\sqrt{4m}e^{-\chi(a)}\leq 1$, and the equality holds if and only if
$a=\frac{1}{2m}$.

######  Proposition 2.9.

There exists $n_{0}\geq 1$ such that for all $a\in(0,\frac{1}{2m}]$ and all
$n\geq n_{0}$ we have

$|\\{R=n+2\\}|\leq n^{-\frac{3}{2}}(\sqrt{4m}e^{-\chi(a)})^{n}.$

In particular, if $a\in(0,\frac{1}{2m})$ then $|\\{R=n+2\\}|$ decays
exponentially in $n$.

To prove Proposition 2.9, we estimate the size of each element of the
partition $\\{\Delta^{+}_{0,i}\\}_{i=1}^{\infty}$, and the cardinality of the
set of elements with a given stopping time.

######  Lemma 2.10.

For all $a\in(0,\frac{1}{2m}]$ and all $i\geq 1$,

$|\Delta^{+}_{0,i}|\leq\exp(-\chi(a)(R_{i}-2)).$

###### Proof.

For $i\geq 1$ write $\Delta_{0,i}^{+}=\omega\cap\Delta_{0}^{+}$,
$\omega\in\mathscr{P}$ and put

$\begin{split}R_{i,\alpha}&=\\#\left\\{0\leq k\leq R_{i}-1\colon
f^{k}(\omega)\subset\Omega^{+}_{\alpha}\right\\},\\\
R_{i,\beta}&=\\#\left\\{0\leq k\leq R_{i}-1\colon
f^{k}(\omega)\subset\Omega^{+}_{\beta}\right\\}.\end{split}$

Clearly we have $R_{i,\alpha}+R_{i,\beta}=R_{i}.$ Proposition 2.5 implies the
following:

  * (i)

If $\omega\subset\Omega^{+}_{\alpha}$, then $|\omega|_{y}=\frac{1}{m}$ and
$S_{R_{i}}\phi^{c}=\log m$ on $\omega$. In particular, $R_{i}$ is odd and

$R_{i,\alpha}=\frac{1}{2}(R_{i}-1)\ \text{ and }\
R_{i,\beta}=\frac{1}{2}(R_{i}+1).$

  * (ii)

If $\omega\subset\Omega^{+}_{\beta}$, then $|\omega|_{y}=\frac{1}{m^{2}}$ and
$S_{R_{i}}\phi^{c}=2\log m$ on $\omega$. In particular, $R_{i}$ is even and

$R_{i,\alpha}=\frac{1}{2}(R_{i}-2)\ \text{ and }\
R_{i,\beta}=\frac{1}{2}(R_{i}+2).$

See Figure 6. We also have $\log|\omega|_{x}=R_{i,\alpha}\log
a+R_{i,\beta}\log(1-ma).$ Combining this with the above (i) (ii) yields
$|\omega|_{x}\leq\exp(-\chi(a)(R_{i}-2)).$ Since
$|\Delta^{+}_{0,i}|=|\omega|_{x}|\omega|_{y}$, we obtain the desired
inequality. ∎

Figure 6. The images of $\Delta_{0,i}^{+}$, $i\geq 1$ for $m=2$:
$\Delta_{0,i}^{+}\subset\Omega^{+}_{\alpha}$ (left);
$\Delta_{0,i}^{+}\subset\Omega^{+}_{\beta}$ (right); $|[0,1]^{2}\setminus
f^{R_{i}}(\Delta_{0,i}^{+})|=0$ in both cases.

For each $n\geq 1$, define

$P_{n}^{*}=\begin{cases}\\{v_{0}\cdots v_{n}\in P_{n}\colon l(v_{0})=-1\text{
and }l(v_{n})=0\\}&\text{ if $n$ is odd,}\\\ \\{v_{0}\cdots v_{n}\in
P_{n}\colon l(v_{0})=0\text{ and }l(v_{n})=0\\}&\text{ if $n$ is
even.}\end{cases}$

######  Lemma 2.11.

For all $n\geq 2$ we have

$\\#P_{n}^{*}\leq\frac{2(m+1)}{n+1}\begin{pmatrix}n+2\\\
\lfloor\frac{n+4}{2}\rfloor\end{pmatrix}m^{\lfloor\frac{n+1}{2}\rfloor}.$

###### Proof.

For integers $n$, $s$, $t$ with $n\geq 1$ and $s\leq t$, let

$Z_{n}(s,t)=\\{l_{0}\cdots l_{n}\in Z_{n}\colon l_{0}=s,\ l_{n}=t\\}.$

If $s<t$, then let

$Z_{n}^{*}(s,t)=\\{l_{0}\cdots l_{n}\in Z_{n}(s,t)\colon l_{k}>s\text{ for
}k\in\\{1,\ldots,n\\}\\}.$

Note that $Z_{n}^{*}(s,t)\neq\emptyset$ if and only if $t-s\equiv n$ mod $2$.
The reflection principle of the symmetric random walk on $\mathbb{Z}$ gives

(2.11) $\\#Z_{n}^{*}(s,t)=\frac{t-s}{n}\begin{pmatrix}n\\\
\frac{n+t-s}{2}\end{pmatrix}.$

By Lemma 2.1, for each $v_{0}\cdots v_{n}\in P_{n}^{*}$ we have

(2.12) $\\#\Phi_{n}^{-1}(l(v_{0})\cdots
l(v_{n}))\leq\begin{cases}(m+1)m^{\frac{n+1}{2}}&\text{ if $n$ is odd,}\\\
(m+1)m^{\frac{n}{2}}&\text{ if $n$ is even.}\end{cases}$

Suppose $n\geq 2$ is odd. For each $v_{0}\cdots v_{n}\in P_{n}^{*}$ we have
$l(v_{0})\cdots l(v_{n})\in Z_{n}(-1,0).$ Since the map $l_{0}\cdots l_{n}\in
Z_{n}(-1,0)\mapsto(-2)l_{0}\cdots l_{n}\in Z_{n+1}^{*}(-2,0)$ is injective,
(2.11) gives

(2.13)
$\\#\Phi_{n}(P_{n}^{*})\leq\\#Z_{n+1}^{*}(-2,0)\leq\frac{2}{n+1}\begin{pmatrix}n+1\\\
\frac{n+3}{2}\end{pmatrix}.$

Suppose $n\geq 2$ is even. Similarly, for each $v_{0}\cdots v_{n}\in
P_{n}^{*}$ we have $l(v_{0})\cdots l(v_{n})\in Z_{n}(0,0).$ Since the map
$l_{0}\cdots l_{n}\in Z_{n}(0,0)\mapsto(-2)(-1)l_{0}\cdots l_{n}\in
Z_{n+2}^{*}(-2,0)$ is injective, (2.11) gives

(2.14)
$\\#\Phi_{n}(P_{n}^{*})\leq\\#Z_{n+2}^{*}(-2,0)\leq\frac{2}{n+2}\begin{pmatrix}n+2\\\
\frac{n+4}{2}\end{pmatrix}.$

Combining (2.12), (2.13) and (2.14) yields the desired inequality in the
lemma. ∎

###### Proof of Proposition 2.9.

Let $n\geq 2$. For each $i\geq 1$ with $R_{i}=n+2$, there exists a unique path
$v_{0}\cdots v_{n}\in P_{n}^{*}$ such that $f^{k}(\Delta^{+}_{0,i})\subset
v_{k}$ for $0\leq k\leq n$. By Lemmas 2.10 and 2.11 we have

$\begin{split}|\\{R=n+2\\}|&=\sum_{i\geq 1\colon
R_{i}=n+2}|\Delta^{+}_{0,i}|\leq\\#P_{n}^{*}e^{-\chi(a)n}\\\
&\leq\frac{2(m+1)}{n+1}\begin{pmatrix}n+2\\\
\lfloor\frac{n+4}{2}\rfloor\end{pmatrix}m^{\lfloor\frac{n+1}{2}\rfloor}e^{-\chi(a)n}\leq
n^{-\frac{3}{2}}(\sqrt{4m}e^{-\chi(a)})^{n},\end{split}$

provided $n$ is large enough. To deduce the last inequality we have evaluated
the binomial coefficient using Stirling’s formula for factorials. ∎

###  2.9. Symbolic dynamics

For $a,b\in(0,\frac{1}{m})$ we put

$\Lambda_{a}=\bigcap_{n=0}^{\infty}f_{a}^{-n}\left(\bigcup_{\gamma\in D}{\rm
int}(\Omega_{\gamma}^{+})\right)\quad\text{ and
}\quad\Lambda_{a,b}=\bigcap_{n=-\infty}^{\infty}f_{a,b}^{-n}\left(\bigcup_{\gamma\in
D}{\rm int}(\Omega_{\gamma})\right).$

Define coding maps
$\pi_{a}\colon(x,y)\in\Lambda_{a}\mapsto(\omega_{n})_{n\in\mathbb{Z}_{+}}\in
D^{\mathbb{Z}_{+}}$ and
$\pi_{a,b}\colon(x,y,z)\in\Lambda_{a,b}\mapsto(\omega_{n})_{n\in\mathbb{Z}}\in
D^{\mathbb{Z}}$ by

$(x,y)\in\bigcap_{n=0}^{\infty}f_{a}^{-n}({\rm int}(\Omega_{\omega_{n}}^{+}))\
\text{ and }\ (x,y,z)\in\bigcap_{n=-\infty}^{\infty}f_{a,b}^{-n}({\rm
int}(\Omega_{\omega_{n}})).$

Note that $\sigma_{+}\circ\pi_{a}=\pi_{a}\circ f_{a}$ and
$\sigma\circ\pi_{a,b}=\pi_{a,b}\circ f_{a,b}.$ The coding maps are not
injective. The next connection between the heterochaos baker maps and the Dyck
system lies at the basis of the proof of Theorem C.

######  Theorem 2.12 ([32], Theorem 1.1).

For all $a,b\in(0,\frac{1}{m})$ we have

$\Sigma_{D}^{+}=\overline{\pi_{a}(\Lambda_{a})}\ \text{ and }\
\Sigma_{D}=\overline{\pi_{a,b}(\Lambda_{a,b})}.$

##  3\. Mixing of all orders for the heterochaos baker maps

In this section we prove Theorem A. In Section 3.1 we show a certain
invariance of correlations, which implies the sufficiency to consider only
$a\in(0,\frac{1}{2m}]$. In Section 3.2 we establish the exactness and
$K$-property of the corresponding maps, and in Section 3.3 complete the proof
of Theorem A.

###  3.1. Invariance of correlations

Let $(X,\mathscr{B},\nu)$ be a probability space and let $T\colon X\to X$ be a
measurable map preserving $\nu$. For an integer $k\geq 2$ and functions
$\phi_{0},\phi_{1},\ldots,\phi_{k-1}\in L^{k}(\nu)$, consider their
correlations

${\rm
Cor}_{n_{1},\ldots,n_{k-1}}(T;\phi_{0},\ldots,\phi_{k-1};\nu)=\left|\int\prod_{j=0}^{k-1}(\phi_{j}\circ
T^{n_{j}}){\rm d}\nu-\prod_{j=0}^{k-1}\int\phi_{j}{\rm d}\nu\right|,$

where $0=n_{0}\leq n_{1}\leq\cdots\leq n_{k-1}$. It is well-known that
$(T,\nu)$ is $k$-mixing if and only if ${\rm
Cor}_{n_{1},\ldots,n_{k-1}}(T;\phi_{0},\ldots,\phi_{k-1};\nu)\to 0$ as
$n_{1},n_{2}-n_{1},\ldots,n_{k-1}-n_{k-2}\to\infty$ for all
$\phi_{0},\phi_{1},\ldots,\phi_{k-1}\in L^{k}(\nu)$.

Define an involution $\iota\colon[0,1]^{3}\to[0,1]^{3}$ by

$\iota(x,y,z)=(1-z,1-y,1-x).$

######  Proposition 3.1.

Let $a\in(0,\frac{1}{m})$. For all $k\geq 2$, $\phi_{0},\ldots,\phi_{k-1}\in
L^{k}({\rm Leb})$ and integers $n_{0},\ldots,n_{k-1}$ with $0=n_{0}\leq
n_{1}\leq n_{2}\leq\cdots\leq n_{k-1}$ we have

$\begin{split}{\rm
Cor}_{n_{1},\ldots,n_{k-1}}&(g_{a};\phi_{0},\ldots,\phi_{k-1};{\rm Leb})\\\
&={\rm
Cor}_{n_{k-1},n_{k-1}-n_{1},n_{k-1}-n_{2},\ldots,0}(g_{\frac{1}{m}-a};\phi_{0}\circ\iota^{-1},\ldots,\phi_{k-1}\circ\iota^{-1};{\rm
Leb}).\end{split}$

###### Proof.

Define $g_{a}^{*}\colon[0,1]^{3}\to[0,1]^{3}$ by

(3.1) $g_{a}^{*}=\iota^{-1}\circ g_{\frac{1}{m}-a}\circ\iota.$

For Lebesgue a.e. $p\in[0,1]^{3}$ we have $g_{a}^{*}(g_{a}(p))=p$. Moreover,
for Lebesgue a.e. $p\in[0,1]^{3}$ we have

(3.2) $\begin{split}\phi_{j}\circ
g_{a}^{n_{j}}(p)&=\phi_{j}\circ(g_{a}^{*})^{n_{k-1}-n_{j}}\circ
g_{a}^{n_{k-1}-n_{j}}\circ g_{a}^{n_{j}}(p)\\\
&=\phi_{j}\circ(g_{a}^{*})^{n_{k-1}-n_{j}}\circ g_{a}^{n_{k-1}}(p)\ \text{ for
every }j\in\\{0,\ldots,k-1\\}.\end{split}$

Using (3.2) and the $g_{a}$-invariance of ${\rm Leb}$, (3.1) and the
$\iota$-invariance of ${\rm Leb}$ yield

$\begin{split}\int\prod_{j=0}^{k-1}(\phi_{j}\circ g_{a}^{n_{j}}){\rm d}{\rm
Leb}&=\int\prod_{j=0}^{k-1}(\phi_{j}\circ(g_{a}^{*})^{n_{k-1}-n_{j}}){\rm
d}{\rm Leb}\\\ &=\int\prod_{j=0}^{k-1}(\phi_{j}\circ\iota^{-1}\circ
g_{\frac{1}{m}-a}^{n_{k-1}-n_{j}}\circ\iota){\rm d}{\rm Leb}\\\
&=\int\prod_{j=0}^{k-1}(\phi_{j}\circ\iota^{-1}\circ
g_{\frac{1}{m}-a}^{n_{k-1}-n_{j}}){\rm d}{\rm Leb}.\end{split}$

We also have $\int\phi_{j}{\rm d}{\rm Leb}=\int\phi_{j}\circ\iota^{-1}{\rm
d}{\rm Leb}$, and so the desired equality holds. ∎

###  3.2. Exactness and $K$-property

Let $(X,\mathscr{B},\nu)$ be a probability space. For sub-sigma-algebras
$\mathscr{C}$, $\mathscr{D}$ of $\mathscr{B}$ we write
$\mathscr{C}\stackrel{{\scriptstyle\circ}}{{\subset}}\mathscr{D}$ if for every
$C\in\mathscr{C}$ there exists $D\in\mathscr{D}$ such that $\nu(C\ominus
D)=0$, where $\ominus$ denotes the symmetric difference of sets. We write
$\mathscr{C}\stackrel{{\scriptstyle\circ}}{{=}}\mathscr{D}$ if
$\mathscr{C}\stackrel{{\scriptstyle\circ}}{{\subset}}\mathscr{D}$ and
$\mathscr{D}\stackrel{{\scriptstyle\circ}}{{\subset}}\mathscr{C}$. If
$\\{\mathscr{B}_{n}\\}_{n=1}^{\infty}$ is a family of sub-sigma-algebras of
$\mathscr{B}$, let $\bigvee_{n=1}^{\infty}\mathscr{B}_{n}$ denote the smallest
sub-sigma-algebra that contains all the $\mathscr{B}_{n}$. If
$\\{\mathscr{B}_{n}\\}_{n=1}^{\infty}$ is a family of partitions of $X$ into
measurable sets, let $\bigvee_{n=1}^{\infty}\mathscr{B}_{n}$ denote the
smallest sub-sigma-algebra that contains all the $\mathscr{B}_{n}$.

Let $T\colon X\to X$ be a measurable map preserving $\nu$. For a sub-sigma-
algebra $\mathscr{C}$ of $\mathscr{B}$ and $n\geq 1$, let
$T^{-n}\mathscr{C}=\\{T^{-n}(B)\colon B\in\mathscr{C}\\}$. If $T$ has a
measurable inverse, let $T^{n}\mathscr{C}=\\{T^{n}(B)\colon
B\in\mathscr{C}\\}$. We say $(T,\nu)$ is exact if $T$ has no measurable
inverse and
$\bigcap_{n=0}^{\infty}T^{-n}\mathscr{B}\stackrel{{\scriptstyle\circ}}{{=}}\\{X,\emptyset\\}$.
We say $(T,\nu)$ has $K$-property if $T$ has a measurable inverse and there
exists a sub-sigma-algebra $\mathscr{K}$ of $\mathscr{B}$ such that:

  * (i)

$\mathscr{K}\subset T\mathscr{K}$.

  * (ii)

$\bigvee_{n=0}^{\infty}T^{n}\mathscr{K}\stackrel{{\scriptstyle\circ}}{{=}}\mathscr{B}$.

  * (iii)

$\bigcap_{n=0}^{\infty}T^{-n}\mathscr{K}\stackrel{{\scriptstyle\circ}}{{=}}\mathscr{\\{}X,\emptyset\\}$.

If $(T,\nu)$ has $K$-property, $T$ is usually called a Kolmogorov automorphism
[33]. Exactness or $K$-property implies mixing of all orders [26, Section
2.6].

######  Proposition 3.2.

For any $a\in(0,\frac{1}{2m}]$, $(f_{a},{\rm Leb})$ is exact and
$(g_{a}|_{\Lambda_{a},\frac{1}{m}-a},{\rm Leb}|_{\Lambda_{a,\frac{1}{m}-a}})$
has $K$-property.

For a proof of this proposition we need some notation and one preliminary
lemma. For $a\in(0,\frac{1}{m})$ define $\mathscr{A}=\mathscr{A}_{a}$ by

$\mathscr{A}=\bigcup_{n=1}^{\infty}\\{B_{0}\cap(f^{R})^{-1}(B_{1})\cap\cdots\cap(f^{R})^{-n+1}(B_{n-1})\colon
B_{0},\ldots,B_{n-1}\in\mathscr{P}|_{\Delta_{0}^{+}}\\}.$

Let $\mathscr{B}(X)$ denote the Borel sigma-algebra on a topological space
$X$.

######  Lemma 3.3.

If $a\in(0,\frac{1}{2m}]$, then for any $A\in\mathscr{B}([0,1]^{2})$ with
positive Lebesgue measure and any $\varepsilon\in(0,1)$, there exists
$\omega\in\mathscr{A}$ such that $|A\cap\omega|\geq(1-\varepsilon)|\omega|$.

###### Proof.

Since the collection of unions of countably many elements of $\mathscr{A}$ is
an algebra on $\Delta^{+}_{0}$, for any $\varepsilon>0$ there exist finitely
many elements $\omega_{1},\ldots,\omega_{k}$ of $\mathscr{A}$ such that
$|\bigcup_{j=1}^{k}\omega_{j}\ominus A|<\varepsilon^{2}$. If $\varepsilon$ is
sufficiently small, there exists $j_{0}\in\\{1,\ldots,k\\}$ such that
$|A\cap\omega_{j_{0}}|\geq(1-\varepsilon)|\omega_{j_{0}}|$, for otherwise we
obtain the following contradiction:
$|A|<(1-\varepsilon)\sum_{j=1}^{k}|\omega_{j}|+|A\setminus\bigcup_{j=1}^{k}\omega_{j}|<(1-\varepsilon)(|A|+\varepsilon^{2})+\varepsilon^{2}<|A|$.
∎

###### Proof of Proposition 3.2.

Let $A\in\bigcap_{n=0}^{\infty}f^{-n}\mathscr{B}([0,1]^{2})$ satisfy $|A|>0$.
To verify the exactness of $(f,{\rm Leb})$ it suffices to show that $|A|=1$.
By Lemma 3.3, for any $\varepsilon>0$ there exist $n\geq 1$ and an affine
pullback of $\omega$ of $(0,1)^{2}$ by $f^{n}$ such that
$|A\cap\omega|>(1-\varepsilon)|\omega|$. There exists
$A^{\prime}\in\mathscr{B}([0,1]^{2})$ such that $A=f^{-n}(A^{\prime})$, and
therefore $A^{\prime}\supset f^{n}(A)$ and

$|A|=|A^{\prime}|\geq|f^{n}(A)|\geq|f^{n}(A\cap\omega)|\geq(1-\varepsilon)|f^{n}(\omega)|=1-\varepsilon.$

The last inequality is because $f^{n}|_{\omega}$ is affine. Since
$\varepsilon$ is arbitrary we obtain $|A|=1$.

Note that ${\rm Leb}(\Lambda)=1$, $g|_{\Lambda}$ has a measurable inverse, and
the restriction ${\rm Leb}|_{\Lambda}$ is $g|_{\Lambda}$-invariant. Let
$\mathscr{K}$ denote the smallest sub-sigma-algebra of $\mathscr{B}(\Lambda)$
that contains $\\{(A\times[0,1])\cap\Lambda\colon A\in\mathscr{A}\\}$. From
the definition of the map (1.4) we obtain $\mathscr{K}\subset g\mathscr{K}$
and
$\bigvee_{n=0}^{\infty}g^{n}\mathscr{K}\stackrel{{\scriptstyle\circ}}{{=}}\mathscr{B}(\Lambda)$.
The argument in the previous paragraph shows
$\bigcap_{n=0}^{\infty}g^{-n}\mathscr{K}\stackrel{{\scriptstyle\circ}}{{=}}\mathscr{\\{}\Lambda,\emptyset\\}$.
We have verified that $(g|_{\Lambda},{\rm Leb}|_{\Lambda})$ has $K$-property.
∎

######  Remark 3.4.

The exactness of $(f_{a},{\rm Leb})$ for $a\in(0,\frac{1}{2m})$ can be shown
by applying [36, Theorem 1 (iii)] (see also [35, Lemma 5]) to the tower map
$F$ introduced in Section 4.1. This argument does not work for
$a=\frac{1}{2m}$ since the lift of the Lebesgue measure to the tower becomes
an infinite measure. For more details on this point, see Remark 4.3.

###  3.3. Proof of Theorem A

Let $a\in(0,\frac{1}{2m}]$. By Proposition 3.2, $(f_{a},{\rm Leb})$ is exact
and so mixing of all orders. By Proposition 3.2, $(g_{a},{\rm Leb})$ is mixing
of all orders. By Proposition 3.1, $(g_{\frac{1}{m}-a},{\rm Leb})$ is mixing
of all orders too, and so is $(f_{\frac{1}{m}-a},{\rm Leb})$. The proof of
Theorem A is complete. ∎

##  4\. Exponential mixing for the heterochaos baker maps

In this section we prove Theorem B. In Section 4.1 we introduce towers with
exponential tail associated with the heterochaos baker maps. In Section 4.2 we
apply the results in [35] to the towers and use large deviations to complete
the proof of Theorem B.

###  4.1. Towers

Put $\Delta_{0}=\Delta^{+}_{0}\times[0,1]$. We extend the stopping time to a
function on $\Delta_{0}$ in the obvious way, and still denote the extension by
$R$: $R(x,y,z)=R(x,y)$ for $(x,y,z)\in\Delta_{0}$. We define a tower $\Delta$
associated with $g=g_{a}$ by

$\Delta=\\{(p,\ell)\colon p\in\Delta_{0},\ \ell=0,1,\ldots,R(p)-1\\}.$

For each $\ell\geq 1$, the $\ell$-th floor is the set

$\Delta_{\ell}=\\{(p,\ell)\in\Delta\colon p\in\Delta_{0}\\}.$

We identify the ground floor $\Delta_{0}\times\\{0\\}$ with $\Delta_{0}$. Note
that $\Delta=\bigcup_{\ell=0}^{\infty}\Delta_{\ell}$. Define a tower map
$G\colon\Delta\to\Delta$ by

$G(p,\ell)=\begin{cases}(p,\ell+1)&\ \text{ if }\ell+1<R(p),\\\
(g^{R(p)}(p),0)&\ \text{ if }\ell+1=R(p).\end{cases}$

Collapsing the $z$-coordinate, we obtain the quotient tower $\Delta^{+}$ and
the tower map $F\colon\Delta^{+}\to\Delta^{+}$. The maps $g$ and $G$, $f$ and
$F$ are semiconjugated by the maps $\theta\colon(p,\ell)\in\Delta\mapsto
g^{\ell}(p)\in[0,1]^{3}$ and $\theta^{+}\colon(p,\ell)\in\Delta^{+}\mapsto
f^{\ell}(p)\in[0,1]^{2}$ respectively. Let ${\rm pr}\colon\Delta\to\Delta^{+}$
denote the canonical projection. In summary, the following diagram commutes:

$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{split}{\rm
pr}\end{split}}$$\scriptstyle{\begin{split}&\\\
&\theta\end{split}}$$\scriptstyle{\begin{split}G\end{split}}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{split}{&\\\
&\theta}\end{split}}$$\scriptstyle{\begin{split}{\rm
pr}\end{split}}$$\textstyle{\Delta^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{split}&\\\
&\theta^{+}\end{split}}$$\scriptstyle{\begin{split}\\!\\!\\!\\!\\!\\!\\!\\!\\!F\end{split}}$$\textstyle{\Delta^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{split}&\\\
&\theta^{+}\end{split}}$$\textstyle{[0,1]^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{split}\
\ \ \ \ \
g\end{split}}$$\textstyle{[0,1]^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[0,1]^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{split}f\end{split}}$$\textstyle{[0,1]^{2}.}$

We fix a sigma-algebra on $\Delta$ that is obtained by naturally transplanting
the Borel sigma-algebra on $[0,1]^{3}$. Each floor $\Delta_{\ell}$, $\ell\geq
0$ is identified with $\\{R>\ell\\}$, and so equipped with the restriction of
the Lebesgue measure. If $a\in(0,\frac{1}{2m})$, then in view of Proposition
2.9 let $\mu$ denote the probability measure on $\Delta$ given by

$\mu(A)=\frac{1}{\int R{\rm d}{\rm
Leb}}\sum_{\ell=0}^{\infty}|A\cap\Delta_{\ell}|\ \text{ for any measurable set
}A\subset\Delta.$

Since the Lebesgue measure on $[0,1]^{3}$ is $g$-invariant, $\mu$ is
$G$-invariant. The measure $\mu\circ\theta^{-1}$ is $f$-invariant, and
absolutely continuous with respect to the Lebesgue measure, and hence
$\mu\circ\theta^{-1}={\rm Leb}$. The measure $\mu^{+}=\mu\circ{\rm pr}^{-1}$
is $F$-invariant.

###  4.2. Proof of Theorem B

Let $a\in(0,\frac{1}{2m})$. Let $\eta\in(0,1]$ and let $\varphi$,
$\psi\in\mathscr{H}_{\eta}([0,1]^{3})$. For a function $\phi$ on $[0,1]^{3}$,
let $\tilde{\phi}$ denote its lift to the tower $\Delta$ associated with
$g=g_{a}$, i.e. $\tilde{\phi}=\phi\circ\theta$. Note that ${\rm
Cor}_{n}(g;\varphi,\psi;{\rm Leb})={\rm
Cor}_{n}(G;\tilde{\varphi},\tilde{\psi};\mu)$.

The rest of the proof of Theorem B breaks into three steps, much in parallel
to [35, Section 4.1] with one important difference in Step 2. In Step 1 we
begin by approximating $\tilde{\varphi}\circ G^{k}$, $\tilde{\psi}\circ
G^{k}$, $k\geq 1$ by functions $\varphi_{k}$, $\psi_{k}$ on $\Delta$ which do
not depend on the $z$-coordinate. In Step 2 we provide error bounds of these
approximations. Since [35, p.608 Sublemma] does not hold as a result of the
breakdown of the condition (P4) (backward contraction) in [35], we estimate
errors of these approximations using large deviations for the map $\tau_{a}$
in (1.1). In Step 3 we view $\varphi_{k}$, $\psi_{k}$ as functions on
$\Delta^{+}$, and show exponential decay of their correlations by applying
[35]. We then unify all these estimates.

Step 1: Approximations. Put $\Delta_{0,i}=\Delta^{+}_{0.i}\times[0,1]$ for
$i\geq 1$. Each floor $\Delta_{\ell}$, $\ell\geq 0$ is partitioned into
$\\{\Delta_{\ell,i}\\}_{i\geq 1\colon R_{i}>\ell}$ where $\Delta_{\ell,i}$ is
a copy of $\Delta_{0,i}$. Let $\mathscr{D}_{0}$ denote the partition of
$\Delta$ into $\Delta_{\ell,i}$-components. It has the Markov property: For
every $A\in\mathscr{D}_{0}$, $G(A)$ is the union of elements of
$\mathscr{D}_{0}$. For $k\geq 1$ we put
$\mathscr{D}_{k}=\bigvee_{j=0}^{k-1}G^{-j}\mathscr{D}_{0}$. For $k\geq 0$, let
$\mathscr{D}_{k}^{+}$ denote the partition of $\Delta^{+}$ which is obtained
as the canonical projection of $\mathscr{D}_{k}$. For
$\phi\in\mathscr{H}_{\eta}([0,1]^{3})$ and $k\geq 1$, define
$\phi_{k}\colon\Delta\to\mathbb{R}$ by

$\phi_{k}|_{A}=\inf\\{\tilde{\phi}(w)\colon w\in G^{k}(A)\\}\quad\text{for
every }A\in\mathscr{D}_{2k}.$

Clearly $\phi_{k}$ is constant on each element of $\mathscr{D}_{2k}$, and
$|\phi_{k}|_{\infty}\leq|\phi|_{\infty}$. Since $\phi_{k}$ does not depend on
the $z$-coordinate, we may view $\phi_{k}$ as a function on $\Delta^{+}$. Let
$n>k$. By the identities ${\rm
Cor}_{n}(G;\tilde{\varphi},\tilde{\psi};\mu)={\rm
Cor}_{n-k}(G;\tilde{\varphi},\tilde{\psi}\circ G^{k};\mu)$ and ${\rm
Cor}_{n}(G;\varphi_{k},\psi_{k};\mu)={\rm
Cor}_{n}(F;\varphi_{k},\psi_{k};\mu^{+})$, we have ${\rm
Cor}_{n}(G;\tilde{\varphi},\tilde{\psi};\mu)\leq I+I\\!I+I\\!I\\!I$ where

$\begin{split}I&=|{\rm Cor}_{n-k}(G;\tilde{\varphi},\tilde{\psi}\circ
G^{k};\mu)-{\rm Cor}_{n-k}(G;\tilde{\varphi},\psi_{k};\mu)|,\\\ I\\!I&=|{\rm
Cor}_{n-k}(G;\tilde{\varphi},\psi_{k};\mu)-{\rm
Cor}_{n-k}(G;\varphi_{k},\psi_{k};\mu)|,\\\ I\\!I\\!I&={\rm
Cor}_{n-k}(F;\varphi_{k},\psi_{k};\mu^{+}).\end{split}$

Step 2: Estimates of $I$, $I\\!I$. A direct calculation shows $I\leq
2|\varphi|_{\infty}\int|\tilde{\psi}\circ G^{k}-\psi_{k}|{\rm d}\mu,$ and a
similar argument to the one in [35, p.608] shows $I\\!I\leq
2|\psi|_{\infty}\int|\tilde{\varphi}\circ G^{k}-\varphi_{k}|{\rm d}\mu.$
Hence, upper bounds of $I$, $I\\!I$ follow from the next lemma. For
$\phi\in\mathscr{H}_{\eta}([0,1]^{3})$ let $|\phi|$ denote the $\eta$-Hölder
norm of $\phi$, namely

$|\phi|={\sup}_{\stackrel{{\scriptstyle p,q\in[0,1]^{3}}}{{p\neq
q}}}\frac{|\phi(p)-\phi(q)|}{|p-q|^{\eta}},$

where $|p-q|$ denotes the Euclidean distance between $p$ and $q$.

######  Lemma 4.1.

For any $\eta\in(0,1]$ and any $\phi\in\mathscr{H}_{\eta}([0,1]^{3})$, there
exist constants $C=C(\phi)>0$ and $\xi=\xi(\eta)\in(0,1)$ such that for every
$k\geq 1$,

$\begin{split}\int|\tilde{\phi}\circ G^{k}-\phi_{k}|{\rm d}\mu\leq
C\xi^{k}.\end{split}$

###### Proof.

Put

$\chi^{u}=-\log\max\left\\{a,1-ma\right\\},\
\chi^{s}=\log\max\left\\{b,1-mb\right\\},\ \chi^{c}=(1-2ma)\log m.$

The minimal expansion rate by $g$ in the $x$-direction is $\exp(\chi^{u})$ and
the maximal contraction rate by $g$ in the $z$-direction is $\exp(\chi^{s})$.
Since $a\in(0,\frac{1}{2m})$ we have $\chi^{c}>0$. Fix
$\varepsilon\in(0,\chi^{c})$, and recall the definition of a block and the
notation introduced in the beginning of Section 2.4. For each
$A\in\mathscr{D}_{k}$, $\theta(A)$ is a block satisfying
$|\theta(A)|_{x}\leq\exp(-\chi^{u}k)$ and
$|\theta(A)|_{z}\leq\exp(\chi^{s}k).$ Define

$\mathscr{D}_{k}^{\prime}=\left\\{A\in\mathscr{D}_{k}\colon|\theta(A)|_{y}<\exp(-(\chi^{c}-\varepsilon)k)\right\\}.$

Put
$\xi_{0}=\exp(\max\\{-\chi^{u},-\chi^{c}+\varepsilon,\chi^{s}\\})\in(0,1).$
For every $A\in\mathscr{D}_{2k}$ with $G^{k}(A)\in\mathscr{D}_{k}^{\prime}$,
the Euclidean diameter of $\theta(G^{k}(A))$ does not exceed $3\xi_{0}^{k}$.
Since $\phi\in\mathscr{H}_{\eta}([0,1]^{3})$, this bound implies

(4.1) $\sup_{A}|\tilde{\phi}\circ G^{k}-\phi_{k}|\leq
3^{\eta}|\phi|\xi_{0}^{\eta k}.$

For each $w\in\Delta^{+}$ let $\mathscr{D}_{k}^{+}(w)$ denote the element of
$\mathscr{D}_{k}^{+}$ that contains $w$. Then $\mathscr{D}_{k}^{+}(w)$ is a
rectangle satisfying
$|\mathscr{D}_{k}^{+}(w)|_{y}\leq\exp(-S_{k}\phi^{c}(\theta^{+}(w)))$, which
implies

(4.2)
$\begin{split}\sum_{A\in\mathscr{D}_{k}\setminus\mathscr{D}_{k}^{\prime}}\mu(A)&\leq\mu^{+}\left\\{w\in\Delta^{+}\colon
S_{k}\phi^{c}(\theta^{+}(w))\leq(\chi^{c}-\varepsilon)k\right\\}\\\
&=\left|\left\\{p\in[0,1]^{2}\colon
S_{k}\phi^{c}(p)\leq(\chi^{c}-\varepsilon)k\right\\}\right|.\end{split}$

Recall that $g$ is a skew product over $\tau$ in (1.1) that preserves the
Lebesgue measure on $[0,1]$, and $\int_{[0,1]}\phi^{c}(x,0){\rm d}x=\chi^{c}$.
From the large deviations applied to the sequence of independently identically
distributed random variables $x\in[0,1]\mapsto\phi^{c}(f^{k}(x,0))$
$(k=0,1,\ldots)$, the sum in (4.2) decays exponentially in $k$. There exist
$C>0$ and $\xi_{1}\in(0,1)$ such that

(4.3)
$\sum_{A\in\mathscr{D}_{k}\setminus\mathscr{D}_{k}^{\prime}}\mu(G^{-k}(A))\leq\sum_{A\in\mathscr{D}_{k}\setminus\mathscr{D}_{k}^{\prime}}\mu(A)\leq
C\xi_{1}^{k}.$

The first inequality is due to the $G$-invariance of $\mu$. Let $D_{k}$ denote
the union of elements of $\mathscr{D}_{k}^{\prime}$. For any
$A\in\mathscr{D}_{2k}$, either $G^{k}(A)\subset D_{k}$ or
$G^{k}(A)\subset\Delta\setminus D_{k}$ holds. We have

(4.4) $\sum_{A\in\mathscr{D}_{2k}\colon G^{k}(A)\subset\Delta\setminus
D_{k}}\mu(A)\leq\sum_{A\in\mathscr{D}_{k}\setminus\mathscr{D}_{k}^{\prime}}\mu(G^{-k}(A)).$

From (4.1), (4.2), (4.3) and (4.4) we obtain

$\begin{split}\int|\tilde{\phi}\circ G^{k}-\phi_{k}|{\rm
d}\mu=&\sum_{A\in\mathscr{D}_{2k}\colon G^{k}(A)\subset
D_{k}}\int_{A}|\tilde{\phi}\circ G^{k}-\phi_{k}|{\rm d}\mu\\\
&+\sum_{A\in\mathscr{D}_{2k}\colon G^{k}(A)\subset\Delta\setminus
D_{k}}\int_{A}|\tilde{\phi}\circ G^{k}-\phi_{k}|{\rm d}\mu\\\
\leq&3^{\eta}|\phi|\xi_{0}^{\eta
k}\sum_{A\in\mathscr{D}_{2k}}\mu(A)+2|\phi|_{\infty}\sum_{A\in\mathscr{D}_{k}\setminus\mathscr{D}_{k}^{\prime}}\mu(A)\\\
\leq&3^{\eta}|\phi|\xi_{0}^{\eta k}+2C|\phi|_{\infty}\xi_{1}^{k}.\end{split}$

Taking $C(\phi)=3^{\eta}|\phi|+2C|\phi|_{\infty}$ and
$\xi=\max\\{\xi_{0}^{\eta},\xi_{1}\\}$ yields the desired inequality in Lemma
4.1. ∎

Step3: Estimate of $I\\!I\\!I$ and an overall estimate. It is clear that the
greatest common divisor of $\\{R_{i}\colon i\geq 1\\}$ is $1$. Using the
Perron-Frobenius operator $\mathcal{P}\colon L^{1}(\mu^{+})\to L^{1}(\mu^{+})$
given by $F_{*}(\phi\mu^{+})=\mathcal{P}(\phi)\mu^{+},$ we write

$\begin{split}I\\!I\\!I&={\rm
Cor}_{n-2k}(F;\mathcal{P}^{2k}(\varphi_{k}),\psi_{k};\mu^{+}).\end{split}$

We introduce a separation time
$s\colon\Delta_{0}^{+}\times\Delta_{0}^{+}\to\mathbb{Z}$ by defining
$s(p,p^{\prime})$ to be the smallest integer $n\geq 0$ such that
$(f^{R})^{n}(p)$, $(f^{R})^{n}(p^{\prime})$ lie in different elements of
$\mathscr{P}|_{\Delta_{0}^{+}}$. For general points $w=(p,\ell)$,
$w^{\prime}=(p^{\prime},\ell^{\prime})\in\Delta^{+}$, define
$s(w,w^{\prime})=s(p,p^{\prime})$ if $\ell=\ell^{\prime}$ and
$s(w,w^{\prime})=0$ otherwise. This defines a separation time
$s\colon\Delta^{+}\times\Delta^{+}\to\mathbb{Z}$. For $\beta\in(0,1]$ define

$C_{\beta}(\Delta^{+})=\\{\phi\colon\Delta^{+}\to\mathbb{R}\colon\exists C>0\
\forall w,w^{\prime}\in\Delta^{+},\ |\phi(w)-\phi(w^{\prime})|\leq
C\beta^{s(w,w^{\prime})}\\}.$

For each $\phi\in C_{\beta}(\Delta^{+})$ define

$\|\phi\|=|\phi|_{\infty}+{\rm esssup}_{\stackrel{{\scriptstyle
w,w^{\prime}\in\Delta^{+}}}{{w\neq
w^{\prime}}}}\frac{|\phi(w)-\phi(w^{\prime})|}{\beta^{s(w,w^{\prime})}}.$

######  Lemma 4.2.

For all $\beta\in(0,1)$ and all $k\geq 1$, we have
$\mathcal{P}^{2k}(\varphi_{k})\in C_{\beta}(\Delta^{+})$ and
$\|\mathcal{P}^{2k}(\varphi_{k})\|\leq 3|\varphi|_{\infty}$.

###### Proof.

For each $A\in\mathscr{D}_{k}^{+}$, $F^{k}|_{A}$ has a measurable inverse, and
the pullback of the measure $\mu^{+}|_{F^{k}(A)}$ by $F^{k}|_{A}$ is
absolutely continuous with respect to $\mu^{+}|_{A}$. The Radon-Nikodým
derivative ${\rm d}(\mu^{+}|_{F^{k}(A)}\circ F^{k}|_{A})/{\rm d}\mu^{+}|_{A}$
is constant on $A$, which we denote by $JF^{k}(A)$. Note that

$\mathcal{P}^{2k}(\varphi_{k})(w)=\sum_{A\in\mathscr{D}_{2k}^{+}\colon w\in
F^{2k}(A)}\frac{\varphi_{k}|_{A}}{JF^{2k}(A)}.$

Moreover, $JF^{2k}(A)=\mu^{+}(F^{2k}(A))/\mu^{+}(A)\geq 1/\mu^{+}(A)$ for all
$A\in\mathscr{D}_{2k}^{+}$, and so
$\sum_{A\in\mathscr{D}_{2k}^{+}}1/JF^{2k}(A)\leq 1.$ From this and
$|\varphi_{k}|_{\infty}\leq|\varphi|_{\infty}$ we obtain
$|\mathcal{P}^{2k}(\varphi_{k})|_{\infty}\leq|\varphi|_{\infty}$.

For all $w,w^{\prime}\in\Delta^{+}$ with $w\neq w^{\prime}$ we have
$\left|\mathcal{P}^{2k}(\varphi_{k})(w)-\mathcal{P}^{2k}(\varphi_{k})(w^{\prime})\right|\leq
2|\varphi|_{\infty}.$ If $s(w,w^{\prime})\geq 1$, then for each
$A\in\mathscr{D}_{2k}^{+}$ either $w,w^{\prime}\in F^{2k}(A)$ or $w$,
$w^{\prime}\notin F^{2k}(A)$, which implies
$\left|\mathcal{P}^{2k}(\varphi_{k})(w)-\mathcal{P}^{2k}(\varphi_{k})(w^{\prime})\right|=0.$
Hence the conclusion of the lemma holds. ∎

By [35, Section 4.2] (see also [24, Theorem 1.5]), there exist $C>0$ and
$\lambda_{0}\in(0,1)$ such that

$I\\!I\\!I\leq C\lambda_{0}^{n-2k}.$

Combining this with the estimates of $I$, $I\\!I$ in Step 2, and then
substituting $k=\lfloor n/2\rfloor/2$, we obtain constants
$C=C(\varphi,\psi)>0$ and $\lambda\in(0,1)$ such that

${\rm Cor}_{n}(g;\varphi,\psi;{\rm Leb})={\rm
Cor}_{n}(G;\tilde{\varphi},\tilde{\psi};\mu)\leq C\lambda^{n}\text{ for all
}n\geq 1,$

namely, the exponential mixing for $(g_{a},{\rm Leb})$. From this and
Proposition 3.1 we obtain the exponential mixing for $(g_{\frac{1}{m}-a},{\rm
Leb})$. Taking a pair of Hölder continuous functions on $[0,1]^{3}$ which are
constant on sets $\\{(x,y)\\}\times[0,1]$, $(x,y)\in[0,1]^{2}$, we obtain the
exponential mixing for $(f_{a},{\rm Leb})$ and $(f_{\frac{1}{m}-a},{\rm
Leb})$. The proof of Theorem B is complete. ∎

######  Remark 4.3.

In the case $a=\frac{1}{2m}$ which is excluded in Theorem B, it is plausible
that correlations for Hölder continuous functions decay only subexponentially.
It would be nice to prove this. A close inspection into the counting argument
in Section 2.2 and the stopping time estimate in Section 2.7 reveals
$|\\{R=n\\}|\asymp n^{-3/2}$, and as a result $\sum_{n}|\\{R>n\\}|=\infty$. It
follows that the lift of the Lebesgue measure to the tower becomes an infinite
measure, and the general result in [36] to draw subexponential decay of
correlations is not applicable.

##  5\. Exponential mixing for the Dyck system

In this last section we prove Theorem C. In Section 5.1, we identify the two
ergodic measures of maximal entropy for the Dyck system. In Section 5.2 we
show the invariance of correlations (modulo involution) under the replacement
of the two measures. Section 5.3 provides two preliminary lemmas needed to
transfer invariant measures on the two different spaces. In Section 5.4 we
clarify a connection between one of the two ergodic measures of maximal
entropy and the Lebesgue measure on $[0,1]^{3}$. In Section 5.5 we complete
the proof of Theorem C.

###  5.1. Measures of maximal entropy

Let $T$ be a Borel map acting on a topological space. For each $T$-invariant
Borel probability measure $\mu$, let $h(T,\mu)$ denote the measure-theoretic
entropy of $\mu$ with respect to $T$. If $\sup\\{h(T,\mu)\colon\text{$\mu$ is
$T$-invariant}\\}$ is finite, a measure which attains this supremum is called
a measure of maximal entropy.

Following [20, Section 4], for each $i\in\mathbb{Z}$ we define
$H_{i}\colon\Sigma_{D}\to\mathbb{Z}$ by

$H_{i}(\omega)=\begin{cases}\sum_{j=0}^{i-1}\sum_{l=1}^{m}(\delta_{{\alpha_{l}},\omega_{j}}-\delta_{\beta_{l},\omega_{j}})&\text{
for }i\geq 1,\\\
\sum_{j=i}^{-1}\sum_{l=1}^{m}(\delta_{{\beta_{l}},\omega_{j}}-\delta_{\alpha_{l},\omega_{j}})&\text{
for }i\leq-1,\\\ 0&\text{ for }i=0,\end{cases}$

where the delta denotes Kronecker’s delta. These functions are used to
indicate whether a bracket in a prescribed position in a sequence in
$\Sigma_{D}$ is closed or not. Consider three shift invariant Borel sets

(5.1)
$\begin{split}A_{0}&=\bigcap_{i=-\infty}^{\infty}\left(\left(\bigcup_{l=1}^{\infty}\\{H_{i+l}=H_{i}\\}\right)\cap\left(\bigcup_{l=1}^{\infty}\\{H_{i-l}=H_{i}\\}\right)\right),\\\
A_{\alpha}&=\left\\{\omega\in\Sigma_{D}\colon\lim_{i\to\infty}H_{i}(\omega)=\infty\
\text{ and }\ \lim_{i\to-\infty}H_{i}(\omega)=-\infty\right\\},\\\
A_{\beta}&=\left\\{\omega\in\Sigma_{D}\colon\lim_{i\to\infty}H_{i}(\omega)=-\infty\
\text{ and }\ \lim_{i\to-\infty}H_{i}(\omega)=\infty\right\\}.\end{split}$

Any shift invariant ergodic measure on $\Sigma_{D}$ gives measure $1$ to one
of these three sets (see [20, pp.102–103]).

######  Theorem 5.1 ([20], Section 4).

There exist two ergodic measures $\nu_{\alpha}$, $\nu_{\beta}$ of entropy
$\log(m+1)$ which are Bernoulli and satisfy $\nu_{\alpha}(A_{\alpha})=1$ and
$\nu_{\beta}(A_{\beta})=1$. These two measures are precisely all the ergodic
measures of maximal entropy for the two-sided Dyck shift.

Let $\nu_{\alpha}^{+}$ (resp. $\nu_{\beta}^{+}$) denote the shift invariant
measure on $\Sigma_{D}^{+}$ which is the push-forward of $\nu_{\alpha}$ (resp.
$\nu_{\beta}$) under the canonical projection $\Sigma_{D}\to\Sigma_{D}^{+}$.
These two measures are precisely all the ergodic measures of maximal entropy
for the one-sided Dyck shift.

We recall the result in [32] on measures of maximal entropy for the
heterochaos baker maps. Let $c_{1}=\frac{1}{m(m+1)}$ and
$c_{2}=\frac{1}{m+1}$.

######  Theorem 5.2 ([32], Theorem 1.2).

For any $a,b\in(0,\frac{1}{m})$, there exist two $f_{a,b}$-invariant ergodic
Borel probability measures $\mu_{\alpha}$, $\mu_{\beta}$ of entropy
$\log(m+1)$ which are Bernoulli, give positive weight to any non-empty open
subset of $[0,1]^{3}$, and satisfy

$\mu_{\alpha}\left(\bigcup_{i=1}^{m}\Omega_{\alpha_{i}}\right)=\mu_{\beta}\left(\bigcup_{i=1}^{m}\Omega_{\beta_{i}}\right)=\frac{m}{m+1}.$

Moreover, if $a\in[c_{1},c_{2}]$ or $b\in[c_{1},c_{2}]$ then $\mu_{\alpha}$,
$\mu_{\beta}$ are measures of maximal entropy for $f_{a,b}$. If
$a\in(c_{1},c_{2})$ or $b\in(c_{1},c_{2})$ then there is no ergodic measure of
maximal entropy for $f_{a,b}$ other than $\mu_{\alpha}$, $\mu_{\beta}$.

###  5.2. Invariance of correlations

We define an involution $\rho\colon D\to D$ by $\rho(\alpha_{i})=\beta_{i}$
and $\rho(\beta_{i})=\alpha_{i}$ for $i\in\\{1,\ldots,m\\}$, and
$\iota_{D}\colon\Sigma_{D}\to D^{\mathbb{Z}}$ by
$\iota_{D}((\omega_{n})_{n\in\mathbb{Z}})=(\rho(\omega_{-n}))_{n\in\mathbb{Z}}$.
Clearly $\iota_{D}$ is injective, and $\iota_{D}(\Sigma_{D})=\Sigma_{D}$ as in
Lemma 5.4 below.

######  Proposition 5.3.

For all $\varphi,\psi\in L^{2}(\nu_{\beta})$ and all $n\geq 1$ we have

${\rm Cor}_{n}(\sigma;\varphi,\psi;\nu_{\beta})={\rm
Cor}_{n}(\sigma;\psi\circ\iota_{D},\varphi\circ\iota_{D};\nu_{\alpha}).$

###### Proof.

By Lemma 5.4 below, for any shift invariant measure $\nu$ on $\Sigma_{D}$ we
have $h(\sigma,\nu)=h(\sigma,\nu\circ\iota_{D}^{-1})$ and
$\nu(\bigcup_{i=1}^{m}[\alpha_{i}])=\nu\circ\iota_{D}^{-1}(\bigcup_{i=1}^{m}[\beta_{i}])$
where $[\gamma]=\\{(\omega_{n})_{n\in\mathbb{Z}}\colon\omega_{0}=\gamma\\}$
for $\gamma\in D$. This implies $\nu_{\beta}=\nu_{\alpha}\circ\iota_{D}^{-1}$.
We have

$\begin{split}\int\varphi(\psi\circ\sigma^{n}){\rm
d}\nu_{\beta}&=\int\varphi\circ\iota_{D}(\psi\circ\sigma^{n}\circ\iota_{D}){\rm
d}\nu_{\alpha}\\\
&=\int\varphi\circ\iota_{D}(\psi\circ\iota_{D}\circ\sigma^{-n}){\rm
d}\nu_{\alpha}\\\
&=\int\varphi\circ\iota_{D}\circ\sigma^{n}(\psi\circ\iota_{D}){\rm
d}\nu_{\alpha},\end{split}$

and $\int\varphi{\rm d}\nu_{\beta}=\int\varphi\circ\iota_{D}{\rm
d}\nu_{\alpha}$ and $\int\psi{\rm d}\nu_{\beta}=\int\psi\circ\iota_{D}{\rm
d}\nu_{\alpha}$. Hence the desired equality holds. ∎

######  Lemma 5.4.

We have $\iota_{D}(\Sigma_{D})=\Sigma_{D}$, and
$\iota_{D}\circ\sigma=\sigma^{-1}\circ\iota_{D}$.

###### Proof.

The second assertion of the lemma is a consequence of the first one, which was
proved in [32, Section 3.5]. We include the proof here for the reader’s
convenience. For $n\geq 1$ and $\lambda=\gamma_{1}\cdots\gamma_{n}\in
L(\Sigma_{D})$ we set
$\rho^{\ast}(\lambda)=\rho(\gamma_{n})\cdots\rho(\gamma_{1})$. It is enough to
show that $\rho^{\ast}(\lambda)\in L(\Sigma_{D})$, namely ${\rm
red}(\rho^{\ast}(\lambda))\neq 0$. By the relations (1.5), either (i) ${\rm
red}(\lambda)=1$, or (ii) ${\rm red}(\lambda)=\xi\eta$ for some $\xi\in
L(\\{\beta_{1},\ldots,\beta_{m}\\}^{\mathbb{Z}})$ and $\eta\in
L(\\{\alpha_{1},\ldots,\alpha_{m}\\}^{\mathbb{Z}})$. In case (i), clearly we
have ${\rm red}(\rho^{\ast}(\lambda))=1$. In case (ii), we have
$\rho^{\ast}(\eta)\in L(\\{\beta_{1},\ldots,\beta_{m}\\}^{\mathbb{Z}})$ and
$\rho^{\ast}(\xi)\in L(\\{\alpha_{1},\ldots,\alpha_{m}\\}^{\mathbb{Z}})$, and
so ${\rm red}(\rho^{\ast}(\lambda))=\rho^{\ast}(\eta)\rho^{\ast}(\xi)\neq 0$.
∎

###  5.3. Transferring invariant measures

In order to transfer invariant measures on the two different spaces under the
coding map, we need two lemmas. We extend the central Jacobian to a function
on $[0,1]^{3}$ in the obvious way, and still denote the extension by
$\phi^{c}$: $\phi^{c}(x,y,z)=\phi^{c}(x,y)$ for $(x,y,z)\in[0,1]^{3}$.

######  Lemma 5.5.

Let $a,b\in(0,\frac{1}{m})$ and let $\mu$ be an $f_{a,b}$-invariant ergodic
Borel probability measure satisfying $\mu(\Lambda_{a,b})=1$.

  * (a)

If $\int\phi^{c}{\rm d}\mu=0$ then $\mu\circ\pi^{-1}(A_{0})=1$.

  * (b)

If $\int\phi^{c}{\rm d}\mu<0$ then $\mu\circ\pi^{-1}(A_{\alpha})=1$.

  * (c)

If $\int\phi^{c}{\rm d}\mu>0$ then $\mu\circ\pi^{-1}(A_{\beta})=1$.

###### Proof.

Since shift invariant ergodic measures on $\Sigma_{D}$ give measure $1$ to one
of the three sets in (5.1) (see [20, pp.102–103]), the assertions are
consequences of the definition of $H_{i}$, $\phi^{c}$ and Birkhoff’s ergodic
theorem. ∎

Let

$A_{\alpha,\beta}=\left\\{\omega\in\Sigma_{D}\colon\liminf_{i\to\infty}H_{i}(\omega)=-\infty\
\text{ or }\ \liminf_{i\to-\infty}H_{i}(\omega)=-\infty\right\\}.$

Note that $A_{\alpha,\beta}$ is shift invariant and contains $A_{\alpha}\cup
A_{\beta}$.

######  Lemma 5.6 ([32], Lemma 3.1).

For all $a,b\in(0,\frac{1}{m})$, the restriction of $\pi$ to
$\pi^{-1}(A_{\alpha,\beta})$ is a homeomorphism onto its image.

###  5.4. Connection to the Lebesgue measure

Since $c_{1}+c_{2}=\frac{1}{m}$, $f_{c_{1},c_{2}}$ and $f_{c_{2},c_{1}}$
preserve the Lebesgue measure on $[0,1]^{3}$.

######  Proposition 5.7.

We have $h(g_{c_{1}},{\rm Leb})=h(g_{c_{2}},{\rm Leb})=\log(m+1)$, ${\rm
Leb}\circ\pi_{c_{2},c_{1}}^{-1}=\nu_{\alpha}$ and ${\rm
Leb}\circ\pi_{c_{1},c_{2}}^{-1}=\nu_{\beta}.$

###### Proof.

Recall that the restriction of $g_{c_{1}}$ to $\Lambda_{c_{1},c_{2}}$ is
invertible, uniformly expanding in the $x$-direction and uniformly contracting
in the $z$-direction. Using Proposition 2.6 to deal with the dynamics in the
$y$-direction, it is easy to see that

$\bigvee_{n=-\infty}^{\infty}g^{-n}\\{\Omega_{\gamma}\cap\Lambda_{c_{1},c_{2}}\\}_{\gamma\in
D}\stackrel{{\scriptstyle\circ}}{{=}}\mathscr{B}(\Lambda_{c_{1},c_{2}}).$

Moreover, $(g_{c_{1}},{\rm Leb})$ is ergodic by Theorem A. Therefore, Shannon-
McMillan-Breimann’s theorem yields

$h(g_{c_{1}},{\rm
Leb})=-\lim_{n\to\infty}\frac{1}{n}\log\left|\bigcap_{k=0}^{n-1}g_{c_{1}}^{-k}\left(\Omega_{\omega_{k}}\right)\right|$

for Lebesgue a.e. $p\in\Lambda_{c_{1},c_{2}}$ where
$\pi_{c_{1},c_{2}}(p)=(\omega_{n})_{n=-\infty}^{\infty}\in\Sigma_{D}$. Clearly
we have

$\begin{split}-\lim_{n\to\infty}\frac{1}{n}\log\left|\bigcap_{k=0}^{n-1}g_{c_{1}}^{-k}\left(\Omega_{\omega_{k}}\right)\right|\geq&mc_{1}\log\frac{1}{c_{1}}+(1-mc_{1})\log\frac{1}{1-mc_{1}}\\\
&+(1-2mc_{1})\log m=\log(m+1),\end{split}$

and hence $h(g_{c_{1}},{\rm Leb})\geq\log(m+1)$. The reverse inequality is a
consequence of Theorem 5.2. We have verified $h(g_{c_{1}},{\rm
Leb})=\log(m+1)$. By [32, Lemma 3.9] and ${\rm Leb}\circ\iota^{-1}={\rm Leb}$
we have $h(g_{c_{2}},{\rm Leb})=h(g_{c_{1}},{\rm Leb})$.

By Theorem A and Theorem 2.12, the measure ${\rm
Leb}\circ\pi_{c_{2},c_{1}}^{-1}$ on $\Sigma_{D}$ is shift invariant and
ergodic. By $\int\phi^{c}{\rm d}{\rm Leb}=-\frac{m-1}{m+1}\log 2<0$ and Lemma
5.5, it gives measure $1$ to $A_{\alpha}$, and by Lemma 5.6 has entropy equal
to $h(g_{c_{2}},{\rm Leb})$, which equals $\log(m+1)$ as already proved. Hence
we obtain ${\rm Leb}\circ\pi_{c_{2},c_{1}}^{-1}=\nu_{\alpha}$. A proof of the
last equality in the proposition is analogous. ∎

###  5.5. Proof of Theorem C

First we show exponential mixing for $(\sigma,\nu_{\alpha})$,
$(\sigma,\nu_{\beta})$. To this end, by virtue of Proposition 5.3 it suffices
to show that for any pair $\varphi$, $\psi$ of Hölder continuous functions on
$\Sigma_{D}$, their correlation ${\rm
Cor}_{n}(\sigma;\varphi,\psi;\nu_{\beta})$ decays exponentially in $n$. This
does not immediately follow from Theorem B. Indeed, Proposition 5.7 implies
${\rm Cor}_{n}(\sigma;\varphi,\psi;\nu_{\beta})={\rm
Cor}_{n}(g_{c_{1}};\varphi\circ\pi_{c_{1},c_{2}},\psi\circ\pi_{c_{1},c_{2}};{\rm
Leb})$, but $\varphi\circ\pi_{c_{1},c_{2}}$ and $\psi\circ\pi_{c_{1},c_{2}}$
may not be Hölder continuous. Our strategy is to transfer to the Dyck system
the towers associated with $f_{c_{1}}$, $g_{c_{1}}$ constructed in Section
4.1, and mimic the proof of Theorem B.

We restrict ourselves to the shift invariant set $A_{\beta}$ in (5.1). Bear in
mind that $\nu_{\beta}(A_{\beta})=1$. By Lemma 5.6, the restriction of
$\pi_{c_{1},c_{2}}$ to $\pi_{c_{1},c_{2}}^{-1}(A_{\beta})$ is a homeomorphism
onto its image. We put

$\hat{\Delta}_{0}=\pi_{c_{1},c_{2}}(\Delta_{0})\cap A_{\beta}\ \text{ and }\
\hat{\Delta}_{0,i}=\pi_{c_{1},c_{2}}(\Delta_{0,i})\cap A_{\beta}.$

Define $\hat{R}\colon\pi_{c_{1},c_{2}}([0,1]^{3})\cap
A_{\beta}\to\mathbb{Z}_{+}\cup\\{\infty\\}$ by
$\hat{R}(\omega)=R(\pi_{c_{1},c_{2}}^{-1}(\omega))$. We define a tower

$\hat{\Delta}=\\{(\omega,\ell)\colon\omega\in\hat{\Delta}_{0},\
\ell=0,1,\ldots,\hat{R}(\omega)-1\\},$

and define a tower map $\hat{G}\colon\hat{\Delta}\to\hat{\Delta}$ by

$\hat{G}(\omega,\ell)=\begin{cases}(\omega,\ell+1)&\ \text{ if
}\ell+1<\hat{R}(\omega),\\\ (\sigma^{\hat{R}(\omega)}(\omega),0)&\ \text{ if
}\ell+1=\hat{R}(\omega).\end{cases}$

Collapsing the negative coordinate, we obtain the quotient tower
$\hat{\Delta}^{+}$ and the tower map
$\hat{F}\colon\hat{\Delta}^{+}\to\hat{\Delta}^{+}$. We fix a sigma-algebra on
$\hat{\Delta}$ that is obtained by naturally transplanting the Borel sigma-
algebra on $\Sigma_{D}$.

By Proposition 2.6, $\hat{R}$ is finite $\nu_{\beta}$-a.e. and
$\nu_{\beta}\\{\hat{R}=n\\}=|\\{R=n\\}|$. Each floor

$\hat{\Delta}_{\ell}=\\{(p,\ell)\in\hat{\Delta}\colon
p\in\hat{\Delta}_{0}\\},\ \ell\geq 0$

can be identified with $\\{\hat{R}>\ell\\}$, and so equipped with the
restriction of the measure $\nu_{\beta}$. Let $\hat{\mu}$ denote the
probability measure on $\hat{\Delta}$ given by

$\hat{\mu}(A)=\frac{1}{\int\hat{R}{\rm
d}\nu_{\beta}}\sum_{\ell=0}^{\infty}\nu_{\beta}(A\cap\hat{\Delta}_{\ell})\
\text{ for any measurable set }A\subset\hat{\Delta}.$

We repeat Steps 1 to 3 in the proof of Theorem B in Section 3.3. Each floor
$\hat{\Delta}_{\ell}$ is partitioned into $\\{\hat{\Delta}_{\ell,i}\\}_{i\geq
1\colon\hat{R}_{i}>\ell}$ where $\hat{\Delta}_{\ell,i}$ is a copy of
$\hat{\Delta}_{0,i}$. Let $\hat{\mathscr{D}}_{0}$ denote the partition of
$\hat{\Delta}$ into $\hat{\Delta}_{\ell,i}$-components. For $k\geq 1$, let
$\hat{\mathscr{D}}_{k}=\bigvee_{j=0}^{k-1}{\hat{G}}^{-j}\hat{\mathscr{D}}_{0}$.
For $\phi\in\mathscr{H}_{\eta}(\Sigma_{D})$ and $k\geq 1$, define
$\phi_{k}\colon\hat{\Delta}\to\mathbb{R}$ by
$\phi_{k}|_{A}=\inf\\{\tilde{\phi}(w)\colon w\in\hat{G}^{k}(A)\\}$ for
$A\in\hat{\mathscr{D}}_{2k}.$ Define
$\hat{\theta}\colon(\omega,\ell)\in\hat{\Delta}\mapsto\sigma^{\ell}(\omega)\in\Sigma_{D}$.
For every $A\in\hat{\mathscr{D}}_{2k}$, the diameter of the set
$\hat{\theta}({\hat{G}}^{k}(A))$ with respect to the Hamming metric $d$ does
not exceed $e^{-k}$. As a counterpart of (4.1) we have
$\sup_{A}|\tilde{\phi}\circ\hat{G}^{k}-\phi_{k}|\leq|\phi|e^{-\eta k}$, where
$|\phi|$ denotes the $\eta$-Hölder norm of $\phi$ with respect to $d$.
Therefore, as a counterpart of Lemma 4.1 we obtain
$\int|\tilde{\phi}\circ{\hat{G}}^{k}-\phi_{k}|{\rm
d}\hat{\mu}\leq|\phi|e^{-\eta k}$. To finish, the rest of the argument is
completely analogous to Step 3 in the proof of Theorem B.

We have verified exponential mixing for both $(\sigma,\nu_{\alpha})$ and
$(\sigma,\nu_{\beta})$. Taking a pair of Hölder continuous functions on
$\Sigma_{D}$ which depend only on positive coordinates, we obtain exponential
mixing for both $(\sigma_{+},\nu_{\alpha}^{+})$ and
$(\sigma_{+},\nu_{\beta}^{+})$. The proof of Theorem C is complete. ∎

### Acknowledgments

The author thanks Yoshitaka Saiki, Toshi Sugiyama, Masato Tsujii, Kenichiro
Yamamoto, James A. Yorke for fruitful discussions. This research was supported
by the JSPS KAKENHI 19K21835, 20H01811.

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