# A Dichotomy for the dimension of solenoidal attractors on high dimensional
space

Haojie Ren  School of Mathematical Sciences, Fudan University, No 220 Handan
Road, Shanghai, China 200433<EMAIL_ADDRESS>
###### Abstract.

We study dynamical systems generated by skew products:

$T:[0,1)\times\mathbb{C}\to[0,1)\times\mathbb{C}\quad\quad T(x,y)=(bx\mod
1,\gamma y+\phi(x))$

where integer $b\geq 2$, $\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$, and
$\phi$ is a real analytic $\mathbb{Z}$-periodic function. Let $\Delta\in[0,1)$
such that $\gamma=|\gamma|e^{2\pi i\Delta}$. For the case
$\Delta\notin\mathbb{Q}$ we prove the following dichotomy for the solenoidal
attractor $K^{\phi}_{b,\,\gamma}$ for $T$: Either $K^{\phi}_{b,\,\gamma}$ is a
graph of real analytic function, or the Hausdorff dimension of
$K^{\phi}_{b,\,\gamma}$ is equal to $\min\\{3,1+\frac{\log b}{\log
1/|\gamma|}\\}$. Furthermore, given $b$ and $\phi$, the former alternative
only happens for countable many $\gamma$ unless $\phi$ is constant.

##  1\. introduction

One of main purposes of dynamical system is to study the strange attractors.
In this paper, we concern the Hausdorff dimension of solenoidal attractors on
skew products.

For any positive integer $d$, let
$\mathbf{B}_{d}:=\\{\,z\in\mathbb{R}^{d}:\,|z|\leq 1\\}.$ For any functions
$f:[0,1)\to[0,1)$ and $g:[0,1)\times\mathbf{B}_{d}\to\mathbf{B}_{d}$, define
the map

$F:\,[0,1)\times\mathbf{B}_{d}\to[0,1)\times\mathbf{B}_{d}\quad\quad
F(x,\,y)=\big{(}f(x),\,g(x,y)\big{)}.$

Thus

(1.1)
$K_{F}:=\bigcap_{n=0}^{\infty}F^{n}\big{(}\,[0,1)\times\mathbf{B}_{d}\,\big{)}$

is an invariant set and called solenoidal attractor or solenoid for $F$.
Denote $\text{dim}_{H}(K_{F})$ as the Hausdorff dimension of $K_{F}$.

Historical remarks. The study of solenoidal attractors on skew products has
long history. For the case $d=1$, Alexander and Yorke [12] introduced a class
maps called generalized baker’s transformation:

$B:[-1,1]\times[-1,1]\circlearrowleft\quad\quad
B(x,y)=\begin{cases}(2x-1,\gamma y+(1-\gamma))&\,x\geq 0\\\ (2x+1,\gamma
y-(1-\gamma))&\,x<0.\end{cases}$

Solomyak [8, 6] proved $\text{dim}_{H}(K_{B})=2$ for Lebesgue-a.e.
$\gamma\in(1/2,1]$. Later Hochman [3] showed the packing dimension of the
exception set is zero and for every $\gamma$ transcendental in Varjú [13]. See
[14] for some $\gamma$ algebraic.

To generalize $B$ to the nonlinear case, Tsujii [9] introduced dynamical
systems generated by maps:

(1.2)
$\tilde{T}:[0,1)\times\mathbb{R}\to[0,1)\times\mathbb{R}\quad\quad\tilde{T}(x,y)=(bx\mod
1,\gamma y+\phi(x))$

where integer $b\geq 2$, $0<\gamma<1$ and $\phi$ is a real
$\mathbb{Z}$-periodic Lipschitz function. ( Note that
$\tilde{T}\big{(}\,[0,1)\times\mathbf{B}(0,\,\frac{\parallel\phi\parallel_{\infty}}{1-\gamma}\,)\,\big{)}\subset[0,1)\times\mathbf{B}(0,\,\frac{\parallel\phi\parallel_{\infty}}{1-\gamma}\,)$,
thus we can define the solenoids $K_{\tilde{T}}$ by (1.1)). For
$\gamma\in(1/b,1)$, Tsujii [9] proved that $\text{dim}(K_{\hat{T}})=2$ for
$C^{2}$ generic $\phi$. Later the author [22] gave a complete answer for all
$\gamma\in(0,1)$ when $\phi$ is a real analytic $\mathbb{Z}$-periodic
function. See e.g. [2, 7, 15, 16] for results about the dimension of solenoids
when $d=1$.

For the case $d=2$, studying the dimension of solenoids got harder. A well-
known example is Smale-Williams-type solenoids generated by the map

$Q:[0,1)\times\mathbb{R}^{2}\circlearrowleft\quad\quad Q(x,y)=(bx\mod 1,\gamma
y+\phi(x),\lambda y+\psi(x))$

where integer $b\geq 2$, $0<\gamma,\,\lambda<1$ and $\phi,\,\psi$ are real
$\mathbb{Z}$-periodic Lipschitz functions. In particular $K_{Q}$ is called
Smale-Williams solenoid  if $\phi(x)=\cos(2\pi x)$, $\psi(x)=\sin(2\pi x)$ and
$b=2$. When $\gamma,\,\lambda<1/b,$ $K_{Q}$ is called thin solenoid. [30, 31]
gave the positive answers for the dimension of thin Smale-Williams solenoids.
Later Simon [35] calculated the the Hausdorff dimension of thin Smale-Williams
solenoids when $Q$ is one-to-one map. Recently Mohammadpour et al.[33]
consider a general class of functions $Q^{\prime}$. Under the transversality
assumption, [33] calculated the the Hausdorff dimension of thin solenoids for
$Q^{\prime}$ when $Q^{\prime}$ is injective on $K_{Q^{\prime}}$. More results
about the Hausdorff dimension of solenoids can be found in [34, 35, 36, 37,
38].

Main findings. It is well-known that transversality is vital to study the
dimension of solenoids. One of our goals in this paper is to find ways to
remove the transversality assumption for the case $d\geq 2$. Another goal is
to explore ways to use the methods of additive combination[3, 27, 2, 17] to
study the dimension of strange attractors for nonlinear maps as [11, 22] did.
Thus we introduce the following nonlinear maps

(1.3) $T:[0,1)\times\mathbb{C}\to[0,1)\times\mathbb{C}\quad\quad
T(x,y)=(bx\mod 1,\gamma y+\phi(x))$

where integer $b\geq 2$, $\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$, and
$\phi$ is a real $\mathbb{Z}$-periodic Lipschitz function. Note that map $T$
is a generalization of $\tilde{T}$, see (1.2). Let $K^{\phi}_{b,\,\gamma}$ be
the solenoid for $T$. For any $\gamma\in\mathbb{C}\setminus\\{0\\}$, let
$\Delta=\Delta(\gamma)\in[0,1)$ be the unique number such that

(1.4) $\gamma=|\gamma|e^{2\pi i\Delta(\gamma)}.$

The main result of this paper is following.

######  Main Theorem.

Let $b\geq 2$ be an integer, $\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$.
Let $\phi$ be a $\mathbb{Z}$-periodic real analytic function. Then exactly one
of the following holds:

  1. (i)

$K^{\phi}_{b,\gamma}$ is a graph of a real analytic function;

  2. (ii)

$dim_{H}(K^{\phi}_{b,\gamma})=\min\\{3,1+\frac{\log b}{\log 1/|\gamma|}\\}$
when $\Delta\notin\mathbb{Q}$;

  3. (iii)

$dim_{H}(K^{\phi}_{b,\gamma})\geq\min\\{2,1+\frac{\log b}{\log 1/|\gamma|}\\}$
when $\Delta\in\mathbb{Q}$.

Moreover given $b$ and non-constant $\phi$, the first alternative only holds
for countable many $0<|\gamma|<1$.

Organization. In Sect. 2, we will introduce Theorem A which is used for
proving Main Theorem. The rest parts of this paper is devoted to prove Theorem
A. In Sect. 3, we will introduce the classical Ledrappier-Young theory for the
map $T$, and recall some basic properties of entropy that serve as the basic
tools in this paper. The main difficulty in proving Theorem A is proving
Theorem 5.1, since the rest parts can be proved by the similar method of [22]
(See Sect.2.2 for explanations). Thus we shall show the dimension conservation
property of $m_{x}$ in Sect. 4, which is used to prove Theorem 5.1 in Sect. 5.
Finally we will use Theorem 5.1 to give a sketchy proof for Theorem A in Sect.
6.

Acknowledgment. We would like to thank Feliks Przytycki for useful
communication.

##  2\. Theorem A and proof of Main Theorem

In this section, we shall first give the expression of solenoidal attractors
$K^{\phi}_{b,\,\gamma}$. Later we will introduce Theorem A and give some
explanations about the ideas of proof, which is our main finding in this
paper. Finally we will use Theorem A to prove the Main Theorem as we did in
[22].

###  2.1. Expression of $K^{\phi}_{b,\,\gamma}$

Let $\mathbb{Z}_{+}$ denote the set of positive integers and $\mathbb{R}_{+}$
denote the set of nonnegative real numbers. Let $\mathbb{N}$ be the set of
nonnegative integers. Let $\varLambda=\\{0,1,...,b-1\\}$,
$\varLambda^{\\#}=\bigcup_{n=1}^{\infty}\varLambda^{n}$,
$\Sigma=\varLambda^{\mathbb{Z}_{+}}$. For any word
$\textbf{j}\in\varLambda^{\\#}\cup\Sigma$ let $|\textbf{j}|=t$ if
$\textbf{j}=j_{1}j_{2}\ldots j_{t}\in\varLambda^{\\#}$, and
$|\textbf{j}|=\infty$ if $\textbf{j}\in\Sigma$. For any $x\in[0,1]$ define
function

(2.1)
$S(x,\textbf{j})=S_{b,\,\gamma}^{\phi}(x,\textbf{j})=\sum\limits_{n=1}^{|\textbf{j}|}{\gamma^{n-1}\phi\left(\frac{x}{b^{n}}+\frac{j_{1}}{b^{n}}+\frac{j_{2}}{b^{n-1}}+\cdot\cdot\cdot+\frac{j_{n}}{b}\right)},$

thus we have

$K^{\phi}_{b,\,\gamma}=\bigg{\\{}(x,S(x,\textbf{j})):x\in[0,1),\,\textbf{j}\in\Sigma\bigg{\\}}.$

See [9] for details, the proof in [9, Sect. 2] works for all
$\gamma\in\mathbb{C}$ such that $|\gamma|\in(0,1)$.

It is necessary to recall the following formulas from [9], which serve as the
basic tools in our paper. For any $x\in[0,1]$, $\textbf{w}=w_{1}w_{2}\ldots
w_{m}\in\varLambda^{\\#}$ and each
$\textbf{i},\,\textbf{j}\in\varLambda^{\\#}\cup\Sigma,$ let

(2.2) $\textbf{w}(x):=\frac{x+w_{1}+\ldots+w_{m}b^{m-1}}{b^{m}}$

and

(2.3) $\textbf{w}\textbf{i}=w_{1}\ldots w_{m}i_{1}i_{2}\ldots$

as usual, so we have

(2.4)
$S(x,\textbf{w}\textbf{i})=S(x,\textbf{w})+\gamma^{m}S(\textbf{w}(x),\textbf{i})$

by the definition of function $S(\cdot,\cdot)$. Therefore we have

(2.5)
$S(x,\,\textbf{w}\textbf{i})-S(x,\,\textbf{w}\textbf{j})=\gamma^{m}\bigg{(}S(\textbf{w}(x),\,\textbf{i})-S(\textbf{w}(x),\,\textbf{j})\bigg{)}.$

###  2.2. Theorem A

Let $\nu$ denote even distributed probability measure on $\varLambda$ and let
$\nu^{\mathbb{Z}_{+}}$ be product measure on $\Sigma.$ Denote the Lebesgue
measure on $[0,1)$ as $\mathcal{m}$. Define the probability measure on
$[0,1)\times\mathbb{C}$

$\omega:=G(\mathcal{m}\times\nu^{\mathbb{Z}_{+}})$

where the map

$G:[0,1)\times\Sigma\to[0,1)\times\mathbb{C}\quad\quad
G(x,\textbf{j})=(x,S(x,\textbf{j})).$

In fact the measure $\omega$ is the SRB measure for the map $T$. For every
$x\in[0,1]$ define the map

(2.6) $S_{x}:\Sigma\to\mathbb{C}\quad\quad
S_{x}(\,\textbf{j}\,)=S(x,\textbf{j})$

and measure

(2.7) $m_{x}:=S_{x}(\nu^{\mathbb{Z}_{+}})$

as in [9, 22]. Note that $\omega=\int_{[0,1)}\big{(}\delta_{x}\times
m_{x}\big{)}\,dx.$ For any $n\geq 1$, we have

(2.8)
$m_{x}=\frac{1}{b^{n}}\sum_{\textbf{j}\in\varLambda^{n}}T^{n}(m_{\textbf{j}(x)})$

where probability measures

(2.9) $T^{n}(m_{\textbf{j}\,(x)})=f_{x,\,\textbf{j}}(m_{\textbf{j}(x)})$

and
$f_{x,\,\textbf{j}}(y)=\gamma^{n}y+S(x,\,\textbf{j}),\,\forall\,y\in\mathbb{C}$.
See [9, Section 2] for details.

A probability measure $\mu$ in a metric space $X$ is called exact-dimensional
if there exists a constant $\kappa\geq 0$ such that for $\mu-\text{a.e.}$ $x$,

(2.10) $\lim_{r\to 0}\frac{\log\mu\big{(}\mathbf{B}(x,r)\big{)}}{\log
r}=\kappa.$

In this situation, we write $\text{dim}(\mu)=\kappa$ and call it the dimension
of $\mu$.

To study the dimension of the SRB measure $\omega$ as [22] did, we shall
recall the following gentle transversality condition (H). Shen and the author
[11] put it and studied the case $\gamma\in(1/b,1)$. Recently Gao and Shen
[18] studied the case $\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$.

######  Definition 2.1.

Given an integer $b\geq 2$ and $\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$,
we say that a real $\mathbb{Z}$-periodic $C^{1}$ function $\phi(x)$ satisfies
the condition (H) if

$S(x,\textbf{j})-S(x,\textbf{i})\not\equiv
0,\quad\forall\,\textbf{j}\neq\textbf{i}\in\Sigma.$

The following theorem is our main findings in this paper.

######  Theorem A.

Let $\phi(x)$ be a real analytic $\mathbb{Z}$-periodic function satisfies the
condition (H) for an integer $b\geq 2$, $\gamma\in\mathbb{C}$ such that
$0<|\gamma|<1$. If $\Delta\in\mathbb{R}\setminus\mathbb{Q}$, then

$dim(\omega)=\min\\{1+\frac{\log b}{\log 1/|\gamma|},3\\}.$

Recall that $\Delta$ is defined by (1.4).

Explanations of Theorem A. Inspired by the recent works [3, 2, 11], the author
[22] has studied Theorem A for the case $\gamma\in(0,1)$, thus it is nature to
consider whether these methods can be used to study the general case
$\gamma\in\mathbb{C}$. In fact, most parts of proof in [22] also work for the
case $\gamma\in\mathbb{C}$ such that $|\gamma|\in(0,1)$. Therefore, if the
inverse theorem for convolution on $\mathbb{R}$ of Hochman [3, Theorem 2.7]
also holds for all measures in $\mathscr{P}(\mathbb{C})$, there will be little
difficulty to prove Theorem A. Unfortunately that is not true, studying the
entropy of convolutions on $\mathbb{R}^{d}$ gets more tough.

To study the selfsimilar measures on $\mathbb{R}^{d}$, Hochman [27]
generalized the inverse theorem on $\mathbb{R}$ [3, Theorem 2.7] to high-
dimension spaces [27, Theorem 2.8]. However the latter is not easy to use
directly. Thus the main difficulty in our paper is to give a corollary of [27,
Theorem 2.8], whose condition is easier to verify in our system (See Theorem
5.1).

Indeed, The classical Ledrappier-Young theory implies that: for Lebesgue-a.e.
$x\in[0,1]$

$\text{dim}(m_{x})=\alpha.$

Thus it suffice to prove $\alpha=\min\\{2,\frac{\log b}{\log 1/|\gamma|}\\}$
(See Theorem 3.1 for details).

As mentioned before, studying the dimension of measures $m_{x}$ directly is
hard, thus we hope to understand the properties of projection measures
$\pi_{\theta}m_{x}$ and the conditional measures
$\\{\,(m_{x})_{\pi_{\theta}^{-1}(z)}\\}_{z\in\mathbb{R}}$, see Definition 4.1.
Thanks to the methods in [25, 26], we build a new Ledrappier-Young theory for
the nonlinear system generated by the map $T$, which establishes the
connections between $\text{dim}(m_{x})$ and $\text{dim}(\pi_{\theta}m_{x})$,
$\text{dim}\big{(}(m_{x})_{\pi_{\theta}^{-1}(z)}\big{)}$ for the case
$\Delta\notin\mathbb{Q}$, see Theorem 4.1. Therefore inspired by the work of
Hochman [27], we can use the skills in [3, 27, 2, 22] to build Theorem 5.1
which is a corollary of [27, Theorem 2.8]. The rest of proofs are similar to
[22] and we will omit some details for convenience.

###  2.3. Proof of Main Theorem

We shall first introduce the following Theorem 2.1, which is an immediate
consequence of [18, Theorem 2.1]. Note that the proof of [22, Theorem 2.1]
also works for $\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$, thus we omit
the proof of Theorem 2.1.

######  Theorem 2.1.

Fix $b\geq 2$ integer and $\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$.
Assume that $\phi$ is analytic $\mathbb{Z}$-periodic function. Then exactly
one of the following holds:

  1. (H.1)

$K^{\phi}_{b,\,\gamma}$ is a graph of real analytic function;

  2. (H.2)

$\phi$ satisfies the condition (H).

For any $\theta\in\mathbb{R}$, let $\pi_{\theta}:\mathbb{C}\to\mathbb{R}$ be
the projection function such that

(2.11) $z\mapsto\text{Re}(ze^{-2\pi i\theta})$

where $\text{Re}(ze^{-2\pi i\theta})$ is the real part of $ze^{-2\pi
i\theta}$.

###### Proof of the Main Theorem.

If $\phi$ is a non-constant function, then there exists $m\in\mathbb{Z}_{+}$
and $x^{\prime}\in[0,1)$ such that
$\phi(\frac{x^{\prime}}{b^{m}})\neq\phi(\frac{x^{\prime}+1}{b^{m}})$. Let
$10_{\infty}=100\ldots,\,0_{\infty}=00\ldots\in\Sigma$. Thus the function
$g(\gamma):=S^{\phi}_{b,\,\gamma}(x^{\prime},10_{\infty})-S^{\phi}_{b,\,\gamma}(x^{\prime},0_{\infty})$
has at most countable zero points in $\mathbf{B}(0,1)$, which implies: (i)
holds for at most countable many $\gamma\in\mathbb{C}$ such that
$0<|\gamma|<1$ and we only need to consider the case that condition (H) holds
by Theorem 2.1.

For the case $\Delta\in\mathbb{R}\setminus\mathbb{Q}$, it suffice to show
$dim(\omega)=\min\\{1+\frac{\log b}{\log 1/|\gamma|},3\\}$ since
$\text{supp}(\omega)=K^{\phi}_{b,\,\gamma}$ and
$\text{dim}_{B}(K^{\phi}_{b,\,\gamma})\leq\min\\{1+\frac{\log b}{\log
1/|\gamma|},3\\}.$ Thus we have (ii) holds by Theorem A.

For the case $\Delta\in\mathbb{Q}$, we may assume that $\Delta\neq 0$ by [22,
Main Theorem]. Let $\Delta=k/m\in(0,1)$ for some $k,\,m\in\mathbb{Z}_{+}.$
Since the condition (H) holds, there exist $\theta\in[0,1)$ such that

(2.12) $\pi_{\theta}\circ
S^{\phi}_{b,\,\gamma}(x,1_{\infty})-\pi_{\theta}\circ
S^{\phi}_{b,\,\gamma}(x,0_{\infty})\not\equiv 0.$

define the function $\overline{\phi}:\mathbb{R}\to\mathbb{R}$ be such that

$\overline{\phi}(x)=\pi_{\theta}\bigg{(}\,\sum_{t=0}^{m-1}\gamma^{m-1-t}\phi(b^{t}x)\,\bigg{)}.$

Thus (2.12) implies $\overline{\phi}$ is a real analytic $\mathbb{Z}$-periodic
function such that the condition (H) holds for $b^{m}$ and $|\gamma|^{m}.$
Therefore we have

(2.13)
$\text{dim}\big{(}\,\omega^{\overline{\phi}}_{b^{m},\,|\gamma|^{m}}\,\big{)}=\min\\{2,1+\frac{\log
b}{\log 1/|\gamma|}\\}$

by [22, Theorem A]. Define the function
$\hat{\pi}_{\theta}:[0,1)\times\mathbb{C}\to[0,1)\times\mathbb{C}$ such that
$\hat{\pi}_{\theta}(x,z)=(x,\,\pi_{\theta}(z))$. Therefore we have

$\hat{\pi}_{\theta}(\,\omega^{\phi}_{b,\,\gamma}\,)=\omega^{\overline{\phi}}_{b^{m},\,|\gamma|^{m}},$

which implies
$\text{dim}\big{(}\,\omega^{\phi}_{b,\,\gamma}\,\big{)}\geq\min\\{2,1+\frac{\log
b}{\log 1/|\gamma|}\\}$ by
$\text{dim}\big{(}\,\omega^{\phi}_{b,\,\gamma}\,\big{)}\geq\text{dim}\big{(}\,\hat{\pi}_{\theta}(\omega^{\phi}_{b,\,\gamma})\,\big{)}$
and (2.13). Therefore (iii) holds. ∎

##  3\. Ledrappier-Young theory and entropy

In this section we shall first recall the classical Ledrappier-Young formula
for $T$. Then we will give a brief introduce the conditional measure theorem
of Rohlin [21] and recall the definition of entropy. Finally we will introduce
some basic properties of the entropy of measures, which serve as basic tools
in our paper.

###  3.1. Ledrappier-Young formula

The following Theorem 3.1 is the classical Ledrappier-Young formula for SRB
measure $\omega$, which can be proved by exactly the same methods in [22,
Theorem 3.1]. Thus we omit the proof, see also [4, 5, 19].

######  Theorem 3.1.

Let $b\geq 2$ be an integer and $\gamma\in\mathbb{C}$ such that
$0<|\gamma|<1$. If $\phi:\mathbb{R}\to\mathbb{R}$ is a $\mathbb{Z}$-periodic
Lipschitz function, then

  1. (1)

$\omega$ is exact dimensional;

  2. (2)

there is a constant $\alpha\in[0,2]$ such that for Lebesgue-a.e. $x\in[0,1)$,
$m_{x}$ is exact dimensional and $\dim(m_{x})=\alpha$ and

(3.1) $\dim(\omega)=1+\alpha.$

Recall that the definition of $m_{x}$ is from (2.7).

###  3.2. Conditional measure and entropy

Let $(\Omega,\mathscr{B},\mu)$ be a Lebesgue space (see [23, Definition 2.3]).
A partition $\eta$ of $\Omega$ is called measurable partition if, up to a set
of measure zero, the quotient space is separated by a countable number of
measurable sets $\\{B_{i}\\}$. Let $\hat{\eta}$ be the sub-$\sigma$-algebra of
$\mathscr{B}$ whose elements are unions of elements of $\eta$.

######  Theorem 3.2 (Rohlin [21]).

Let $\eta$ be a measurable partition of a Lebesgue space
$(\Omega,\mathscr{B},\mu)$. Then for every $x$ in the set of full
$\mu$-measure, there is a probability measure $\mu^{\eta}_{x}$ defined on
$\eta(x)$, the element of $\eta$ containing $x$. These measures are uniquely
characterized (up to set of $\mu$-measure 0) by the following properties: if
$A\subset\Omega$ is a measurable set, then $x\mapsto\mu^{\eta}_{x}(A)$ is
$\hat{\eta}$-measurable and

$\mu(A)=\int\mu^{\eta}_{x}(A)d\mu(x).$

These properties imply that for any $g\in L^{1}(\Omega,\mathscr{B},\mu)$,
$\mu^{\eta}_{x}(g)=\mathbf{E}_{\mu}(g|\,\hat{\eta})(x)$ for
$\mu-\text{a.e.\,}x$, and $\mu(g)=\int\mathbf{E}_{\mu}(g|\,\hat{\eta})d\mu.$

In Theorem 3.2, we call $\\{\,\mu^{\eta}_{x}\,\\}_{x\in\Omega}$ the canonical
system of conditional measure with $\eta$. A countable partition $\mathcal{P}$
is a countable collection of pairwise disjoint measurable subsets of $\Omega$
whose union is equal to $\Omega$. For any sub-$\sigma$-algebra $\mathcal{M}$
of $\mathscr{B}$, any countable $\mathscr{B}$-measurable partition
$\mathcal{P}$ of $\Omega$ we define the conditional information

$\mathbf{I}_{\mu}(\,\mathcal{P}\,|\,\mathcal{M})=-\sum_{I\in\mathcal{P}}\chi_{I}\text{log}_{b}\mathbf{E}_{\mu}(\chi_{I}|\mathcal{M}),$

and the conditional entropy

$H_{\mu}(\,\mathcal{P}\,|\,\mathcal{M})=\int_{\Omega}\mathbf{I}_{\mu}(\,\mathcal{P}\,|\,\mathcal{M})d\mu.$

If $\mathcal{M}=\\{\emptyset,\,\Omega\\}$, let
$\mathbf{I}_{\mu}(\mathcal{P})=\mathbf{I}_{\mu}(\,\mathcal{P}\,|\,\mathcal{M})$
and $H(\mu,\mathcal{P})=H_{\mu}(\,\mathcal{P}\,|\,\mathcal{M})$ for
convenience.

For any countable $\mathscr{B}$-measurable partition $\mathcal{Q}$ of
$\Omega$. Let $\mathcal{Q}(x)$ be the member of $\mathcal{Q}$ that contains
$x$. If $\mu(\mathcal{Q}(x))>0$, we call the conditional measure

$\mu_{\mathcal{Q}(x)}(A)=\frac{\mu(A\cap\mathcal{Q}(x))}{\mu(\mathcal{Q}(x))}$

a $\mathcal{Q}$-component of $\mu$. Let function
$f_{b}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a function such that

$f_{b}(p)=-p\log_{b}p$

where the common convention $0\log 0=0$ is adopted. In particular we have

$H(\mu,\mathcal{Q})=\sum_{Q\in\mathcal{Q}}f_{b}\big{(}\mu(Q)\big{)}.$

For another countable partition $\mathcal{P}$, we also have

$H(\mu,\mathcal{Q}|\mathcal{P})=\sum_{P\in\mathcal{P},\,\mu(P)>0}\mu(P)H(\mu_{P},\mathcal{Q}).$

When $\mathcal{Q}$ is a refinement of $\mathcal{P}$, i.e.,
$\mathcal{Q}(x)\subset\mathcal{P}(x)$ for every $x\in\Omega$, we have

$H(\mu,\mathcal{Q}|\mathcal{P})=H(\mu,\mathcal{Q})-H(\mu,\mathcal{P}).$

###  3.3. Entropy of measures

In this subsection we shall introduce some notations whose were used
extensively in [3, 2, 17, 27, 22], which also serve as the basic language in
this paper. Latter we will recall some basic facts about the Entropy of
measures.

Let $(\Omega,\mathscr{B},\mu)$ be a Lebesgue space. If there exists a sequence
of partitions $\mathcal{Q}_{i}$, $i=0,1,2,\cdots$, such that
$\mathcal{Q}_{i+1}$ is a refinement of $\mathcal{Q}_{i}$, we shall denote
$\mu_{x,\,i}=\mu_{\mathcal{Q}_{i}(x)}$, and call it a $i$-th component measure
of $\mu$. For a finite set $I$ of integers, if for every $i\in I$, there is a
random variable $Y_{i}$ defined over $(\Omega,\hat{\mathcal{Q}_{i}},\mu)$, see
Section 3.2 for the definition of $\hat{\mathcal{Q}_{i}}$. Then we shall use
the following notation

$\mathbb{P}_{i\in I}(K_{i})=\mathbb{P}_{i\in
I}^{\,\mu}(K_{i}):=\frac{1}{\\#I}\sum_{i\in I}\mu(K_{i}),$

where $K_{i}$ is an event for $Y_{i}$. If $Y_{i}$’s are $\mathbb{R}$-valued
random variable, we shall also use the notation

$\mathbb{E}_{i\in I}(Y_{i})=\mathbb{E}^{\,\mu}_{i\in
I}(Y_{i}):=\frac{1}{\\#I}\sum_{i\in I}\mathbb{E}(Y_{i}).$

Therefore the following holds

$H(\mu,\mathcal{Q}_{m+n}|\mathcal{Q}_{n})=\mathbb{E}(H(\mu_{x,\,n},\mathcal{Q}_{m+n}))=\mathbb{E}^{\,\mu}_{i=n}(H(\mu_{x,\,i},\mathcal{Q}_{i+m})).$

In most cases of this paper, we shall consider the situation that
$\Omega=\mathbb{R},\,\mathbb{C}$. Write $\mathbb{R}^{d}$ for $\mathbb{R}$ or
$\mathbb{C}$. Let $\mathcal{L}^{\mathbb{R}}_{n}$ be the partition of
$\mathbb{R}$ into $b$-adic intervals of level $n$, i.e., the intervals
$[j/b^{n},(j+1)/b^{n})$, $j\in\mathbb{Z}$. In this paper we also regard
$\mathbb{C}$ as $\mathbb{R}^{2}$. Similar let

$\mathcal{L}^{\mathbb{C}}_{n}:=\bigg{\\{}\,[j_{1}/b^{n},(j_{1}+1)/b^{n})\times[j_{2}/b^{n},(j_{2}+1)/b^{n})\,\bigg{\\}}_{j_{1},\,j_{2}\in\mathbb{Z}}.$

In the rest of paper, we may write
$\mathcal{L}^{\mathbb{R}}_{n},\,\mathcal{L}^{\mathbb{C}}_{n}$ as
$\mathcal{L}_{n}$ for convenience. For any $I\in\mathcal{L}_{n}$ and
$\mu(I)>0$ let

$\mu^{I}:=g_{I}(\mu_{I})$

where map $g_{I}(z):=b^{n}\cdot z+c_{I},\,\forall\,z\in\mathbb{R}^{d}$ and
$g_{I}(I)=[0,1)^{d}$ holds for some $c_{I}\in\mathbb{R}^{d}$. As [3] did,
denote

(3.2) $\mu^{x,\,i}=\mu^{\mathcal{L}_{i}(x)}.$

Let $\mathscr{P}(\mathbb{R}^{d})$ denote the collection of all Borel
probability measures in $\mathbb{R}^{d}$. If a probability measure
$\mu\in\mathscr{P}(\mathbb{R}^{d})$ is exact dimensional, its dimension is
closely related to the entropy. In fact we have the following result [10,
Theorem 4.4]. See also [20, Theorem 1.3].

######  Proposition 3.1.

If $\mu\in\mathscr{P}(\mathbb{R}^{d})$ is exact dimensional, then

$\dim(\mu)=\lim\limits_{n\to\infty}\frac{1}{n}H(\mu,\mathcal{L}_{n}).$

The following are some well-known facts about entropy and conditional entropy,
which will be used a lot in our work. See [3, Section 3.1] for details. Define
function

(3.3) $H:[0,1]\to\mathbb{R}_{+}\quad\quad H(p)=f_{b}(t)+f_{b}(1-t).$

######  Lemma 3.1 (Concavity and convexity).

Consider a measurable space $(\Omega,\mathscr{B})$ which is endowed with
partitions $\mathcal{Q}$ and $\mathcal{P}$ such that $\mathcal{P}$ is a
refinement of $\mathcal{Q}$. Let $\mu,\mu^{\prime}$ be probability measures in
$(\Omega,\mathscr{B})$. The for any $t\in(0,1)$,

$(\text{concavity})\quad
tH(\mu,\mathcal{Q})+(1-t)H(\mu^{\prime},\mathcal{Q})\leq
H(t\mu+(1-t)\mu^{\prime},\mathcal{Q}),$ $\qquad\qquad
tH(\mu,\mathcal{P}|\mathcal{Q})+(1-t)H(\mu^{\prime},\mathcal{P}|\mathcal{Q})\leq
H(t\mu+(1-t)\mu^{\prime},\mathcal{P}|\mathcal{Q}),$ $(\text{convexity})\quad
H(t\mu+(1-t)\mu^{\prime},\mathcal{Q})\leq
tH(\mu,\mathcal{Q})+(1-t)H(\mu^{\prime},\mathcal{Q})+H(t).$

######  Lemma 3.2.

Let $\mu\in\mathscr{P}(\mathbb{R}^{d})$. There is a constant $C_{d}>0$ such
that for any affine map $f(x)=ax+c$,
$a,\in\mathbb{R}\setminus\\{0\\},\,c\in\mathbb{R}^{d}$ and for any
$n\in\mathbb{N}$ we have

$\left|H(f\mu,\,\mathcal{L}_{n+[\log_{b}|a|]})-H(\mu,\,\mathcal{L}_{n})\right|\leq
C_{d}.$

######  Lemma 3.3.

Given a probability space $(\Omega,\mathscr{B},\mu)$, if
$f,g:\Omega\to\mathbb{R}^{d}$ are measurable and $\sup_{x}|f(x)-g(x)|\leq
b^{-n}$ then

$\left|H(f\mu,\mathcal{L}_{n})-H(g\mu,\mathcal{L}_{n})\right|\leq C_{d},$

where $C_{d}$ is an absolute constant.

The following lemma is from [3, Lemma 3.4].

######  Lemma 3.4.

For $R\geq 1$ and $\mu\in\mathscr{P}([-R,R]^{2})$ and integers $m\leq n$,

$\frac{1}{n}H(\mu,\mathcal{L}_{n})=\mathbb{E}^{\,\mu}_{0\leq
i<n}\bigg{(}\frac{1}{m}H(\mu^{x,\,i},\mathcal{L}_{m})\bigg{)}+O\big{(}\frac{m}{n}+\frac{\log
R}{n}\big{)}.$

See (3.2) for the definition of $\mu^{x,\,i}$.

##  4\.  dimension conservation

In this section, we shall first introduce the definition of dimension
conservation. Later we will give the dimension conservation property of
measures $m_{x}$. The following notation is introduced by Furstenberg [24].

######  Definition 4.1.

For any $\theta\in[0,1]$, a Borel probability measure
$\mu\in\mathscr{P}(\mathbb{C})$ is called $\theta$-dimension conservation if
we have

  1. (1)

$\mu,\,\pi_{\theta}\mu$ are exact-dimensional;

  2. (2)

for $\mu\text{-a.e.}\,z\in\mathbb{C}$, $\mu^{\eta_{\theta}}_{z}$ is exact-
dimensional and

$\text{dim}(\mu)=\text{dim}(\pi_{\theta}\mu)+\text{dim}(\mu^{\eta_{\theta}}_{z})$

where $\pi_{\theta}$ is from (2.11), partition
$\eta_{\theta}:=\big{\\{}\,\pi_{\theta}^{-1}(\,\\{x\\}\,):\,x\in\mathbb{R}\,\big{\\}}$
and $\\{\,\mu^{\eta_{\theta}}_{z}\,\\}_{z\in\mathbb{C}}$ is the canonical
system of conditional measure with $\eta_{\theta}$ (See Theorem 3.2).

Feng and Hu [25] studied the exact-dimensional properties of selfsimilar
measures by Ledrappier-Young theory [5]. The difference is that they consider
the system in symbolic space instead of Riemann space. They also provided some
useful tools for this approach. Based on these, Falconer and Jin [26] studied
the dimension conservation properties of random self-similar. Following this
strategy we have the following result, which is important to prove Theorem
5.1.

######  Theorem 4.1.

Let $b\geq 2$ be an integer and $\gamma\in\mathbb{C}$ such that
$0<|\gamma|<1$. Let $\phi:\mathbb{R}\to\mathbb{R}$ be a $\mathbb{Z}$-periodic
Lipschitz function. If $\Delta\in\mathbb{R}\setminus\mathbb{Q}$, then there
exists nonnegative constants $\beta,\,\upsilon$ such that the following holds.

For Lebesgue-$\text{a.e.}\,(x,\theta)\in[0,1)^{2},$ the measure $m_{x}$ is
$\theta$-dimension conservation. Furthermore we have

  1. (D.1)

$\text{dim}(\pi_{\theta}m_{x})=\beta$;

  2. (D.2)

for $m_{x}\text{-a.e.}\,z\in\mathbb{C}$,
$\text{dim}\big{(}\,(m_{x})^{\eta_{\theta}}_{z}\,\big{)}=\upsilon$.

Recall $\Delta=\Delta(\gamma)$ is defined by (1.4). We only consider the case
$\Delta\notin\mathbb{Q}$ in the rest of paper. The following is a corollary of
Theorem 4.1 by the observation of Hochman [27, Lemma 3.21(5)]. We offer the
proof for the completeness.

######  Corollary 4.1.

If $\alpha\geq\beta+1$, then $\alpha\geq 2.$

###### Proof.

By Theorem 4.1 there exists $x,\,\theta_{1},\,\theta_{2}\in[0,1)$ such that
$\theta_{1}\neq\theta_{2}$ and the following holds:
$m_{x},\,\pi_{\theta_{j}}m_{x}$ are exact-dimensional and
$\text{dim}(m_{x})=\alpha$, $\text{dim}(\pi_{\theta_{j}}m_{x})=\beta$,
$j=1,2.$ For any $n\in\mathbb{Z}_{+}$, since each atom of
$\pi_{\theta_{1}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\bigvee\pi_{\theta_{2}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})$
intersects $\boldmath{O}_{\theta_{1},\,\theta_{2}}(1)$ atoms of
$\mathcal{L}^{\mathbb{C}}_{n}$ and vice versa, we have

(4.1) $\displaystyle H(m_{x},\,\mathcal{L}^{\mathbb{C}}_{n})$
$\displaystyle=H\big{(}m_{x},\,\pi_{\theta_{1}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\vee\pi_{\theta_{2}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\big{)}+\boldmath{O}_{\theta_{1},\,\theta_{2}}(1)$
$\displaystyle=H(\,m_{x},\,\pi_{\theta_{1}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n}))+H\big{(}m_{x},\,\pi_{\theta_{2}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\mid\pi_{\theta_{1}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\big{)}+\boldmath{O}_{\theta_{1},\,\theta_{2}}(1)$
$\displaystyle\geq
H\big{(}m_{x},\,\pi_{\theta_{1}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\mid\pi_{\theta_{2}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\big{)}+H\big{(}m_{x},\,\pi_{\theta_{2}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\mid\pi_{\theta_{1}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\big{)}+\boldmath{O}_{\theta_{1},\,\theta_{2}}(1).$

The above also implies

$H\big{(}m_{x},\,\pi_{\theta_{1}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\mid\pi_{\theta_{2}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\big{)}=H(m_{x},\,\mathcal{L}^{\mathbb{C}}_{n})-H(\pi_{\theta_{2}}m_{x},\,\mathcal{L}^{\mathbb{R}}_{n})+\boldmath{O}_{\theta_{1},\,\theta_{2}}(1)$

and

$H\big{(}m_{x},\,\pi_{\theta_{2}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\mid\pi_{\theta_{1}}^{-1}(\mathcal{L}^{\mathbb{R}}_{n})\big{)}=H(m_{x},\,\mathcal{L}^{\mathbb{C}}_{n})-H(\pi_{\theta_{1}}m_{x},\,\mathcal{L}^{\mathbb{R}}_{n})+\boldmath{O}_{\theta_{1},\,\theta_{2}}(1).$

Combining these with (4.1), we have

$\frac{1}{n}H(m_{x},\,\mathcal{L}^{\mathbb{C}}_{n})\geq\sum_{k=1}^{2}\bigg{(}\frac{1}{n}H(m_{x},\,\mathcal{L}^{\mathbb{C}}_{n})-\frac{1}{n}H(\pi_{\theta_{k}}m_{x},\,\mathcal{L}^{\mathbb{R}}_{n})\bigg{)}+\boldmath{O}_{\theta_{1},\,\theta_{2}}(\frac{1}{n}).$

Therefore when $n$ goes to infinity, we have

$\alpha\geq 2(\alpha-\beta)$

by Proposition 3.1, which implies $\alpha\geq 2.$ ∎

######  Remark 4.1.

Under the observation of Corollary 4.1, we only need Theorem 4.1 (D.1) to
prove Theorem 5.1. For completeness and the readers to understand our ideas,
we still introduce Theorem 4.1 (D.2) and give the proof.

###  4.1. Group extension

By the classical Ledrappier-Young theory [4, 5, 19], we can only get the
formula (3.1). To study the dimension of $m_{x}$, it is important to
understand the projection measures $\pi_{\theta}m_{x}$ and conditional
measures $(m_{x})^{\eta_{\theta}}_{z}$. Thus it is nature to consider the map

$\displaystyle\hat{T}:$
$\displaystyle\Sigma\times[0,1)^{2}\,\to\,\Sigma\times[0,1)^{2}$
$\displaystyle(\,\textbf{j},\,x,\,\theta)\,\mapsto\,(\,\lfloor
bx\rfloor\,\textbf{j},\,bx\mod 1,\,\theta-\Delta\mod 1)$

where $\lfloor bx\rfloor$ is the largest integer not greater than $bx$ and see
(2.3) for $\lfloor bx\rfloor\,\textbf{j}$. Recall $\mathcal{m}$ is the
Lebesgue measure on $[0,1)$. Let

$\tilde{\omega}:=\nu^{\mathbb{Z}_{+}}\times\mathcal{m}\times\mathcal{m}$

and $\mathcal{B}$ be the Borel set of $\Sigma\times[0,1)^{2}$. For any metric
space $(X,\,\rho)$, denote $\mathcal{B}_{X}$ as the Borel set of $X$. Let us
consider the probability space
$(\Sigma\times[0,1)^{2},\,\mathcal{B},\,\tilde{\omega})$ and we have the
following Lemma 4.1.

######  Lemma 4.1.

If $\Delta\notin\mathbb{Q}$, then $\hat{T}$ is ergodic.

###### Proof.

Let
$F:\varLambda^{\mathbb{Z}}\times[0,1)\to\varLambda^{\mathbb{Z}}\times[0,1)$ be
the map such that
$(\textbf{j},\,\theta)\,\mapsto(\tau(\textbf{j}),\,\theta-\Delta\mod 1)$ where
$\tau$ is the left shift map on $\varLambda^{\mathbb{Z}}$. Let us consider the
probability space
$(\varLambda^{\mathbb{Z}}\times[0,1),\,\mathcal{B}_{\varLambda^{\mathbb{Z}}\times[0,1)},\,\nu^{\mathbb{Z}}\times\mathcal{m})$.
It suffice to proof that $F$ is ergodic, since $\Pi\circ F=\hat{T}\circ\Pi$
where
$\Pi:\,\varLambda^{\mathbb{Z}}\times[0,1)\,\to\Sigma\times[0,1)\times[0,1)$ is
the map such that

$(\textbf{j},\,\theta)\,\mapsto\,\bigg{(}j_{0}j_{-1}\ldots,\,\sum_{k=1}^{\infty}\frac{j_{k}}{b^{k}},\,\theta\bigg{)}.$

Regard $[0,1)$ as a group with the addition

$a+b:=(a+b)\mod 1\quad\quad\forall\,a,b\in[0,1)$

as usual. Assume $F$ is not ergodic, then there exists a proper closed
subgroup $H$ of $[0,1)$ and functions $f^{\prime}\in
C(\varLambda^{\mathbb{Z}};\,H),\,h\in C(\varLambda^{\mathbb{Z}};\,[0,1))$ such
that $\Delta=h\circ\tau-f^{\prime}-h$ by [28, Theorem 5.1]. Let
$1_{\infty}=\ldots 111\ldots\in\Lambda^{\mathbb{Z}}$. Thus
$f^{\prime}(1_{\infty})=-\Delta\in H$, which contradicts with that $H$ is a
proper closed subgroup of $[0,1)$. ∎

###  4.2. Proof of Theorem 4.1 (D.1)

In the rest of paper we denote $\log$ as $\log_{b}$ for convenience. Let
$R^{\phi}_{\gamma}:=\frac{2\parallel\phi\parallel_{\infty}}{1-|\gamma|}.$
Following [25], for any function $g:\Sigma\to\mathbb{R}^{d}$ and every
$\textbf{j}\in\Sigma$, $n\in\mathbb{N}$, let

(4.2)
$\mathbf{B}_{g}(\textbf{j},\,n):=g^{-1}\bigg{(}\,\mathbf{B}\big{(}g(\textbf{j}),\,R^{\phi}_{\gamma}|\gamma|^{n}\,\big{)}\,\bigg{)}.$

Let $\tau$ be the left shift on $\Sigma$ as usual. Let $\mathcal{P}$ be the
partition of $\Sigma$ such that

$\mathcal{P}=\\{\,[k]\,:\,k\in\varLambda\,\\}$

where $[k]=\\{\,\textbf{j}\in\Sigma:j_{1}=k\,\\}.$ It is necessary to recall
the following basic facts about $\pi_{\theta}$. For any $\theta\in\mathbb{R}$,
we have

(4.3) $\pi_{\theta}(\gamma z)=|\gamma|\pi_{\theta-\Delta}(z)\quad\quad\forall
z\in\mathbb{C}.$

Therefore (2.9) implies: for each $n\in\mathbb{Z}_{+}$ and
$\textbf{w}\in\varLambda^{n}$, we have

(4.4)
$\pi_{\theta}(T^{n}m_{\textbf{w}(x)})=g_{\textbf{w},\,\theta}(\pi_{\theta-n\Delta}m_{\textbf{w}(x)})$

where function $g_{\textbf{w},\,\theta}(z):=|\gamma|^{n}z+\pi_{\theta}\circ
S_{x}(\textbf{w})$ for each $z\in\mathbb{R}$.

######  Lemma 4.2.

For any $\theta,x\in[0,1],$ and $\textbf{j}\in\Sigma$, $n\in\mathbb{N}$, we
have

(4.5) $\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n+1)\cap\mathcal{P}(\textbf{j})=\tau^{-1}\bigg{(}\,\mathbf{B}_{\pi_{\theta-\Delta}\circ
S_{(x+j_{1})/b}}(\tau(\textbf{j}),\,n)\,\bigg{)}\cap\mathcal{P}(\textbf{j}).$

###### Proof.

For any $\textbf{i}\in\Sigma$ such that $i_{1}=j_{1}$ and $|\pi_{\theta}\circ
S_{x}(\textbf{j})-\pi_{\theta}\circ S_{x}(\textbf{i})|\leq
R^{\phi}_{\gamma}|\gamma|^{n+1}$, we have

$\bigg{|}\,\pi_{\theta}\bigg{(}\gamma
S_{(x+j_{1})/b}\circ\tau(\textbf{j})-\gamma
S_{(x+j_{1})/b}\circ\tau(\textbf{i})\bigg{)}\,\bigg{|}\leq
R^{\phi}_{\gamma}|\gamma|^{n+1}$

by the definition of function $S_{x}$ and (2.5). This implies

$\bigg{|}\,\pi_{\theta-\Delta}\bigg{(}S_{(x+j_{1})/b}\circ\tau(\textbf{j})-S_{(x+j_{1})/b}\circ\tau(\textbf{i})\bigg{)}\,\bigg{|}\leq
R^{\phi}_{\gamma}|\gamma|^{n}$

by (4.3). Combining the above with (4.2), we have

$\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n+1)\cap\mathcal{P}(\textbf{j})\subset\tau^{-1}\bigg{(}\mathbf{B}_{\pi_{\theta-\Delta}\circ
S_{(x+j_{1})/b}}(\tau(\textbf{j}),\,n)\bigg{)}\cap\mathcal{P}(\textbf{j}).$

Finally the other side can be proved by the same method. ∎

Let $\eta$ be the partition of $\Sigma\times[0,1)^{2}$ such that

$\eta:=\big{\\{}\,\Sigma\times\\{x\\}\times\\{\theta\\}:\,\theta,x\in[0,1)\,\big{\\}}$

and $\hat{\eta}$ be the sub-$\sigma$-algebra of $\mathcal{B}$ generated by
$\eta$. Let $\tilde{\mathcal{P}}$ be the sub-$\sigma$-algebra of $\mathcal{B}$
generated by partition

$\big{\\{}\,[k]\times[0,1)^{2}:\,k\in\varLambda\,\big{\\}}.$

Define the function

$\Phi:\Sigma\times[0,1)^{2}\to\mathbb{R}\quad\quad\Phi(\,\textbf{j},\,x,\,\theta)=\pi_{\theta}\circ
S_{x}(\textbf{j}).$

Let $\mathcal{B}_{\Phi}:=\Phi^{-1}\big{(}\mathcal{B}_{\mathbb{R}}\big{)}$ be
the sub-$\sigma$-algebra of $\mathcal{B}$. The following is an immediate
consequence of [25, Proposition 3.5].

######  Lemma 4.3.

For $\hat{\omega}$-a.e. $(\,\textbf{j},x,\theta)\in\Sigma\times[0,1)^{2}$ we
have

(4.6)
$\lim_{n\to\infty}\log\frac{\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\cap\mathcal{P}(\textbf{j})\big{)}}{\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\big{)}}=-\mathbf{I}_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\Phi}\big{)}(\,\textbf{j},x,\theta).$

Furthermore, set

$g(\textbf{j},x,\theta)=\sup_{n\in\mathbb{N}}-\log\frac{\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\cap\mathcal{P}(\textbf{j})\big{)}}{\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\big{)}}.$

Then $g\geq 0$ and $g\in
L^{1}(\Sigma\times[0,1)^{2},\,\mathcal{B},\,\hat{\omega}).$

We also need the following ergodic theorem due to Maker [29].

######  Theorem 4.2.

Let $(\Omega,\,\mathcal{M},\,\mu,\,G)$ be a measure-preserving system and let
$\\{g_{n}\\}$ be integrable functions on $(\Omega,\,\mathcal{M},\,\mu)$. If
$g_{n}(x)\to g(x)$ a.e. and if $sup_{n}|g_{n}(x)|=\overline{g}(x)$ is
integrable, then for a.e. $x$,

$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}g_{n-k}\circ
G^{k}(x)=g_{\infty}(x),$

where $g_{\infty}(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}g\circ
G^{k}(x).$

For any $\textbf{j}\in\Sigma$ and $n\in\mathbb{Z}_{+}$, let
$\textbf{j}_{n}=j_{1}j_{2}\ldots j_{n}\in\varLambda^{n}.$

######  Lemma 4.4.

For Lebesgue-$\text{a.e.}\,(x,\theta)\in[0,1)^{2}$ the following holds. For
$\pi_{\theta}m_{x}$-a.e. $z\in\mathbb{R}$, we have

$\lim_{r\to
0}\frac{\log\bigg{(}\pi_{\theta}m_{x}\big{(}\mathbf{B}(z,\,r)\big{)}\bigg{)}}{\log
r}=\frac{H_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\Phi}\big{)}-\log
b}{\log|\gamma|}.$

###### Proof.

For any $(\textbf{j},\,x,\,\theta)\in\Sigma\times[0,1)^{2}$ and
$n\in\mathbb{Z}_{+}$, we have

(4.7) $\displaystyle\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\big{)}$
$\displaystyle=\prod_{k=0}^{n-1}\frac{\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta-k\Delta}\circ
S_{\textbf{j}_{k}(x)}}(\tau^{k}(\textbf{j}),\,n-k)\big{)}}{\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta-(k+1)\Delta}\circ
S_{\textbf{j}_{k+1}(x)}}(\tau^{k+1}(\textbf{j}),\,n-k-1)\big{)}}$
$\displaystyle=\prod_{k=0}^{n-1}\frac{\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta-k\Delta}\circ
S_{\textbf{j}_{k}(x)}}(\tau^{k}(\textbf{j}),\,n-k)\big{)}}{b\nu^{\mathbb{Z}_{+}}\big{(}\,\mathbf{B}_{\pi_{\theta-k\Delta}\circ
S_{\textbf{j}_{k}(x)}}(\tau^{k}(\textbf{j}),\,n-k)\cap\mathcal{P}\big{(}\tau^{k}(\textbf{j})\big{)}\,\big{)}}$

by the definition of $R^{\phi}_{\gamma}$ and (4.5) where
$\textbf{j}_{0}(x)=x$. For any $n\in\mathbb{N}$, let function
$g_{n}:\Sigma\times[0,1)^{2}\to\mathbb{Z}_{+}$ be such that

$g_{n}(\textbf{j},x,\theta)=-\log\frac{\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\cap\mathcal{P}(\textbf{j})\big{)}}{\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\big{)}}$

Thus we have

$\log\bigg{(}\nu^{\mathbb{Z}_{+}}\big{(}\,\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\,\big{)}\bigg{)}=n\log
1/b+\sum_{k=0}^{n-1}g_{n-k}\circ\hat{T}^{k}(\textbf{j},x,\theta),$

which implies: for $\hat{\omega}$-a.e.
$(\textbf{j},\,x,\,\theta)\in\Sigma\times[0,1)^{2}$ we have

$\lim_{n\to\infty}\frac{\log\bigg{(}\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\big{)}\bigg{)}}{n\log|\gamma|}=\frac{H_{\hat{\omega}}\big{(}\hat{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\Phi}\big{)}-\log
b}{\log|\gamma|}$

by Lemma 4.3 and Theorem 4.2. Combining this with

$\nu^{\mathbb{Z}_{+}}\big{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\big{)}=\pi_{\theta}m_{x}\bigg{(}\,\mathbf{B}\big{(}\,\pi_{\theta}\circ
S_{x}(\textbf{j}),\,R^{\phi}_{\gamma}|\gamma|^{n}\,\big{)}\,\bigg{)},$

thus Lemma 4.4 holds. ∎

###  4.3. Proof of Theorem 4.1

This subsection is devoted to give a sketchy proof for the rest of Theorem
4.1, since the details are similar to Sect. 4.2 and [26, Sect. 3.1, Sect.
3.3]. Define function $\hat{S}:\Sigma\times[0,1)^{2}\to\mathbb{C}$ such that

$(\textbf{j},\,x,\,\theta)\mapsto S(x,\,\textbf{j})$

and
$\mathcal{B}_{\hat{S}}:=\hat{S}^{-1}\big{(}\mathcal{B}_{\mathbb{C}}\big{)}$.
For any $x,\theta\in[0,1)$, define the partition of $\Sigma$ be such that

$\tilde{\eta}_{x,\,\theta}:=\big{\\{}\,(\pi_{\theta}\circ
S_{x})^{-1}(\,\\{\,z\,\\}\,)\,:\,z\in\mathbb{R}\,\big{\\}}.$

By [25, Proposition 3.5] we have the following result.

######  Lemma 4.5.

For $\hat{\omega}$-a.e. $(\textbf{j},x,\theta)\in\Sigma\times[0,1)^{2}$ we
have

(4.8)
$\lim_{n\to\infty}\log\frac{\nu^{\mathbb{Z}_{+}}_{(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{x}}(\textbf{j},\,n)\cap\mathcal{P}(\textbf{j})\big{)}}{\nu^{\mathbb{Z}_{+}}_{(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{x}}(\textbf{j},\,n)\big{)}}=-\mathbf{I}_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\hat{S}}\big{)}(\textbf{j},x,\theta)$

where
$\nu^{\mathbb{Z}_{+}}_{(\textbf{j},\,x,\,\theta)}=(\nu^{\mathbb{Z}_{+}})^{\tilde{\eta}_{x,\,\theta}}_{\textbf{j}}$
and
$\big{\\{}\,(\nu^{\mathbb{Z}_{+}})^{\tilde{\eta}_{x,\,\theta}}_{\textbf{j}}:\,\textbf{j}\in\Sigma\,\big{\\}}$
is the canonical system of conditional measure with
$\tilde{\eta}_{x,\,\theta}$. Furthermore, set

$h(\textbf{j},x,\theta)=\sup_{n\in\mathbb{N}}-\log\frac{\nu^{\mathbb{Z}_{+}}_{(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{x}}(\textbf{j},\,n)\cap\mathcal{P}(\textbf{j})\big{)}}{\nu^{\mathbb{Z}_{+}}_{(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{x}}(\textbf{j},\,n)\big{)}}.$

Then $h\geq 0$ and $h\in
L^{1}(\Sigma\times[0,1)^{2},\,\mathcal{B},\,\hat{\omega}).$

######  Lemma 4.6.

For any $x\in[0,1)$ and $\textbf{j}\in\Sigma$, $n\in\mathbb{N}$, we have

(4.9)
$\mathbf{B}_{S_{x}}(\textbf{j},\,n+1)\cap\mathcal{P}(\textbf{j})=\tau^{-1}\bigg{(}\mathbf{B}_{S_{(x+j_{1})/b}}(\tau(\textbf{j}),\,n)\bigg{)}\cap\mathcal{P}(\textbf{j}).$

###### Proof.

The proof is similar to Lemma 4.2. ∎

######  Lemma 4.7.

For every $x,\theta\in[0,1)$ and $A\in\mathcal{B}_{\Sigma}$, for
$\nu^{\mathbb{Z}_{+}}$-a.e. $\textbf{j}\in\Sigma,$

$(\nu^{\mathbb{Z}_{+}})^{\tilde{\eta}_{x,\,\theta}}_{\textbf{j}}(A)=\lim_{n\to\infty}\frac{\nu^{\mathbb{Z}_{+}}\bigg{(}A\cap\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\bigg{)}}{\nu^{\mathbb{Z}_{+}}\bigg{(}\mathbf{B}_{\pi_{\theta}\circ
S_{x}}(\textbf{j},\,n)\bigg{)}}$

###### Proof.

The proof is similar to [26, Lemma 3.1]. ∎

######  Lemma 4.8.

For Lebesgue-$\text{a.e.}\,(x,\theta)\in[0,1)^{2}$, for $m_{x}$-a.e.
$z\in\mathbb{C}$,

$\lim_{r\to
0}\frac{\log\bigg{(}(m_{x})^{\eta_{\theta}}_{z}(\mathbf{B}(z,\,r))\bigg{)}}{\log
r}=\frac{H_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\hat{S}}\big{)}-H_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\Phi}\big{)}}{\log|\gamma|}.$

We only give a sketchy proof here, see [26, Section 3.3] for details if the
readers are unfamiliar with Ledrappier-Young theory.

###### Proof.

Combining Lemma 4.5, Lemma 4.3 with Lemma 4.6 and Lemma 4.7 the following
holds: for every $k\in\mathbb{Z}_{+}$, for $\hat{\omega}$-a.e.
$(\textbf{j},x,\theta)\in\Sigma\times[0,1)^{2},$ we have

$\nu^{\mathbb{Z}_{+}}_{(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{x}}(\textbf{j},\,k)\cap\mathcal{P}(\textbf{j})\big{)}=\nu^{\mathbb{Z}_{+}}_{\hat{T}(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{\frac{x+j_{1}}{b}}}(\tau(\textbf{j}),\,k-1)\big{)}\cdot\text{exp}\bigg{(}-\mathbf{I}_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\Phi}\big{)}(\textbf{j},x,\theta)\bigg{)}.$

Thus we have

$\displaystyle\nu^{\mathbb{Z}_{+}}_{(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{x}}(\textbf{j},\,k)\cap\mathcal{P}(\textbf{j})\big{)}$
$\displaystyle=\prod_{k=0}^{n-1}\bigg{(}\frac{\nu^{\mathbb{Z}_{+}}_{\hat{T}(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{\textbf{j}_{k}(x)}}(\tau^{k}(\textbf{j}),\,n-k)\big{)}}{\nu^{\mathbb{Z}_{+}}_{\hat{T}(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{\textbf{j}_{k}(x)}}(\tau^{k}(\textbf{j}),\,n-k)\cap\mathcal{P}(\tau^{k}(\textbf{j}))\big{)}}\bigg{)}\cdot\text{exp}\bigg{(}-\mathbf{I}_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\Phi}\big{)}\circ\hat{T}^{k}(\textbf{j},x,\theta)\bigg{)},$

which implies, for $\hat{\omega}$-a.e.
$(\textbf{j},\,x,\,\theta)\in\Sigma\times[0,1)^{2},$ we have

$\lim_{n\to\infty}\frac{\log\nu^{\mathbb{Z}_{+}}_{(\textbf{j},\,x,\,\theta)}\big{(}\mathbf{B}_{S_{x}}(\textbf{j},\,k)\cap\mathcal{P}(\textbf{j})\big{)}}{n\log|\gamma|}=\frac{H_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\hat{S}}\big{)}-H_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\Phi}\big{)}}{\log|\gamma|}$

by Theorem 4.2, Lemma 4.8 and Birkhoff Ergodic Theorem. ∎

######  Lemma 4.9.

For Lebesgue-$\text{a.e.}\,(x,\theta)\in[0,1)^{2}$, for $m_{x}$-a.e.
$z\in\mathbb{C}$,

$\lim_{r\to 0}\frac{\log\big{(}m_{x}(\mathbf{B}(z,\,r))\big{)}}{\log
r}=\frac{H_{\hat{\omega}}\big{(}\tilde{\mathcal{P}}|\hat{\eta}\vee\mathcal{B}_{\hat{S}}\big{)}-\log
b}{\log|\gamma|}.$

###### Proof.

The proof is similar to Lemma 4.4 by Lemma 4.6. ∎

###### The proof of Theorem 4.1.

Combining Lemma 4.9 and Lemma 4.4 with Lemma 4.8, we have Theorem 4.1 holds. ∎

##  5\. The inverse theorem for entropy

The main result of this section is Theorem 5.1, which is the corollary of [27,
Theorem 2.8] by Theorem 4.1. We will use Theorem 5.1 to study the convolution
of $m_{x}$ and any probability measures on $\mathbb{C}$ in Sect. 6.

######  Theorem 5.1.

Let integer $b\geq 2$ and $\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$. Let
$\phi:\mathbb{R}\to\mathbb{R}$ be a $\mathbb{Z}$-periodic Lipschitz function
such that condition (H) holds. If $\alpha<2$ and
$\Delta\in\mathbb{R}\setminus\mathbb{Q}$, then the following holds.

For any $\varepsilon>0$ and $R>0$, there are
$\delta_{6}=\delta_{6}(\varepsilon,\,R,\,\phi,\,\gamma),\,\delta_{7}=\delta_{7}(\varepsilon,\,R,\,\phi,\,\gamma)>0$
with the following properties: for every
$m>M(\varepsilon,\,R,\,\phi,\,\gamma)$,
$n>N(\varepsilon,\,R,\,\phi,\,\gamma,\,m)$ and $x\in[0,1)$, if
$\mu\in\mathscr{P}\big{(}[-R,R]^{2}\big{)}$ is a measure such that

$\frac{1}{n}H(\mu,\,\mathcal{L}_{n})>\varepsilon,$

and if

$\mathbb{P}^{\,m_{x}}_{0\leq
i<n}\left(\frac{1}{m}H((m_{x})^{x,\,i},\mathcal{L}^{\mathbb{C}}_{m})<\alpha+\delta_{6}\right)>1-\delta_{6},$

then we have

$\frac{1}{n}H(\mu\ast
m_{x},\,\mathcal{L}_{n})\geq\frac{1}{n}H(m_{x},\,\mathcal{L}_{n})+\delta_{7}.$

###  5.1. The projection theorem for entropy

This subsection is devoted to analyze the uniform low bound of
$\frac{1}{n}H(\pi_{\theta}m_{x},\,\mathcal{L}_{n})$ for all
$x,\,\theta\in[0,1]$ when $n$ is large enough. See Lemma 5.2 for details. We
shall first introduce the following Lemma 5.1, whose idea is inspired by [22,
Lemma 5.1].

######  Lemma 5.1.

If condition (H) holds and $\Delta\in\mathbb{R}\setminus\mathbb{Q}$, then the
measure $\pi_{\theta}m_{x}$ has no atom for each $x,\,\theta\in[0,1].$

###### Proof.

Assume the lemma fails. Since the support of $\pi_{\theta}m_{x}$ is contained
in the ball $\mathbf{B}(0,\frac{\parallel\phi\parallel_{\infty}}{1-|\gamma|})$
for all $x,\,\theta\in[0,1]$, these are $x_{1},\,\theta_{1}\in[0,1]$ and
$p\in\mathbb{R}$ such that

(5.1)
$\pi_{\theta_{1}}m_{x_{1}}(\\{p\\})=\max_{x,\,\theta\in[0,1],\,z\in\mathbb{R}}\pi_{\theta}m_{x}(\\{z\\})>0$

by the compactness of the probability measures $\pi_{\theta}m_{x}$ in the weak
star topology and the continuity of function $\pi_{\theta}\circ S_{x}$.
Therefore, for any $n\in\mathbb{N}$, (2.8) and (4.4) imply

(5.2)
$\pi_{\theta_{1}}m_{x_{1}}(\\{p\\})=\frac{1}{b^{n}}\sum_{\textbf{w}\in\varLambda^{n}}\pi_{\theta_{1}-n\Delta}m_{\textbf{w}(x_{1})}(\\{p_{\textbf{w}}\\}).$

where $p_{\textbf{w}}=\frac{p-\pi_{\theta_{1}}\circ
S_{x_{1}}(\textbf{w})}{|\gamma|^{n}}.$ Combining this with (5.1) and (5.2) we
have

$\pi_{\theta_{1}-n\Delta}m_{\textbf{w}(x_{1})}(\\{p_{\textbf{w}}\\})=\pi_{\theta_{1}}m_{x_{1}}(\\{p\\})>0\quad\quad\forall\,\textbf{w}\in\varLambda^{n}.$

This implies

(5.3) $M_{0}:=\sup_{\textbf{w}\in\varLambda^{\\#}}|P_{\textbf{w}}|<\infty,$

since the supports of the family of probability measures
$\\{\,\pi_{\theta}m_{x}\,\\}_{\theta,\,x\in[0,1]}$ have uniform bound in
$\mathbb{R}$.

Since $S(x,0_{\infty})-S(x,1_{\infty})\not\equiv 0$, there are
$x_{2},\,\theta_{2}\in[0,1)$ such that

(5.4) $\pi_{\theta_{2}}\circ S_{x_{2}}(0_{\infty})-\pi_{\theta_{2}}\circ
S_{x_{2}}(1_{\infty})\neq 0.$

Let $n_{t}\in\mathbb{N}$ and
$\textbf{w}^{n_{t}}\in\varLambda^{n_{t}},\,t=1,2,\ldots$ be such that
$\lim_{t\to\infty}(\,\theta_{1}-n_{t}\Delta\mod 1)=\theta_{2}$ and
$\lim_{t\to\infty}\textbf{w}^{n_{t}}(x_{1})=x_{2}$, since
$\Delta\notin\mathbb{Q}$ and $x_{2}$ is $1/b^{n_{t}}$-approximate by the set
$\\{\,\textbf{w}(x_{1}):\,\textbf{w}\in\varLambda^{n_{t}}\\}$. For any
$t,\,m\in\mathbb{Z}_{+}$, by the definition of $p_{\textbf{w}}$ we have

$\pi_{\theta_{1}}\circ
S_{x_{1}}(\textbf{w}^{n_{t}}0_{m})-\pi_{\theta_{1}}\circ
S_{x_{1}}(\textbf{w}^{n_{t}}1_{m})=|\gamma|^{n_{t}+m}(p_{\textbf{w}^{n_{t}}1_{m}}-p_{\textbf{w}^{n_{t}}0_{m}}),$

which implies that

$\big{|}\pi_{(\theta_{1}-n_{t}\Delta\mod 1)}\circ
S_{\textbf{w}^{n_{t}}(x_{1})}(0_{m})-\pi_{(\theta_{1}-n_{t}\Delta\mod 1)}\circ
S_{\textbf{w}^{n_{t}}(x_{1})}(1_{m})\big{|}\leq 2|\gamma|^{m}M_{0}$

where $1_{m}=11\ldots 1,0_{m}=00\ldots 0\in\varLambda^{m}.$ This contradicts
(5.4) when $m,\,t$ goes to infinity. ∎

For any $n\in\mathbb{N}$, let $\tilde{n}$ be the unique integer such that
$b^{-\tilde{n}}\leq|\gamma|^{n}<b^{-\tilde{n}+1}$.

######  Lemma 5.2.

If the condition (H) holds and $\Delta\notin\mathbb{Q}$, we have

$\lim_{n\to\infty}\inf_{x,\,\theta\in[0,1]}\frac{1}{n}H(\pi_{\theta}m_{x},\,\mathcal{L}_{n})=\beta.$

###### Proof.

We shall first prove the following claim.

######  Claim 1.

For any $x,\,\theta\in[0,1]$ and $\ell,\,n\in\mathbb{Z}_{+}$ such that
$\ell,n/\ell$ is large enough, we have

$\frac{1}{n}H(\pi_{\theta}m_{x},\,\mathcal{L}_{n})\geq\bigg{(}1+O\big{(}\frac{1}{\tilde{\ell}}+\frac{\tilde{\ell}}{n}\big{)}\bigg{)}\cdot\frac{1}{t_{n,\ell}}\sum_{k=0}^{t_{n,\ell}-1}\bigg{(}\frac{1}{b^{\tilde{\ell}}}\sum_{\textbf{w}\in\varLambda^{\tilde{\ell}}}\frac{1}{\tilde{\ell}}H(\pi_{\theta-k\ell\Delta}m_{\textbf{w}(0)},\,\mathcal{L}_{\tilde{\ell}})\bigg{)}+O(\frac{1}{\tilde{\ell}}+\frac{\tilde{\ell}}{n})$

where $t_{n,\ell}=\max\\{t\in\mathbb{N}:\,\widetilde{t\ell}\leq n\\}$.

This is enough to conclude the proof. Indeed, for any $\ell\in\mathbb{N}$ such
that $\hat{\ell}>0$, let $f_{\ell}:[0,1)\to\mathbb{R}$ be a function such that

$f_{\ell}(\theta):=\frac{1}{b^{\tilde{\ell}}}\sum_{\textbf{w}\in\varLambda^{\tilde{\ell}}}\frac{1}{\tilde{\ell}}H(\pi_{\theta}m_{\textbf{w}(0)},\,\mathcal{L}_{\tilde{\ell}}).$

Since $\Delta\in\mathbb{R}\setminus\mathbb{Q}$ implies
$\ell\Delta\notin\mathbb{Q}$, therefore we have

(5.5)
$\lim_{n\to\infty}\inf_{\theta\in[0,1)}\frac{1}{n}\sum_{k=0}^{n-1}f_{\ell}(\theta-k\ell\Delta)=\int_{[0,1)}f_{\ell}(s)ds$

by the Unique Ergodic Theorem [23, Theorem 6.19]. Also we have

$\int_{[0,1)}f_{\ell}(s)ds=\int_{[0,1)^{2}}\frac{1}{\tilde{\ell}}H(\pi_{\theta}m_{x},\,\mathcal{L}_{\tilde{\ell}})dxd\theta+O(\frac{1}{\tilde{\ell}}).$

Combining this with

$\lim_{\ell\to\infty}\int_{[0,1)^{2}}\frac{1}{\tilde{\ell}}H(\pi_{\theta}m_{x},\,\mathcal{L}_{\tilde{\ell}})dxd\theta=\beta$

by Proposition 3.1, Theorem 4.1 and Lebesgue Convergence Theorem, we have

$\lim_{\ell\to\infty}\int_{[0,1)}f_{\ell}(s)ds=\beta.$

Combining this with (5.5) and Claim 1, we have Lemma 5.2 holds.

Now let us prove Claim 1. Since
$n-\widetilde{t_{n,\ell}\ell}=O_{b,\,\gamma}(\tilde{\ell})$, we have

$\frac{1}{n}H(\pi_{\theta}m_{x},\,\mathcal{L}_{n})=\frac{1}{n}\sum_{k=0}^{t_{n,\ell}-1}H(\pi_{\theta}m_{x},\,\mathcal{L}_{\widetilde{(k\ell+\ell)}}|\mathcal{L}_{\widetilde{(k\ell)}})+O(\frac{\tilde{\ell}}{n}).$

By the concavity of conditional entropy, this implies

(5.6)
$\frac{1}{n}H(\pi_{\theta}m_{x},\,\mathcal{L}_{n})\geq\frac{1}{n}\sum_{k=0}^{t_{n,\ell}-1}\frac{1}{b^{k\ell}}\sum_{\textbf{w}\in\varLambda^{k\ell}}H(\pi_{\theta}(T^{k\ell}m_{\textbf{w}(x)}),\,\mathcal{L}_{\widetilde{(k\ell+\ell)}}|\mathcal{L}_{\widetilde{k\ell}})+O(\frac{\tilde{\ell}}{n}).$

Also (4.4) implies the measure $\pi_{\theta}(T^{k\ell}m_{\textbf{w}(x)})$ is
supported in an interval of length $O(b^{-\widetilde{k\ell}})$. Thus we have

$H(\pi_{\theta}(T^{k\ell}m_{\textbf{w}(x)}),\,\mathcal{L}_{\widetilde{(k\ell+\ell)}}|\mathcal{L}_{\widetilde{k\ell}})=H(\pi_{\theta}(T^{k\ell}m_{\textbf{w}(x)}),\,\mathcal{L}_{\widetilde{(k\ell+\ell)}})+O(1).$

Combining this with
$\widetilde{(k\ell+\ell)}=\widetilde{k\ell}+\tilde{\ell}+O_{b,\,\gamma}(1)$
and (4.4), we have

$H(\pi_{\theta}(T^{k\ell}m_{\textbf{w}(x)}),\,\mathcal{L}_{\widetilde{(k\ell+\ell)}}|\mathcal{L}_{\widetilde{k\ell}})=H(\pi_{\theta-k\ell\Delta}m_{\textbf{w}(x)},\,\mathcal{L}_{\tilde{\ell}})+O(1).$

Therefore the following holds:

$\frac{1}{n}H(\pi_{\theta}m_{x},\,\mathcal{L}_{n})\geq\frac{1}{n}\sum_{k=0}^{t_{n,\ell}-1}\frac{1}{b^{k\ell}}\sum_{\textbf{w}\in\varLambda^{k\ell}}H(\pi_{\theta-k\ell\Delta}m_{\textbf{w}(x)},\,\mathcal{L}_{\tilde{\ell}})+O(\frac{\tilde{\ell}}{n}+\frac{t_{n,\ell}}{n}).$

Since $\parallel\pi_{\theta-k\ell\Delta}\circ
S_{\textbf{w}(x)}-\pi_{\theta-k\ell\Delta}\circ
S_{(\tau^{k\ell-\tilde{\ell}}\textbf{w})(0)}\parallel_{\infty}=O(b^{-\tilde{\ell}})$
for any $k\geq L_{0}(b,\gamma),$ therefore we have

$\frac{1}{n}H(\pi_{\theta}m_{x},\,\mathcal{L}_{n})\geq\frac{1}{n}\sum_{k=0}^{t_{n,\ell}-1}\frac{1}{b^{\tilde{\ell}}}\sum_{\textbf{w}\in\varLambda^{\tilde{\ell}}}H(\pi_{\theta-k\ell\Delta}m_{\textbf{w}(0)},\,\mathcal{L}_{\tilde{\ell}})+O(\frac{\tilde{\ell}}{n}+\frac{t_{n,\ell}}{n}).$

Combining this with $O(\frac{t_{n,\ell}}{n})=O(\frac{1}{\tilde{\ell}})$ and

$\frac{1}{n}=\frac{t_{n,\ell}\tilde{\ell}}{n}\cdot\frac{1}{t_{n,\ell}}\cdot\frac{1}{\tilde{\ell}}=\bigg{(}1+0(\frac{1}{\tilde{\ell}}+\frac{\tilde{\ell}}{n})\bigg{)}\cdot\frac{1}{t_{n,\ell}}\frac{1}{\tilde{\ell}},$

we have this claim holds. ∎

###  5.2. An inverse theorem for entropy

In this section, we shall introduce the inverse theorem for convolutions on
$\mathbb{R}^{d}$ due to Hochman [27]. Before that it is necessary to recall
the following notations from [27].

For a linear subspace $V\leq\mathbb{C}$, let $W=V^{\perp}$ be the orthogonal
complement of $V$ and let $\pi_{V}$ be the orthogonal projection on $V$. Thus,
if $V=\\{\,te^{2\pi i\theta}\,\\}_{t\in\mathbb{R}}$ for some
$\theta\in\mathbb{R}$, then we have $\pi_{V}=\pi_{\theta}.$ For any set
$A\subset\mathbb{C}$ and $\varepsilon>0$ write $\varepsilon$-neighborhood of
$A$ by

$A^{(\varepsilon)}:=\\{z\in\mathbb{C}:\,d(z,A)<\varepsilon\\}.$

######  Definition 5.1.

Let $V\leq\mathbb{C}$ be a linear subspace and $\varepsilon>0$. A measure
$\mu\in\mathscr{P}(\mathbb{C})$ is $(V,\varepsilon)$-concentrated if there is
a translate $W$ such that $\mu(W^{(\varepsilon)})\geq 1-\varepsilon.$

We also need the following definition.

######  Definition 5.2.

Let $V\leq\mathbb{C}$ be a linear subspace, $W=V^{\perp}$ is the orthogonal
complement, and $\varepsilon>0$. A probability measure
$\mu\in\mathscr{P}(\mathbb{C})$ is $(V,\varepsilon)$-saturated at scale $m$,
or $(V,\varepsilon,m)$-saturated, if

$\frac{1}{m}H(\mu,\mathcal{L}_{m})\geq\frac{1}{m}H(\pi_{W}\mu,\mathcal{L}_{m})+\text{dim}(V)-\varepsilon.$

######  Remark 5.1.

It is worth noting that for the special case $V=\mathbb{C}$,
$(V,\varepsilon,m)$-saturated implies $\frac{1}{m}H(\mu,\mathcal{L}_{m})\geq
2-\varepsilon$. Also $(V,\varepsilon)$-saturated at scale $m$ is trivial if
$V=\\{0\\}$.

The following Theorem 5.2 is the inverse theorem of Hochman, which serves as a
basic tool to study the dimension of $m_{x}$ in our paper. See [27, Theorem
2.8].

######  Theorem 5.2 (Hochman).

For any $R,\varepsilon>0$ and $m\in\mathbb{N}$ there is a
$\delta_{0}=\delta_{0}(\varepsilon,\,R,\,m)>0$ such that for every
$n>N_{0}(\varepsilon,\,R,\,\delta_{0},\,m)$, the following holds: if
$\mu,\zeta\in\mathscr{P}([-R,R]^{2})$ and

$\frac{1}{n}H(\mu\ast\zeta,\mathcal{L}_{n})<\frac{1}{n}H(\mu,\mathcal{L}_{n})+\delta_{0},$

then there exists a sequence $V_{0},\ldots V_{n}\leq\mathbb{C}$ of subspace
such that

(5.7) $\mathbb{P}^{\,\mu\times\zeta}_{0\leq
i<n}\left(\,\begin{matrix}\mu^{x,\,i}\,\text{is}\,(V_{i},\,\varepsilon,\,m)-\text{saturated
and}\\\
\zeta^{y,\,i}\,\text{is}\,(V_{i},\,\varepsilon)-\text{concentrated}\end{matrix}\,\right)>1-\varepsilon.$

Note that the definitions of $\mu^{x,\,i},\,\zeta^{y,\,i}$ are from (3.2).

###  5.3. The proof of Theorem 5.1

We shall first introduce the following Lemma 5.3, which is an important
observation for proving Theorem 5.1.

######  Lemma 5.3.

Assume the condition (H) holds and $\Delta\in\mathbb{R}\setminus\mathbb{Q}.$
Then for any $\delta,\,\varepsilon>0$, $m\geq M_{2}(\varepsilon,\delta),$ and
$k\geq K_{1}(\varepsilon,\delta,m)$, the following holds:

(5.8)
$\inf_{x,\,\theta\in[0,1]}\mathbb{P}^{\,m_{x}}_{i=k}\bigg{(}\,\frac{1}{m}H\big{(}\pi_{\theta}(m_{x})^{z,\,i},\mathcal{L}_{m}\big{)}\geq\beta-\varepsilon\,\bigg{)}\geq
1-\delta.$

Before giving the proof, we shall introduce the following results [2, Lemma
3.3] and [27, Lemma 3.4]. Recall that the total variation distance between
$\mu,\eta\in\mathscr{P}(\mathbb{C})$ is

$\parallel\mu-\eta\parallel=\sup_{A\in\mathscr{B}(\mathbb{C})}|\,\mu(A)-\eta(A)|.$

######  Lemma 5.4.

For every $\varepsilon>0$ there exists a
$\delta_{1}=\delta_{1}(\varepsilon)>0$ with the following property. For any
$0<\delta\leq\delta_{1},$ suppose that a probability measure
$\eta\in\mathscr{P}(\mathbb{C})$ can be written as a convex combination
$\eta=(1-\delta)\eta^{\prime}+\delta\eta^{\prime\prime}$. Then for every $k$,
we have

$\mathbb{P}^{\,\eta}_{i=k}\big{(}z\,:\,\parallel\eta_{z,\,i}-\eta^{\prime}_{z,\,i}\parallel<\varepsilon\big{)}>1-\varepsilon.$

In fact [2, Lemma 3.3] only dealt with the case
$\eta\in\mathscr{P}(\mathbb{R})$, however the proof also works for all
$\eta\in\mathscr{P}(\mathbb{R}^{d})$.

######  Lemma 5.5.

If $\mathcal{A}$ is the partition of $\mathbb{R}^{d}$, and if
$\mu,\eta\in\mathscr{P}(\mathbb{R}^{d})$ are supported at most $k$ atoms of
partition $\mathcal{A}$ and $\parallel\mu-\eta\parallel<\varepsilon,$ then

$|H(\mu,\,\mathcal{A})-H(\eta,\,\mathcal{A})|<2\varepsilon\log_{b}k+2H(\frac{1}{2}\varepsilon)$

where $H(\frac{1}{2}\varepsilon)$ is defined by (3.3).

Notation. For any $n\in\mathbb{N}$, let $\hat{n}$ be the unique integer such
that

(5.9) $|\gamma|^{\hat{n}}\leq b^{-n}<|\gamma|^{\hat{n}-1}$

as [22] did. We also need the following corollary of Lemma 5.1.

######  Corollary 5.1.

If the condition (H) holds, then for any $\varepsilon>0$, there is a
$\delta_{2}=\delta_{2}(\varepsilon)>0$ such that the following holds. For any
$x\in[0,1]$ and $n\in\mathbb{N}$, we have

(5.10)
$m_{x}\big{(}\partial\mathcal{L}_{n}^{(\delta_{2}/b^{n})}\big{)}<\varepsilon$

where set
$\partial\mathcal{L}_{n}:=\bigcup_{k_{1},k_{2}\in\mathbb{Z}}\bigg{\\{}z\in\mathbb{C}:\,\text{Re}(z)=\frac{k_{1}}{b^{n}}\,\text{or}\,\text{Im}(z)=\frac{k_{2}}{b^{n}}\bigg{\\}}.$

###### Proof.

By Lemma 5.1 there is $\delta^{\prime}>0$ such that

(5.11)
$\sup_{x,\,\theta\in[0,1],\,z\in\mathbb{R}}\pi_{\theta}m_{x}\big{(}\mathbf{B}(z,\delta^{\prime})\big{)}\leq\frac{\varepsilon}{2},$

since the probability measures $\pi_{\theta}m_{x}$ are compact in the weak
star topology and the functions $\pi_{\theta}\circ S_{x}(\textbf{j})$ are
continuous. Recall
$R^{\phi}_{\gamma}=\frac{2\parallel\phi\parallel_{\infty}}{1-|\gamma|}.$ There
is constant $\ell\in\mathbb{N}$ and $c_{\ell}>0$ such that

(5.12)
$c_{\ell}>\frac{1}{b^{n}|\gamma|^{\hat{n}+\ell}}>R^{\phi}_{\gamma}\quad\quad\forall
n\in\mathbb{N}.$

For any set $A\subset\mathbb{C}$ and $z_{1},\,z_{2}\in\mathbb{C}$, denote the
set

$z_{1}A+z_{2}=\big{\\{}\,z_{1}\cdot z+z_{2}\,:\,z\in A\,\\}.$

Let $\delta_{2}=\delta^{\prime}/c_{\ell}$. For any $n\in\mathbb{N}$, combining
(2.8) with (2.9) we have

(5.13)
$m_{x}\big{(}\partial\mathcal{L}_{n}^{(\delta_{2}/b^{n})}\big{)}=\frac{1}{b^{\hat{n}+\ell}}\sum_{\textbf{w}\in\varLambda^{\hat{n}+\ell}}m_{\textbf{w}(x)}\bigg{(}\frac{\partial\mathcal{L}_{n}^{(\delta_{2}/b^{n})}-S(x,\textbf{w})}{\gamma^{\hat{n}+\ell}}\bigg{)}.$

Also the definition of $\partial\mathcal{L}_{n}$ and (5.12) imply that: for
each $\textbf{w}\in\varLambda^{\hat{n}+\ell}$ such that

$m_{\textbf{w}(x)}\bigg{(}\frac{\partial\mathcal{L}_{n}^{(\delta_{2}/b^{n})}-S(x,\textbf{w})}{\gamma^{\hat{n}+\ell}}\bigg{)}>0,$

there are $k_{1}^{\textbf{w}},\,k_{2}^{\textbf{w}}\in\mathbb{Z}$ such that

$\bigg{(}\frac{\partial\mathcal{L}_{n}-S(x,\textbf{w})}{\gamma^{\hat{n}+\ell}}\bigg{)}\bigcap\text{supp}\big{(}m_{\textbf{w}(x)}\big{)}\subset\frac{\big{\\{}z\in\mathbb{C}:\,\text{Re}(z)=\frac{k_{1}^{\textbf{w}}}{b^{n}}\,\text{or}\,\text{Im}(z)=\frac{k_{2}^{\textbf{w}}}{b^{n}}\big{\\}}-S(x,\textbf{w})}{\gamma^{\hat{n}+\ell}},$

which implies

$\bigg{(}\frac{\partial\mathcal{L}_{n}^{(\delta_{2}/b^{n})}-S(x,\textbf{w})}{\gamma^{\hat{n}+\ell}}\bigg{)}\bigcap\text{supp}\big{(}m_{\textbf{w}(x)}\big{)}\subset\pi_{\theta_{\textbf{w}}}^{-1}\bigg{(}\mathbf{B}(z_{1}^{\textbf{w}},\frac{\delta_{2}}{b^{n}|\gamma|^{\hat{n}+\ell}})\bigg{)}\bigcup\pi_{\theta^{\prime}_{\textbf{w}}}^{-1}\bigg{(}\mathbf{B}(z_{2}^{\textbf{w}},\frac{\delta_{2}}{b^{n}|\gamma|^{\hat{n}+\ell}})\bigg{)}$

for some $\theta_{\textbf{w}},\,\theta^{\prime}_{\textbf{w}}\in[0,1]$ and
$z_{1}^{\textbf{w}},\,z_{2}^{\textbf{w}}\in\mathbb{R}.$ Therefore we have

$m_{x}\big{(}\partial\mathcal{L}_{n}^{(\delta_{2}/b^{n})}\big{)}<\varepsilon$

by (5.11), (5.12) and (5.13), $\delta_{2}=\delta^{\prime}/c_{\ell}$. ∎

###### The proof of Lemma 5.3.

For any $\delta,\,\varepsilon>0$, choose
$\delta_{1}=\delta_{1}\big{(}\min\\{\delta,\,\varepsilon/8\\}\big{)}\in(0,1)$
by Lemma 5.4. Thus there is a $\delta_{2}=\delta_{2}(\delta_{1})>0$ such that
(5.10) holds by Corollary 5.1. Let $\ell\in\mathbb{N}$ be a constant such that

(5.14)
$2R^{\phi}_{\gamma}|\gamma|^{\hat{n}+\ell}\leq\frac{\delta_{2}}{b^{n}}\quad\quad\forall
n\in\mathbb{N}.$

By Lemma 5.2 there is a number $M_{1}=M_{1}(\varepsilon)\in\mathbb{N}$ such
that, for all $m\geq M_{1}$ we have

(5.15)
$\inf_{x,\,\theta\in[0,1]}\frac{1}{m}H(\pi_{\theta}m_{x},\,\mathcal{L}_{m})\geq\beta-\varepsilon/3.$

For any $x,\,\theta\in[0,1]$ and $k\geq M_{1}$, let
$A_{k,\,\ell}\subset\varLambda^{\hat{k}+\ell}$ be the union of all elements
$\textbf{w}\in\varLambda^{\hat{k}+\ell}$ such that
$\text{supp}(T^{\hat{k}+\ell}m_{\textbf{w}(x)})\subset I_{\textbf{w}}$ for
some $I_{\textbf{w}}\in\mathcal{L}_{k}^{\mathbb{C}}$. Let
$A^{c}_{k,\,\ell}:=\varLambda^{\hat{k}+\ell}\setminus A_{k,\,\ell}.$ Thus for
each $\textbf{w}\in A^{c}_{k,\,\ell}$ we have
$\text{supp}(T^{\hat{k}+\ell}m_{\textbf{w}(x)})\subset\partial\mathcal{L}_{k}^{(\delta_{2}/b^{k})}$
by (5.14), therefore we have

(5.16) $\frac{\\#A^{c}_{k,\,\ell}}{b^{\hat{k}+\ell}}\leq\delta_{1}$

by Corollary 5.1. Note that $\delta_{1}<1$ implies $A_{k}\neq\emptyset$.
Define

$\eta^{\prime}:=\frac{1}{\\#A_{k,\,\ell}}\sum_{\textbf{w}\in
A_{k,\,\ell}}T^{\hat{k}+\ell}m_{\textbf{w}(x)}$

and $\eta^{\prime\prime}\in\mathscr{P}(\mathbb{C})$ such that

$m_{x}=\bigg{(}1-\frac{\\#A^{c}_{k,\,\ell}}{b^{\hat{k}+\ell}}\bigg{)}\,\eta^{\prime}+\bigg{(}\frac{\\#A^{c}_{k,\,\ell}}{b^{\hat{k}+\ell}}\bigg{)}\,\eta^{\prime\prime}.$

Combining this with Lemma 5.4, we have

$\mathbb{P}_{i=k}^{m_{x}}\bigg{(}z\,:\,\parallel(m_{x})_{z,\,i}-\eta^{\prime}_{z,\,i}\parallel<\varepsilon/8\bigg{)}>1-\delta.$

Thus Lemma 5.5 implies that the following holds

(5.17)
$\mathbb{P}_{i=k}^{m_{x}}\bigg{(}z\,:\,\big{|}\frac{1}{m}H\big{(}(m_{x})_{z,\,i},\pi_{\theta}^{-1}(\mathcal{L}_{i+m})\big{)}-\frac{1}{m}H\big{(}\eta^{\prime}_{z,\,i},\pi_{\theta}^{-1}(\mathcal{L}_{i+m})\big{)}\big{|}\leq\frac{1}{4}\varepsilon+\frac{2H(\varepsilon/8)}{m}\,\bigg{)}>1-\delta.$

For any $I\in\mathcal{L}_{k}$, define the set
$B_{I,\,k}:=\big{\\{}\,\textbf{w}\in
A_{k,\,\ell}:\,\text{supp}(T^{\hat{k}+\ell}m_{\textbf{w}(x)})\subset
I\,\big{\\}}$. For the case $B_{I,k}\neq\emptyset$, we have

(5.18)
$\displaystyle\frac{1}{m}H\big{(}\eta^{\prime}_{I},\pi_{\theta}^{-1}(\mathcal{L}_{k+m})\big{)}$
$\displaystyle\geq\frac{1}{\\#B_{I,\,k}}\sum_{\textbf{w}\in
B_{I,\,k}}\frac{1}{m}H\big{(}T^{\hat{k}+\ell}m_{\textbf{w}(x)},\pi_{\theta}^{-1}(\mathcal{L}_{k+m})\big{)}$
$\displaystyle=\frac{1}{\\#B_{I,\,k}}\sum_{\textbf{w}\in
B_{I,\,k}}\frac{1}{m}H\big{(}\pi_{\theta-(\hat{k}+\ell)\Delta}m_{\textbf{w}(x)},\mathcal{L}_{m}\big{)}+O(\frac{\ell}{m})$
$\displaystyle\geq\beta-\varepsilon/3+O(\frac{\ell}{m})$

by the concavity of conditional entropy and (2.9), (5.14). Also we have

(5.19)
$\frac{1}{m}H\big{(}(m_{x})_{z,\,k},\pi_{\theta}^{-1}(\mathcal{L}_{k+m})\big{)}=\frac{1}{m}H\big{(}\pi_{\theta}(m_{x})^{z,\,k},\mathcal{L}_{m}\big{)}+O(\frac{1}{m}).$

Let $M_{3}\in\mathbb{N}$ be large enough such that
$O(\frac{1}{m})+O(\frac{\ell}{m})+\frac{2H(\varepsilon/8)}{m}\leq\varepsilon/6$
for all $m\geq M_{3}$. Thus for $m\geq M_{2}:=\max\\{M_{1},M_{3}\\}$, we have

$\mathbb{P}_{i=k}^{m_{x}}\bigg{(}z\,:\,\frac{1}{m}H\big{(}\pi_{\theta}(m_{x})^{z,\,k},\mathcal{L}_{m}\big{)}\geq\beta-\varepsilon\bigg{)}>1-\delta$

by (5.17), (5.18) and (5.19). ∎

We still need the following observation from [3].

######  Lemma 5.6.

For any $\varepsilon>0$ and $R>0$, there is
$\delta_{3}=\delta_{3}(\varepsilon,R)>0$ such that the following holds. For
all $n\geq N_{1}(\varepsilon,\,\delta_{3},R)$ and
$\zeta\in\mathscr{P}([-R,R]^{2})$, if

$\mathbb{P}^{\,\zeta}_{0\leq
i<n}\left(\,\begin{matrix}\zeta^{x,\,i}\,\text{is}\,(0,\,\delta_{3})-\text{concentrated}\end{matrix}\,\right)>1-2\delta_{3},$

then

$\frac{1}{n}H(\zeta,\mathcal{L_{n}})<\varepsilon.$

###### Proof.

Without loss generality we may assume $\delta_{3}=1/b^{m}$ for some
$m\in\mathbb{Z}_{+}$. Since by Lemma 3.4 we have

$\frac{1}{n}H(\zeta,\mathcal{L}_{n})=\mathbb{E}^{\,\zeta}_{0\leq
i<n}\bigg{(}\frac{1}{m}H(\zeta^{x,i},\mathcal{L}_{m})\bigg{)}+O\big{(}\frac{m}{n}+\frac{\log
R}{n}\big{)},$

thus it suffice to prove the following claim.

######  Claim 2.

For any $\varepsilon>0$, $m\geq M(\varepsilon)$ and measure
$\mu\in\mathscr{P}([0,1]^{2})$, the following holds. If
$\mu\,\text{is}\,(0,\,1/b^{m})-\text{concentrated}$, then

$\frac{1}{m}H(\mu,\mathcal{L}_{m})\leq\varepsilon/2.$

Indeed we only need to choose $n$ large enough. In the rest we shall prove the
claim. By the definition of $(0,\,1/b^{m})$-concentrated, there are
$\tau^{\prime},\tau^{\prime\prime}\in\mathscr{P}([0,1]^{2})$ such that
$\text{supp}(\tau^{\prime})\subset\boldsymbol{B}(p,1/b^{m})$ for some
$p\in\mathbb{C}$ and

$\mu=(1-t)\tau^{\prime}+t\tau^{\prime\prime}$

where $t\leq 1/b^{m}.$ By the convexity of entropy we have

$\frac{1}{m}H(\mu,\mathcal{L}_{m})\leq(1-t)\,\frac{1}{m}H(\tau^{\prime},\mathcal{L}_{m})+t\,\frac{1}{m}H(\tau^{\prime\prime},\mathcal{L}_{m})+H(t)=O(\frac{1}{m})$

since $\tau^{\prime}$ intersects at most nine elements of $\mathcal{L}_{m}$
and $H(t)=O(\sqrt{t})$. ∎

###### The proof of Theorem 5.1.

Choose $\delta_{3}=\delta_{3}(\varepsilon,R)$ and
$N_{1}(\varepsilon,\delta_{3},R)$ by Lemma 5.6. Also $\alpha<2$ implies that
$\alpha<\beta+1$ by Corollary 4.1. let

$\delta_{6}:=(\frac{1}{9}\min\\{\delta_{3},\beta+1-\alpha,\,2-\alpha,\,1\\})^{2}.$

Also by Lemma 5.3, for any $m\geq M_{2}(\delta_{6},\delta_{6})>0$ and $k\geq
K_{1}(\delta_{6},\delta_{6},m)$ such that

(5.20)
$\inf_{x,\,\theta\in[0,1]}\mathbb{P}^{\,m_{x}}_{i=k}\bigg{(}\,\frac{1}{m}H\big{(}\pi_{\theta}(m_{x})^{z,\,i},\mathcal{L}_{m}\big{)}\geq\beta-\delta_{6}\,\bigg{)}\geq
1-\delta_{6}.$

By Theorem 5.2 choose $\delta_{7}=\delta_{0}(\delta_{6},R,m)$ and
$N_{0}(\delta_{6},R,\delta_{7},m)$. Let $M=M_{2}(\delta_{6},\delta_{6})$ and
$N=N(\varepsilon,\phi,\gamma)=\max\\{N_{0},N_{1},K_{1}/\delta_{3}\\}$.

If Theorem 5.1 fails for some $m\geq M$ and $n\geq N$, then there exists a
sequence $V_{0},\ldots V_{n-1}\leq\mathbb{C}$ of subspace such that (5.7)
holds for measures $m_{x}$ and $\mu$. Thus, by Markov equality there are
subset $I\subset\\{0,1,\ldots,n-1\\}$ such that $\frac{\\#I}{b}\geq
1-\sqrt{\delta_{6}}$ and

(5.21)
$\mathbb{P}^{\,m_{x}\times\mu}_{i=k}\left(\,\begin{matrix}(m_{x})^{z,\,i}\,\text{is}\,(V_{i},\,\delta_{6},\,m)-\text{saturated
and}\\\
\mu^{y,\,i}\,\text{is}\,(V_{i},\,\delta_{6})-\text{concentrated}\end{matrix}\,\right)\geq
1-\sqrt{\delta_{6}}$

for each $k\in I.$ Let $I_{j}$ be the union of all the elements $t\in I$ such
that $\text{dim}(V_{t})=j$ for $j=0,1,2$. Also our condition implies that
there are subset $J\subset\\{0,1,\ldots,n-1\\}$ such that $\frac{\\#J}{b}\geq
1-\sqrt{\delta_{6}}$ and

(5.22)
$\mathbb{P}^{\,m_{x}}_{i=k}\left(\frac{1}{m}H((m_{x})^{x,\,i},\mathcal{L}^{\mathbb{C}}_{i+m})<\alpha+\delta_{6}\right)\geq
1-\sqrt{\delta_{6}}$

for each $k\in J.$ Since $\alpha+\delta_{6}<2-\delta_{6}$ and
$1-\sqrt{\delta_{6}}>1/2$, we have $I_{2}\cap J=\emptyset$ by Remark 5.1,
(5.21) and (5.22). Therefore we have

(5.23) $\frac{\\#I_{2}}{n}\leq\sqrt{\delta_{6}}.$

Also we have $\alpha+\delta_{6}<\beta+1-\delta_{6}$. Thus (5.20), (5.21),
(5.22) and $\frac{K_{1}}{n}\leq\delta_{6}$ imply that

$\frac{\\#I_{1}}{n}\leq 2\sqrt{\delta_{6}}.$

Combining this with (5.23) we have

$\frac{\\#I_{0}}{n}\geq 1-4\sqrt{\delta_{6}}.$

Thus we have

$\mathbb{P}^{\,\mu}_{0\leq
i<n}\left(\,\begin{matrix}\mu^{y,\,i}\,\text{is}\,(0,\,\delta_{6})-\text{concentrated}\end{matrix}\,\right)\geq(1-4\sqrt{\delta_{6}})(1-\sqrt{\delta_{6}})\geq
1-\delta_{3}$

by (5.21). This implies $\frac{1}{n}H(\mu,\mathcal{L}_{n})<\varepsilon$ by
Lemma 5.6, which contradicts our condition. ∎

##  6\. The proof of Theorem A

In this section, we will use Theorem 5.1 to finish the proof of Theorem A.
Since the proof is similar to [22], some details will be omitted. In the rest
of paper, we suppose that $\phi(x)$ is a real analytic $\mathbb{Z}$-periodic
function such that the condition (H) holds for fixed integer $b\geq 2$ and
$\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$.

###  6.1. Entropy Porosity

In this subsection, we first recall the definition of entropy porosity from
[2], then we will show the entropy porosity properties of $m_{x}$ for all
$x\in[0,1]$.

######  Definition 6.1 (Entropy porous).

A measure $\mu\in\mathscr{P}(\mathbb{C})$ is $(h,\delta,m)$- entropy porous
from scale $n_{1}$ to $n_{2}$ if

$\mathbb{P}^{\,\mu}_{n_{1}\leq
i<n_{2}}\left(\frac{1}{m}H(\mu^{x,\,i},\mathcal{L}^{\mathbb{C}}_{m})<h+\delta\right)>1-\delta.$

The main result of this subsection is the following Theorem 6.1. The idea of
the proof is inspired by [2, Proposition 3.2] and [22, Theorem 5.1].

######  Theorem 6.1.

Assume the condition (H) holds and $\Delta\in\mathbb{R}\setminus\mathbb{Q}$.
Then for any $\varepsilon>0$, $m\geq M_{4}(\varepsilon),$ $k\geq
K_{4}(\varepsilon,m)$ and $n\geq N_{4}(\varepsilon,m,k)$, the following holds:

$\nu^{n}\left(\left\\{\textbf{i}\in\varLambda^{n}:m_{\textbf{i}(0)}\text{ is
}(\alpha,\varepsilon,m)-\text{entropy porous from scale }1\text{ to
}k\right\\}\right)>1-\varepsilon.$

Before giving the proof of Theorem 6.1, we shall introduce the following Lemma
6.1. The proof is the same as [22, Lemma 5.2] by Lemma 5.1, thus we omit the
details.

######  Lemma 6.1.

For any $\varepsilon>0,m\geq M_{5}(\varepsilon),n\geq N_{5}(\varepsilon,m)$,

$\inf\limits_{x\in[0,1]}\mathbb{\nu}^{n}\left(\,\left\\{\textbf{i}\in\varLambda^{n}:\alpha-\varepsilon<\frac{1}{m}H(m_{\textbf{i}(x)},\mathcal{L}_{m})<\alpha+\varepsilon\right\\}\,\right)>1-\varepsilon.$

The following is a corollary of Lemma 6.1.

######  Corollary 6.1.

Assume the condition (H) holds and $\Delta\in\mathbb{R}\setminus\mathbb{Q}.$
Then for any $\varepsilon>0$, $m\geq M_{6}(\varepsilon)$ and $k\geq
K_{6}(\varepsilon,m)$, the following holds:

(6.1)
$\inf_{x\in[0,1]}\mathbb{P}^{\,m_{x}}_{i=k}\bigg{(}\,\frac{1}{m}H\big{(}(m_{x})^{z,\,i},\mathcal{L}_{m}\big{)}\geq\alpha-\varepsilon\,\bigg{)}\geq
1-2\varepsilon.$

###### Proof.

For any $k,\,\ell\in\mathbb{Z}_{+}$ large enough, define the set

$B_{k,\,\ell}=B_{k,\,\ell}(\varepsilon,m):=\left\\{\textbf{i}\in\varLambda^{\hat{k}+\ell}:\frac{1}{m}H(m_{\textbf{i}(x)},\mathcal{L}_{m})>\alpha-\varepsilon/8\right\\}$

and $\tilde{A}_{k,\,\ell}:=A_{k,\,\ell}\cap B_{k,\,\ell}$ where $A_{k,\,\ell}$
is from Lemma 5.3. let
$\tilde{A}^{c}_{k,\,\ell}:=\varLambda^{\hat{k}+\ell}\setminus\tilde{A}_{k,\,\ell}$,
thus Corollary 6.1 can be proved by the similar method of Lemma 5.3 with Lemma
6.1. ∎

######  Lemma 6.2.

Assume the condition (H) holds and $\Delta\in\mathbb{R}\setminus\mathbb{Q}.$
For any $\varepsilon>0$, there exists $\delta>0$ such that if $m\geq
M_{7}(\varepsilon)$ and $k\geq K_{7}(\varepsilon,m)$ and if
$\left|\frac{1}{k}H(m_{x},\mathcal{L}_{k})-\alpha\right|<\frac{\delta}{2}$,
then $m_{x}$ is $(\alpha,\varepsilon,m)$-entropy porous from scale $1$ to $k$.

###### Proof.

The method of proof is similar to [2, Lemma 3.7] by Lemma 3.4 and Corollary
6.1, thus we omit the details. ∎

###### The proof of Theorem 6.1.

The proof is same as [22, Theorem 5.1]. ∎

###  6.2. Exponential separation

In this subsection, we deduce from the condition (H) the exponential
separation properties as [22] did.

######  Definition 6.2.

Let $E_{1},E_{2},\ldots$ be subsets of $\mathbb{C}$. For any $\varepsilon>0$
and $Q\subset\mathbb{Z}_{+}$, we say that the sequence
$(E_{n})_{n\in\mathbb{Z}_{+}}$ is $(\varepsilon,Q)$ -exponential separation if

$|p-q|>\sqrt{2}\,\varepsilon^{\hat{n}}\quad\quad\forall p\neq q\in
E_{\hat{n}}$

for each $n\in Q.$

The following is a corollary of [3, Lemma 5.8].

######  Corollary 6.2.

For any $k\in\mathbb{N}$ and compact interval $J\subset\mathbb{R}$, let
$F:J\to\mathbb{C}$ be a $k+1$-times continuously differentiable function. Let
$M=\parallel F\parallel_{J,\,k+1}$, and let $0<d<1$ be such that for every
$x\in J$ there is a $p\in\\{0,1,\ldots,k\\}$ with $|F^{(p)}(x)|>d.$ Then for
every $0<\rho<(d/2)^{2^{k}}$, the set
$F^{-1}\big{(}\mathbf{B}(0,\rho)\big{)}\subset J$ can be cover by
$O_{k,M,|J|}(1/{d^{k+1}})$ intervals of length $\leq 2(\rho/d)^{1/{2^{k}}}$
each.

###### Proof.

Let $J_{t}=[a_{t},b_{t}),$ $t=1,2,\ldots,L$ be the disjoint intervals such
that $J=\bigcup_{t=1}^{L}J_{t}$ and $|J_{t}|<\frac{d}{4M}$ where
$L=\lfloor\frac{4M|J|}{d}\rfloor+1$. For each $t\in\\{1,2,\ldots,L\\}$ there
is a $p_{t}\in\\{0,1,\ldots,k\\}$ with $|F^{(p_{t})}(a_{t})|>d$, thus either
$\big{|}\,\text{Re}(\,F^{(p_{t})}(a_{t})\,)\big{|}>d/2$ or
$\big{|}\,\text{Im}(\,F^{(p_{t})}(a_{t})\,)\big{|}>d/2$. For the former case
let $F_{t}:J\to\mathbb{R}$ be a $p_{t}$-times continuously differentiable
function such that $\text{Re}(F)|_{J_{t}}\equiv F_{t}|_{J_{t}}$, $\parallel
F_{t}\parallel_{J,\,p_{t}}=O_{M,\,|J|,\,p_{t}}(1)$ and
$|F_{t}^{(p_{t})}(x)|\geq d/4$ for each $x\in J$. Do the same thing for the
other case. Thus we have the set $F_{t}^{-1}(-\rho,\rho)\subset J$ can be
cover by $O_{k,M,|J|}(1/{d^{p_{t}}})$ intervals of length $\leq
2(\rho/d)^{1/{2^{p_{t}}}}$ each by [3, Lemma 5.8], which implies the corollary
holds. ∎

The main result of this subsection is the following Theorem 6.2.

######  Theorem 6.2.

There exists $\ell_{0}\in\mathbb{N}$ and $\varepsilon_{0}>0$ such that the
following holds for Lebesgue-a.e.$\,x\in[0,1]$.

For any integer $\ell\geq\ell_{0}$, there exists a set
$Q_{x,\,\ell}\subset\mathbb{Z}_{+}$ such that

  1. (i)

$\\#Q_{x,\,\ell}=\infty$;

  2. (ii)

for any $\textbf{w}\in\varLambda^{\ell}$, the sequence
$(X_{n}^{\textbf{w},\,x})_{n\in\mathbb{Z}_{+}}$ is
$(\varepsilon_{0},Q_{x,\,\ell})$-exponential separation

where
$X_{n}^{\textbf{w},\,x}=\big{\\{}S(x,\,\textbf{j}\textbf{w})\,:\,\textbf{j}\in\varLambda^{n-\ell}\big{\\}}$
for $n>\ell$ and $X_{n}^{\textbf{w},\,x}=\\{0\\}$ for $n\leq\ell$.

###### Proof.

The proof of [11, Lemma 5.2] works for all $\gamma\in\mathbb{C}$ such that
$0<|\gamma|<1$. Combining this with Corollary 6.2 we can prove Theorem 6.2 by
the same method of [22, Theorem 4.1], thus we omit the proof. ∎

###  6.3. Transversality

In this subsection we give some quantified estimates for transversality. For
any $\textbf{i},\,\textbf{j}\in\varLambda^{\\#}\cup\Sigma$ and integer $1\leq
k\leq|\textbf{j}|$, recall $\textbf{j}_{k}=j_{1}j_{2}\ldots j_{k}$. Write
$\textbf{i}<\textbf{j}$ if $\textbf{i}=\textbf{j}_{|\textbf{i}|}$ holds. When
$|\textbf{j}|<\infty$, let $I_{\textbf{j}}$ be an interval in $[0,1]$ such
that

(6.2)
$I_{\textbf{j}}=\bigg{[}\frac{i_{1}+i_{2}b+\dots+i_{n}b^{n-1}}{b^{n}},\frac{1+i_{1}+i_{2}b+\dots+i_{n}b^{n-1}}{b^{n}}\bigg{)}.$

The main result of this subsection is the following Theorem 6.3.

######  Theorem 6.3.

For any $t_{0}>0$, there exists an integer $t>t_{0}$, real number $\Xi_{1}>0$
and $\textbf{h},\,\textbf{h}^{\prime},\,\textbf{a}\in\varLambda^{t}$ with the
following property. For every $z\in I_{\textbf{a}}$ and
$\textbf{i},\,\textbf{j}\in\varLambda^{\\#}$, if $\textbf{h}<\textbf{i}$,
$\textbf{h}^{\prime}<\textbf{j}$, then

  1. (A.1)

$|S^{\prime}(z,\,\textbf{i})|\,,\,|S^{\prime}(z,\,\textbf{j})|>\Xi_{1};$

  2. (A.2)

$|S^{\prime}(z,\,\textbf{i})-S^{\prime}(z,\,\textbf{j})|>\Xi_{1};$

  3. (A.3)

$\frac{\parallel\phi^{\prime}\parallel_{\infty}}{(1-\gamma)b^{t}}<\Xi_{1}/4.$

###### Proof.

Since [22, Lemma 6.1, Lemma 6.2] can be extended the case
$\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$, thus we could prove that (A.1)
and (A.2) hold for some $t^{\prime}>t_{0}$, $\Xi_{1}>0$ and
$\textbf{h},\,\textbf{h}^{\prime},\,\textbf{a}\in\varLambda^{t^{\prime}}$ by
the same method of [22, Theorem 6.1]. Therefore it suffice to choose
$t>t^{\prime}$ large enough to make sure (A.3) also holds and replace
$\textbf{h},\,\textbf{h}^{\prime},\,\textbf{a}\in\varLambda^{t^{\prime}}$ by
$\textbf{h}\textbf{0}_{t-t^{\prime}},\,\textbf{h}^{\prime}\textbf{0}_{t-t^{\prime}},\,\textbf{0}_{t-t^{\prime}}\textbf{a}\in\varLambda^{t}$.
∎

###  6.4. The partitions of the space $\varLambda^{\\#}$

This subsection is devoted to construct a sequence of partitions
$\mathcal{L}_{n}^{\varLambda^{\\#}}$ of the space $\varLambda^{\\#}$ by the
same method of [22, Section.7]. Combining Theorem 3.1 with Theorem 6.2 and
Theorem 6.3, there exists an integer $t>0$ such that (A.3) holds, some
constants $\Xi_{1},\,C>0$, a point $x_{0}\in[0,1)$, set
$M\subset\mathbb{Z}_{+}$ and
$\textbf{a},\,\textbf{h},\,\textbf{h}^{\prime}\in\varLambda^{t}$ with the
following properties.

  1. (B.1)

For each $\textbf{w}\in\varLambda^{t}$, the sequence
$(X_{n}^{\textbf{w},\,x_{0}}\,)_{n\in\mathbb{Z}_{+}}$ is
$(|\gamma|^{C/2},M)$-exponential separation where $X_{n}^{\textbf{w},\,x_{0}}$
is from Theorem 6.2;

  2. (B.2)

$dim(m_{x_{0}})=\alpha$;

  3. (B.3)

For every $z\in I_{\textbf{a}}$ and
$\textbf{i},\,\textbf{j}\in\varLambda^{\\#}$, if $\textbf{h}<\textbf{i}$,
$\textbf{h}^{\prime}<\textbf{j}$ then (A.1), (A.2) hold;

  4. (B.4)

$\\#M=\infty.$

We also fix such elements
$\\{t,\,x_{0},\,C,\,\Xi_{1},\,M,\,\textbf{a},\,\textbf{h},\textbf{h}^{\prime}\\}$.
Let $\overline{\pi}:\varLambda^{\\#}\to\mathbb{N}\times\mathbb{C}^{3}$ be the
map such that

$\textbf{w}\mapsto\bigg{(}|\textbf{w}|,\,S(\textbf{w}(x_{0}),\textbf{h}),\,S(\textbf{w}(x_{0}),\textbf{h}^{\prime}),\,S(x_{0},\textbf{w})\bigg{)}.$

######  Definition 6.3.

For any integer $n\geq 1$, let $\mathcal{L}_{n}^{\varLambda^{\\#}}$ be the
union of all the non-empty subsets of $\varLambda^{\\#}$ of the following form

$\overline{\pi}^{\,-1}\left(\\{m\\}\times I_{1}\times I_{2}\times J\right),$

where
$m\in\mathbb{N},\,I_{1},\,I_{2}\in\mathcal{L}_{n},\,J\in\mathcal{L}^{\mathbb{C}}_{n+[m\log_{b}1/|\gamma|]}.$
The partition $\mathcal{L}_{0}^{\varLambda^{\\#}}$ consists of non-empty
subsets of $\varLambda^{\\#}$ of the following form

$\overline{\pi}^{\,-1}\left(\\{m\\}\times\mathbb{C}\times\mathbb{C}\times
J\right),$

where $m\in\mathbb{N},J\in\mathcal{L}^{\mathbb{C}}_{[m\log_{b}1/|\gamma|]}.$

###  6.5. The proof of Theorem A

We shall recall the following notations from [22]. For any probability measure
$\xi\in\mathscr{P}(\varLambda^{\\#})$ and $\textbf{u}\in\varLambda^{\\#}$,
define the probability measure $A_{\textbf{u}}(\xi)\in\mathscr{P}(\mathbb{C})$
be such that

(6.3)
$A_{\textbf{u}}(\xi)=\sum_{\textbf{w}\in\text{supp}(\xi)}\xi(\\{\textbf{w}\\})\,\delta_{S(x_{0},\textbf{w}\,\textbf{u})}.$

Let $B_{\textbf{q}}(\xi)\in\mathscr{P}(\mathbb{C})$ be the measure such that
for any set $A\in\mathscr{B}(\mathbb{C})$,

(6.4)
$B_{\textbf{q}}(\xi)(A)=\xi\times\nu^{\mathbb{Z}_{+}}\big{(}\,\big{\\{}\,(\textbf{w},\,\textbf{j})\in\varLambda^{\\#}\times\Sigma:S(x_{0},\textbf{w}\,\textbf{q}\,\textbf{j})\in
A\big{\\}}\,\big{)}.$

We also define the discrete measure on $\varLambda^{\\#}$

(6.5)
$\theta^{\,\textbf{u}}_{n}:=\frac{1}{b^{\hat{n}-t}}\sum_{\textbf{w}\in\varLambda^{\hat{n}-t}}\delta_{\textbf{w}\textbf{u}}.$

###### The proof of Theorem A.

[22, Lemma 8.1, corollary 8.1 and Lemma 8.3] can be extended to the case
$\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$ by the same methods. Also by
(B.1), (B.2) and (B.3), the proofs of [22, Lemma 7.1, Lemma 7.2, and Lemma
8.2, Lemma 8.4, Lemma 8.5 ] also work for the case $\gamma\in\mathbb{C}$ such
that $0<|\gamma|<1$. Combining these with Theorem 5.1, we extend the
conclusion of [22, Lemma 8.6] to the case $\gamma\in\mathbb{C}$ if
$\alpha<\min\\{2,\frac{\log b}{\log 1/|\gamma|}\\}$. Assume
$\alpha<\min\\{2,\frac{\log b}{\log 1/|\gamma|}\\}$, then there is a
contradiction as we did in [22]. ∎

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