# The entropy structures of axial products on $\mathbb{N}^{d}$ and Trees

Jung-Chao Ban  Department of Mathematical Sciences, National Chengchi
University, Taipei 11605, Taiwan, ROC.  Math. Division, National Center for
Theoretical Science, National Taiwan University, Taipei 10617, Taiwan. ROC.
<EMAIL_ADDRESS>,  Wen-Guei Hu  College of Mathematics, Sichuan University,
Chengdu, 610064, China<EMAIL_ADDRESS>and  Guan-Yu Lai  Department of
Mathematical Sciences, National Chengchi University, Taipei 11605, Taiwan,
ROC<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first concentrate on the possible values and dense property
of entropies for isotropic and anisotropic axial products of subshifts of
finite type (SFTs) on $\mathbb{N}^{d}$ and $d$-tree $\mathcal{T}_{d}$. We
prove that the entropies of isotropic and anisotropic axial products of SFTs
on $\mathbb{N}^{d}$ are dense in $[0,\infty)$, and the same result also holds
for anisotropic axial products of SFTs on $\mathcal{T}_{d}$. However, the
result is no longer true for isotropic axial products of SFTs on
$\mathcal{T}_{d}$. Next, motivated by the work of Johnson, Kass and Madden
[16], and Schraudner [28], we establish the entropy formula and structures for
full axial extension shifts on $\mathbb{N}^{d}$ and $\mathcal{T}_{d}$.
Combining the aforementioned results with the findings on the surface entropy
for multiplicative integer systems [8] on $\mathbb{N}^{d}$ enables us to
estimate the surface entropy for the full axial extension shifts on
$\mathcal{T}_{d}$. Finally, we extend the results of full axial extension
shifts on $\mathcal{T}_{d}$ to general trees.

###### Key words and phrases:

Full axial extension, entropy, multiplicative integer systems

Ban is partially supported by the Ministry of Science and Technology, ROC
(Contract MOST 111-2115-M-004-005-MY3). Hu is partially supported by the
National Natural Science Foundation of China (Grant 11601355). Lai is
partially supported by the Ministry of Science and Technology, ROC (Contract
MOST 111-2811-M-004-002-MY2).

###### Contents

  1. 1 Introduction
  2. 2 The entropy structure of the axial product on $\mathbb{N}^{2}$ and $\mathcal{T}_{2}$
    1. 2.1 Entropy structure
    2. 2.2 Transitivity and entropy
  3. 3 The entropy structures of full axial extension shifts on $\mathbb{N}^{d}$ and $\mathcal{T}_{d}$
    1. 3.1 Entropy structure
    2. 3.2 Application: surface entropy
  4. 4 General trees

##  1\. Introduction

Let $\mathcal{A}$ be a finite set and
$X_{1},\ldots,X_{d}\subseteq\mathcal{A}^{\mathbb{N}}$ be $d$ shifts, the
associated _axial product of subshifts_ $X_{1},\ldots,X_{d}$_on_
$\mathbb{N}^{d}$, write $\otimes_{i=1}^{d}X_{i}=X_{1}\otimes\cdots\otimes
X_{d}\subset\mathcal{A}^{\mathbb{N}^{d}}$ is defined as follows.

$\otimes_{i=1}^{d}X_{i}=\\{x\in\mathcal{A}^{\mathbb{N}^{d}}:\forall
g\in\mathbb{N}^{d}\text{ }\forall i\in\\{1,\ldots,d\\}\text{,
}x_{g+\mathbb{Z}e_{i}}\in X_{i}\\}\text{,}$

where $x_{g+\mathbb{Z}e_{i}}\in\mathcal{A}^{\mathbb{N}}$ is the sequence
obtained by shifting $x$ by $g$ and $\\{e_{1},\ldots,e_{d}\\}$ denotes the
standard basis of $\mathbb{N}^{d}$. Suppose $\mathcal{T}_{d}$ is a
conventional $d$-tree, that is, $\mathcal{T}_{d}$ is a free semigroup
generated by $\Sigma=\\{f_{1},\ldots,f_{d}\\}$ with the root, say $\epsilon$.
The _axial product of subshifts_
$X_{1},\ldots,X_{d}\subseteq\mathcal{A}^{\mathbb{N}}$_on_ $\mathcal{T}_{d}$,
write $\times_{i=1}^{d}X_{i}=X_{1}\times\cdots\times X_{d}$ is defined
similarly. That is,

$\times_{i=1}^{d}X_{i}=\\{x\in\mathcal{A}^{\mathcal{T}_{d}}:\forall
g\in\mathcal{T}_{d}\text{ }\forall i\in\\{1,\ldots,d\\}\text{,
}x_{g+\mathbb{Z}f_{i}}\in X_{i}\\}\text{.}$

An axial product $\otimes_{i=1}^{d}X_{i}$ (or $\times_{i=1}^{d}X_{i}$) is
called _isotropic_ if $X_{i}=X_{j}$ $\forall 1\leq i\neq j\leq d$, and is
called _anisotropic_ if it is not isotropic111An isotropic axial product space
is also called a _hom-shifts_ on $\mathbb{N}^{d}$ [11] or on $\mathcal{T}_{d}$
[26, 27].

The isotropic axial product of shifts on $\mathbb{N}^{d}$ is introduced in
[21]. Many important physical systems, e.g., the hard square model on
$\mathbb{N}^{2}$ (or $\mathbb{Z}^{2}$), are characterized by this kind of
multidimensional shift. In [11], the authors study the decidability for some
topological properties of $\otimes_{i=1}^{d}X_{i}$ for $d\geq 2$. In [21, 23],
the authors study the relation between the limiting and independence entropy
of $\otimes_{i=1}^{d}X_{i}$ as $d\rightarrow\infty$.

The axial product space on $\mathcal{T}_{d}$ is a sort of tree-shift (cf. [3,
4, 25, 26, 27, 2]), and it is attracting a lot of attention recently since
$\mathcal{T}_{d}$ is not an amenable group and the shifts defined on it
exhibit very rich and different phenomena in the topological (cf. [3]) and
statistical prospects (cf. [5, 6, 27]). In [27], the authors extend the
concept of limiting entropy [22, 23] on $\mathbb{N}^{d}$ to asymptotic
pressure on $\mathcal{T}_{d}$ and study its limiting behavior. The aim of this
paper is to investigate the entropy structures of $\otimes_{i=1}^{d}X_{i}$ and
$\times_{i=1}^{d}X_{i}$, and we introduce the formal definition for the
topological entropy below.

Let $F\subseteq\mathbb{N}^{d}$ be a finite set, we denote by
$\mathcal{P}(F,X):\mathcal{A}^{\mathbb{N}^{d}}\rightarrow\mathcal{A}^{F}$ the
_canonical projection_ of $X\subseteq\mathcal{A}^{\mathbb{N}^{d}}$ into
$\mathcal{A}^{F}$, i.e., $\mathcal{P}(F,X)=\\{(x_{g})_{g\in
F}\in\mathcal{A}^{F}:x\in X\\}$. Denote $F_{n}:=[1,n]^{d}$ and for a subshift
$X\subset\mathcal{A}^{\mathbb{N}^{d}}$, the _entropy_ of $X$ is defined as

(1)
$h(X)=\lim_{n\rightarrow\infty}\frac{\log\mathcal{P}(F_{n},X)}{\left|F_{n}\right|}\text{,}$

where $\left|F\right|$ denotes the number of the elements in $F$. For a
subshift $X\subset\mathcal{A}^{\mathcal{T}_{d}}$, let

$\Delta_{n}=\\{g\in\mathcal{T}_{d}:\left|g\right|\leq n\\}\text{ and
}\mathcal{T}_{n}=\\{g\in\mathcal{T}_{d}:\left|g\right|=n\\}\text{,}$

then the _topological entropy_ of $X\subseteq\mathcal{A}^{\mathcal{T}_{d}}$ is
defined similarly.

(2)
$h^{\mathcal{T}}(X)=\lim_{n\rightarrow\infty}\frac{\log\mathcal{P}(\Delta_{n},X)}{\left|\Delta_{n}\right|}\text{.}$

The primary objective of this paper is to study the entropy structures of
$h(\otimes_{i=1}^{d}X_{i})$ and $h^{\mathcal{T}}(\times_{i=1}^{d}X_{i})$. The
limit (1) exists since $\mathbb{N}^{d}$ is an amenable group (cf. [10]). That
is, if $F_{n}$ is a _Følner sequence_ , i.e.,
$\lim_{n\rightarrow\infty}\frac{\left|gF_{n}\Delta
F_{n}\right|}{\left|F_{n}\right|}=0$, then the limit

$h(X)=\lim_{n\rightarrow\infty}\frac{\log\mathcal{P}(F_{n},X)}{\left|F_{n}\right|}$

exists and is equal to (1). The existence of the limit (2) is proved in
[25]222Since the subadditive property does not hold true for shifts on
$\mathcal{T}_{d}$, the proof of the existence of (2) is quite different than
that in the cases where shifts on $\mathbb{N}^{d}$. We refer the reader to [7]
for the existence of the limit (2) for shifts defined in a large class of
trees., and it appears that the structures of $\\{h(X):X$ is a tree-
SFT$\\}$333For the simplicity of notations, we denote $h^{\mathcal{T}}$ by
$h$. and $\\{h(X):X$ is an $\mathbb{N}^{d}$ SFT$\\}$ are quite different (cf.
[5]). This motivates a systematic study on the sets

$\\{h(\otimes_{i=1}^{d}X_{i}):X_{i}\subseteq\mathcal{A}^{\mathbb{N}}\text{
}\forall i=1,\ldots,d\\}\text{ and
}\\{h(\times_{i=1}^{d}X_{i}):X_{i}\subseteq\mathcal{A}^{\mathbb{N}}\text{
}\forall i=1,\ldots,d\\}\text{.}$

The study consists of four components, and we present the motivations and
findings below.

1). The entropy structure of the axial product on $\mathbb{N}^{2}$ and
$\mathcal{T}_{2}$. It is known that the set
$\\{h(X):X\in\mathcal{A}^{\mathbb{Z}}$ is an mixing SFT$\\}$ is the logarithm
of the numbers in the spectral radii of aperiodic non-negative integral
matrices [19], and the set $\\{h(X):X\in\mathcal{A}^{\mathbb{Z}^{d}}$ is an
SFT$\\}$, $d\geq 2,$ is the class of non-negative right recursively enumerable
numbers [15]. The above results focus on all possible values of the set of
entropies of SFTs on $\mathbb{N}^{d}$, $d\geq 1$ in algebraic and
computational perspectives. Besides, Desai [12] proved that any
$\mathbb{Z}^{d}$ SFT (resp. sofic) $X$ with $h(X)>0$ contains a family of
$\mathbb{Z}^{d}$ subSFTs (subsofics) with entropies dense in the interval
$[0,h(X)]$. This indicates that the possible entropies of $\mathbb{Z}^{d}$
SFTs (or sofic) dense in $[0,\infty)$.

In this paper, we concentrate on the possible values and dense property of
entropies for isotropic and anisotropic axial products of SFTs on
$\mathbb{N}_{d}$ and $\mathcal{T}_{d}$. For simplicity, we consider the cases
where $d=2$, and the case in which $d\geq 2$ can be dealt with in the same
manner. Theorem 2.1 demonstrates that the set of entropy of anisotropic (or
isotropic) axial products of subshifts of finite type (SFTs) on
$\mathbb{N}^{2}$ is dense in $[0,\infty)$. The same result holds true for the
anisotropic axial products of SFTs on $\mathcal{T}_{2}$_._ However, this
outcome no longer holds true for isotropic axial products of SFTs on
$\mathcal{T}_{2}$, Theorem 2.1 reveals that the closure of possible entropy
values of isotropic axial products of SFTs cannot intersect the interval
$(0,\frac{\log 2}{2})$. This phenomenon is new and differs from the case of
the axial products on $\mathbb{N}^{2}$. The possible reason lies in the
inherent difference in the structure of the $\mathbb{N}^{2}$ and
$\mathcal{T}_{2}$ and the isotropic constrain.

2). Whether transitivity implies the positivity of the entropy. Another
viewpoint comes from [18], in Section 9 of [18], the authors describes the
following:

> The question whether transitivity implies the positivity of the entropy is
> challenging. Moreover, if the answer is affirmative, one can ask what is the
> best lower bound for the entropy of transitive maps in the space under
> consideration.

Indeed, there are spaces in which transitive maps have zero topological
entropy, and there is also a class of transitive maps defined in the interval
or circle in which the infimum of the entropies of the class is positive (see
[18] and the references given therein). Thanks to the aforementioned result,
Theorem 2.2 reveals that the infimum of the entropies of the set
$\\{\times_{i=1}^{d}X_{i}:\times_{i=1}^{d}X_{i}$ is transitive and has a
periodic point$\\}$ (or $\\{\otimes_{i=1}^{d}X_{i}:\otimes_{i=1}^{d}X_{i}$ is
transitive and has a periodic point$\\}$) is zero.

3). The entropy structure of full axial extension shifts on $\mathbb{N}^{d}$
and $\mathcal{T}_{d}$. Let $E=\mathcal{A}^{\mathbb{N}}$, that is, the full
shift. It is not difficult to show that the entropy of the axial product of
$E$ with a shift $X$ on $\mathbb{N}^{2}$ equals the entropy of $X$, i.e.,
$h(E\otimes X)=h(X)$. The converse also holds true when the underlined lattice
is $\mathbb{Z}^{2}$ (Theorem 4.1. [16]) or $\mathbb{Z}^{d}$ in which $X$
possesses some mixing assumption (Theorem 2.3. [28]). However, the property is
not generally true for the axial product spaces defined in $\mathcal{T}_{d}$.
Therefore, to calculate the explicit entropy values for $E\times X$ on
$\mathcal{T}_{2}$ and
$E^{r}\times_{i=1}^{d-r}X_{i}:=\left(E^{\times_{r}}\right)\times_{i=1}^{d-r}X_{i}$
on $\mathcal{T}_{d}$ is interesting and is presented in Theorem 3.1, where
$\left(E^{\times_{r}}\right)$ stands for the $r$-axial product of $E$, for
$r\in\mathbb{N}$. In addition, we focus on the lower bound for the values of
$\\{h(E^{r}\times_{i=1}^{d-r}X_{i}):X_{i}$ is an SFT $\forall 1\leq i\leq
d-r\\}$ as well. Theorem 3.2 reveals that different types, e.g., nonempty,
general subshifts or SFTs, of
$\times_{i=1}^{d-r}X_{i}\subseteq\mathcal{A}^{\mathcal{T}_{d}}$ give rise to
different lower bounds of $h\left(E^{r}\times_{i=1}^{d-r}X_{i}\right)$. We
stress that this behavior cannot happen in the full axial extension shifts on
$\mathbb{N}^{d}$. Using the results of surface entropy of multiplicative
systems and full axial product on $\mathcal{T}_{d}$, we give the surface
entropy of full axial extension shifts on $\mathcal{T}_{d}$ (see Section 3.2
for details).

4). General trees. The aforementioned results focus on the axial product of
shifts on $\mathbb{N}^{d}$ and $\mathcal{T}_{d}$. The natural question is, can
these outcomes be extended to broader lattices? Theorem 4.1 gives the explicit
formula for $h(E\times X)$ and $h(X\times E)$, where $E\times X$ and $X\times
E$ are the full axial extensions on golden-mean tree $\mathcal{G}$, and it
appears that both values do not generally coincide. The _golden-mean tree_
$\mathcal{G}$ is a kind of Markov-Cayley tree (defined in Section 4) with the
adjacency matrix is $M=\left[\begin{array}[]{cc}1&1\\\ 1&0\end{array}\right]$.
Generally, let $\mathcal{T}_{n}=\\{g\in\mathcal{T}:\left|g\right|=n\\}$, and
$\gamma=\lim_{n\rightarrow\infty}\frac{|\mathcal{T}_{n+1}|}{|\mathcal{T}_{n}|}$,
Theorem 4.3 demonstrates that if $\gamma>1$, then for all axial product
$X_{1}\times X_{2}$ we have $h(X_{1}\times X_{2})>h(X_{2})$. However, if
$\gamma=1$, there exists an example of axial product $X_{1}\times X_{2}$ on
which $h(X_{1}\times X_{2})=h(X_{2})$. This can be seen an analogous result as
that of Theorem 3.1 for the cases where the underlined space is
$\mathbb{N}^{d}$.

##  2\. The entropy structure of the axial product on $\mathbb{N}^{2}$ and
$\mathcal{T}_{2}$

###  2.1. Entropy structure

Let $\mathbb{N}$ be the set of positive integers. For $d\in\mathbb{N}$,
$\mathbb{N}^{d}=\\{(n_{1},...,n_{d}):n_{i}\in\mathbb{N}\mbox{ for all }1\leq
i\leq d\\}$, and $d$-tree $\mathcal{T}_{d}$ is a semigroup which is generated
by $d$ generators.

Let $1\leq r\leq d$. Define the set of entropies of $d$ dimensional full $r$
extension on $d$-tree by

$\mathcal{H}^{d,r}=\left\\{h(E^{r}\times_{i=1}^{d-r}X_{i}):X_{1},...,X_{d-r}\mbox{
are SFTs}\right\\},$

and if $X_{i}=X$ for all $1\leq i\leq d-r$, we denote $\mathcal{H}^{d,r}$ by
$\mathcal{H}_{i}^{d,r}$. Similarly, we define the set of entropies of $d$
dimensional full $r$ extension on $\mathbb{N}^{d}$ by $\mathbf{H}^{d,r}$ and
if $X_{i}=X$ for all $1\leq i\leq d-r$, we denote $\mathbf{H}^{d,r}$ by
$\mathbf{H}_{i}^{d,r}$. For simplicity, we only consider $d=2$ and $r=0$. The
following notations are also needed. Let $\mathbb{Z}_{m\times
n}=\\{1,...,m\\}\times\\{1,...,n\\}$, and $\mathbb{Z}_{n}=\\{1,...,n\\}$,
$\Delta_{n}=\\{g\in\mathcal{T}:|g|\leq n\\}$ and $\Delta_{0}=\\{\epsilon\\}$.

######  Theorem 2.1.

The closures of $\mathbf{H}^{2,0}_{i},\mathbf{H}^{2,0}$ and
$\mathcal{H}^{2,0}$ equal $[0,\infty)$. Moreover, the closure
$\mathcal{H}^{2,0}_{i}$ does not intersect $(0,\frac{\log 2}{2})$.

###### Proof.

We first prove that the closure of $\mathbf{H}^{2,0}_{i}$ equals $[0,\infty)$.
Let $X=X_{A}$ be an SFT with transition matrix $A$, where $A=[A_{i,j}]$ is a
$2^{m}+n-1$ by $2^{m}+n-1$ matrix and

$A_{i,j}=\left\\{\begin{array}[]{ll}1&\mbox{, if }1\leq i\leq 2^{m}\mbox{ and
}j=2^{m}+1,\\\ 1&\mbox{, if }2^{m}+1\leq i\leq 2^{m}+n-2\mbox{ and }j=i+1,\\\
1&\mbox{, if }i=2^{m}+n-1\mbox{ and }1\leq j\leq 2^{m},\\\ 0&\mbox{,
otherwise.}\end{array}\right.$

See Figure 1 for the directed graph of the transition matrix $A$.

Figure 1. The directed graph of the transition matrix $A$.

In the following, we claim that for $k\geq 1$,

$\left|\mathcal{P}(\mathbb{Z}_{kn\times k},X\otimes
X)\right|=n(2^{m})^{k^{2}}.$

Indeed, due to the rule of $A$, when the symbol on $\mathbb{Z}_{1\times 1}$ is
fixed then the positions of symbols $1\leq i\leq 2^{m}$ are uniquely
determined. Thus, if the symbol on $\mathbb{Z}_{1\times 1}$ is in
$\\{2^{m}+1,...,2^{m}+n-1\\}$, then we have $k^{2}$ positions on
$\mathbb{Z}_{kn\times k}$ having $2^{m}$ choices. If the symbol on
$\mathbb{Z}_{1\times 1}$ is in $\\{1,...,2^{m}\\}$, then we have $k^{2}-1$
positions on $\mathbb{Z}_{kn\times k}$ having $2^{m}$ choices. Hence,

$\displaystyle\left|\mathcal{P}(\mathbb{Z}_{kn\times k},X\otimes X)\right|$
$\displaystyle=|\\{2^{m}+1,...,2^{m}+n-1\\}|(2^{m})^{k^{2}}+|\\{1,...,2^{m}\\}|(2^{m})^{k^{2}-1}$
$\displaystyle=(n-1)(2^{m})^{k^{2}}+(2^{m})(2^{m})^{k^{2}-1}$
$\displaystyle=n(2^{m})^{k^{2}}.$

The proof of claim is complete.

By the existence of the entropy of $X\otimes X$ and the above claim, we have

$\displaystyle h(X\otimes X)$
$\displaystyle=\lim_{k,\ell\to\infty}\frac{\log|\mathcal{P}(\mathbb{Z}_{k\times\ell},X\otimes
X)|}{k\ell}$
$\displaystyle=\lim_{k\to\infty}\frac{\log|\mathcal{P}(\mathbb{Z}_{kn\times
k},X\otimes X)|}{k^{2}n}$ $\displaystyle=\lim_{k\to\infty}\frac{\log
n+k^{2}\log 2^{m}}{k^{2}n}$ $\displaystyle=\frac{m\log 2}{n}.$

Thus, the closure of $\\{h(X\otimes X)=\frac{m\log
2}{n}:m,n\in\mathbb{N}\\}\subseteq\mathbf{H}^{2,0}_{i}$ is $[0,\infty)$. Since
$h(X\otimes X)\geq 0$ for all SFT $X$, we have the closure of
$\mathbf{H}^{2,0}_{i}$ is $[0,\infty)$.

Secondly, we prove that the closure of $\mathbf{H}^{2,0}$ equals $[0,\infty)$.
By Theorem 3.1, we have $\\{h(X_{2}):X_{2}\mbox{ is an
SFT}\\}\subseteq\mathbf{H}^{2,0}$. Since the closure of
$\\{h(X_{2}):X_{2}\mbox{ is an SFT}\\}$ is $[0,\infty)$, and $h(X_{1}\otimes
X_{2})\geq 0$ for all SFTs $X_{1}$ and $X_{2}$, we have the closure of
$\mathbf{H}^{2,0}$ is $[0,\infty)$.

Thirdly, we prove that the closure of $\mathcal{H}^{2,0}$ equals $[0,\infty)$.
Let $X_{1}=X_{A}$ be an SFT with transition matrix $A$ as above, and $X_{2}$
be an identity matrix. For $k\geq 1$, let
$\mathcal{T}_{k}=\\{g\in\mathcal{T}_{2}:|g|=k\\}$, then we have the following
recurrence relation:

For $k\geq 1$ and $0\leq j\leq n-1$, let $a^{\\{1,...,2^{m}\\}}_{k}(j)$ be the
number of positions (labeled the symbol in $\\{1,...,2^{m}\\}$) on
$\mathcal{T}_{k}$ and $a_{k}^{i}(j)$ be the number of positions (labeled the
symbol $i,2^{m}+1\leq i\leq 2^{m}+n-1$) on $\mathcal{T}_{k}$. We have

(3) $\left[\begin{matrix}a^{\\{1,...,2^{m}\\}}_{k}(j)\\\ a^{2^{m}+1}_{k}(j)\\\
\vdots\\\
a^{2^{m}+n-1}_{k}(j)\end{matrix}\right]=\left[\begin{matrix}1&&&&1\\\ 1&1&\\\
&\ddots&\ddots\\\ &&\ddots&1&\\\ &&&1&1\end{matrix}\right]_{n\times
n}\left[\begin{matrix}a^{\\{1,...,2^{m}\\}}_{k-1}(j)\\\
a^{2^{m}+1}_{k-1}(j)\\\ \vdots\\\ a^{2^{m}+n-1}_{k-1}(j)\end{matrix}\right]$

with initial vector
$(a_{0}^{\\{1,...,2^{m}\\}}(j),...,a_{0}^{2^{m}+n-1}(j))^{t}=e_{j+1}$ and
$\\{e_{1},...,e_{n}\\}$ is standard basis in $\mathbb{R}^{n}$. See Figure 2
for the recursive relation (3).

Figure 2. The recurrsive relation of $a^{i}_{k}(j)$ from $\mathcal{T}_{k-1}$
to $\mathcal{T}_{k}$.

Then we have

$\left|\mathcal{P}\left(\Delta_{k},X_{1}\times
X_{2}\right)\right|=\sum_{j=0}^{n-1}(2^{m})^{a_{k}^{\\{1,...,2^{m}\\}}(j)}.$

By Theorem 4.5.12 [20] with left and right eigenvectors
$v=w^{t}=\frac{1}{\sqrt{n}}(1,...,1)$ corresponding to the maximum eigenvalue
$2$, after computation, we have
$a_{k}^{\\{1,...,2^{m}\\}}(j)=\frac{2^{k}}{n}+o(2^{k})$ for all $0\leq j\leq
n-1$. This implies

$\displaystyle h(X_{1}\times X_{2})$
$\displaystyle=\lim_{k\to\infty}\frac{\log\left|\mathcal{P}\left(\Delta_{k},X_{1}\times
X_{2}\right)\right|}{1+2+\cdots+2^{k}}$
$\displaystyle=\lim_{k\to\infty}\frac{\log\sum_{j=0}^{n-1}(2^{m})^{a_{k}(j)}}{2^{k+1}-1}$
$\displaystyle=\lim_{k\to\infty}\frac{\log
n+\left(\frac{m2^{k}}{n}+o(2^{k})\right)\log 2}{2^{k+1}-1}$
$\displaystyle=\frac{m\log 2}{2n}.$

Thus, the closure of $\\{\frac{m\log 2}{2n}:m,n\in\mathbb{N}\\}$ is
$[0,\infty)$. Since $h(X_{1}\times X_{2})\geq 0$ for all SFTs $X_{1}$ and
$X_{2}$, we have the closure of $\mathcal{H}^{2,0}$ is $[0,\infty)$.

Finally, we prove that the closure of $\mathcal{H}^{2,0}_{i}$ does not
intersect $(0,\frac{\log 2}{2})$. We divide the proof into the following three
cases: (1) If the transition matrix of $X$ contains no irreducible component,
then the global pattern does not exist. Thus, $h(X\times X)=0$. (2) If the
transition matrix of $X$ is irreducible and contains no row with a row sum
greater than or equal to 2, then the number of global patterns is less than or
equal to the number of symbols, that is

$\left|\mathcal{P}(\Delta_{n},X\times X)\right|\leq 2$

for all $n\geq 1$. Hence, we have $h(X\times X)=0$. (3) The transition matrix
of $X$ is irreducible and contains a row $i$ with its row sum greater than or
equal to 2. Then we put the symbol $i$ on $\mathcal{T}^{n-1}$, we have

$\left|\mathcal{P}(\Delta_{n},X\times X)\right|\geq 2^{2^{n}}.$

Thus, $h(X_{1}\times X_{1})\geq\frac{\log 2}{2}$. Therefore, the proof of
theorem is complete. ∎

###  2.2. Transitivity and entropy

Let $X\subseteq\mathcal{A}^{\mathbb{N}}$ be a subshift and $\sigma$ be the
shift map. We say $X$ is _transitive_ if for any patterns $u,v$ of $X$, there
exists an $n\in\mathbb{N}$ such that $\sigma^{n}([u])\cap[v]\neq\emptyset$,
where $[w]$ is the cylinder set of the pattern $w$. A point $x\in X$ is called
a _periodic point_ if there is a period $p\in\mathbb{N}$ such that
$\sigma^{p}(x)=x$.

Let $X\subseteq\mathcal{A}^{\mathbb{N}^{2}}$ be a subshift. The _horizontal_
shift map $\sigma_{1}$ and the _vertical_ shift map $\sigma_{2}$ are defined
by $(\sigma_{1}(x))_{i,j}=(x)_{i+1,j}$ and $(\sigma_{2}(x))_{i,j}=(x)_{i,j+1}$
for all $i,j\geq 1,x\in X$. We say $X$ is _transitive_ if for any patterns
$u,v$ of $X$, there exist $n,m\in\mathbb{N}$ such that
$\sigma_{1}^{n}\sigma_{2}^{m}([u])\cap[v]\neq\emptyset$. A point $x\in X$ is
called a _periodic point_ if there exist $p_{1},p_{2}\in\mathbb{N}$ such that
$\sigma_{1}^{p_{1}}\sigma_{2}^{p_{2}}(x)=x$.

Let $T\subseteq\mathcal{A}^{\mathcal{T}_{2}}$ be a 2-tree-shift with the shift
map $\sigma$. We say $T$ is _transitive_ if for any patterns $u,v$ of $T$,
there exists a $w\in\mathcal{T}_{2}$ such that
$\sigma^{w}([u])\cap[v]\neq\emptyset$. A point $t\in T$ is called a _periodic
point_ if there is a $w\in\mathcal{T}_{2}$ such that $\sigma^{w}(t)=t$.

The following theorem is directly obtained from Theorem 2.1.

######  Theorem 2.2.

We have $\inf\mathbf{H}^{2,0}$ (resp. $\inf\mathcal{H}^{2,0}$) equals 0, where
the infimum is taken over all $X_{1}\times X_{2}$ (resp. $X_{1}\otimes X_{2}$)
is transitive and has a periodic point.

###### Proof.

It is easy to check that the cases of $X_{1}\times X_{2}$ and $X_{1}\otimes
X_{2}$ in the proof of the Theorem 2.1 are transitive and have a periodic
point. The proof is complete. ∎

##  3\. The entropy structures of full axial extension shifts on
$\mathbb{N}^{d}$ and $\mathcal{T}_{d}$

In this section, we investigate the entropy structures of the full axial
extension shifts on $\mathbb{N}^{d}$ and $\mathcal{T}_{d}$.

###  3.1. Entropy structure

In this subsection, we provide the rigorous entropy formulas for the full
axial extension shifts on $\mathbb{N}^{d}$ and $\mathcal{T}_{d}$. In the
following theroem, it is easy to see that the formulas are quite different.

######  Theorem 3.1.

Let $d\geq 2$ and $d\geq r\geq 1$ be two positive integers. If
$X_{1},...,X_{d-r}$ are subshifts, then

$h(E^{r}\otimes_{i=1}^{d-r}X_{i})=h(\otimes_{i=1}^{d-r}X_{i})$, and

$h(E^{r}\times_{i=1}^{d-r}X_{i})=r(d-1)\sum_{j=1}^{\infty}\frac{\log\left|\mathcal{P}(\Delta_{j-1},\times_{i=1}^{d-r}X_{i})\right|}{d^{j+1}}$.

###### Proof.

Due to the rule of $E^{r}\otimes_{i=1}^{d-r}X_{i}$, for any
$n_{1},...n_{d}\in\mathbb{N}$, the lattice
$\mathbb{Z}_{n_{1}\times\cdots\times n_{d}}$ can be divided into
$n_{1}\times\cdots\times n_{r}$ many $\mathbb{Z}_{n_{r+1}\times\cdots\times
n_{d}}$ lattices in which

$\left|\mathcal{P}\left(\mathbb{Z}_{n_{1}\times\cdots\times
n_{d}},E^{r}\otimes_{i=1}^{d-r}X_{i}\right)\right|=\left|\mathcal{P}\left(\mathbb{Z}_{n_{r+1}\times\cdots\times
n_{d}},\otimes_{i=1}^{d-r}X_{i}\right)\right|^{n_{1}\cdots n_{r}}.$

Then we have

$\displaystyle h\left(E^{r}\otimes_{i=1}^{d-r}X_{i}\right)$
$\displaystyle=\lim_{n_{1},...,n_{d}\to\infty}\frac{\log\left|\mathcal{P}\left(\mathbb{Z}_{n_{1}\times\cdots\times
n_{d}},E^{r}\otimes_{i=1}^{d-r}X_{i}\right)\right|}{n_{1}\cdots n_{d}}$
$\displaystyle=\lim_{n_{1},...,n_{d}\to\infty}\frac{\log\left|\mathcal{P}\left(\mathbb{Z}_{n_{r+1}\times\cdots\times
n_{d}},\otimes_{i=1}^{d-r}X_{i}\right)\right|^{n_{1}\cdots n_{r}}}{n_{1}\cdots
n_{d}}$
$\displaystyle=\lim_{n_{r+1},...,n_{d}\to\infty}\frac{\log\left|\mathcal{P}\left(\mathbb{Z}_{n_{r+1}\times\cdots\times
n_{d}},\otimes_{i=1}^{d-r}X_{i}\right)\right|}{n_{r+1}\cdots n_{d}}$
$\displaystyle=h\left(\otimes_{i=1}^{d-r}X_{i}\right).$

Due to the rule of $E^{r}\times_{i=1}^{d-r}X_{i}$, for any $n\in\mathbb{N}$,
the $\Delta_{n}$ of $d$-tree can be divided into a $\Delta_{n}$ of
$(d-r)$-tree and $rd^{n-j}$ many $\Delta_{n-j}$ of $(d-r)$-tree for all $1\leq
j\leq n$ in which

$\left|\mathcal{P}\left(\Delta_{n},E^{r}\times_{i=1}^{d-r}X_{i}\right)\right|=\left|\mathcal{P}\left(\Delta_{n},\times_{i=1}^{d-r}X_{i}\right)\right|\prod_{j=1}^{n}\left|\mathcal{P}\left(\Delta_{j-1},\times_{i=1}^{d-r}X_{i}\right)\right|^{rd^{n-j}}.$

We prove the foregoing by induction. For $n=1$, the $\Delta_{1}$ of $d$-tree
can be divided into $r$ many $\Delta_{0}$ and a $\Delta_{1}$ of $(d-r)$-tree
in which

$\left|\mathcal{P}\left(\Delta_{1},E^{r}\times_{i=1}^{d-r}X_{i}\right)\right|=\left|\mathcal{P}\left(\Delta_{1},\times_{i=1}^{d-r}X_{i}\right)\right|\left|\mathcal{P}\left(\Delta_{0},\times_{i=1}^{d-r}X_{i}\right)\right|^{r}.$

Suppose it is true for $n=k$, then for $n=k+1$, the $\Delta_{k+1}$ of $d$-tree
is the $\Delta_{k}$ of $d$-tree that add a $\mathcal{T}_{k+1}$ in the bottom.
Then each partition of $\Delta_{k}$ of $d$-tree become to a partition of
$\Delta_{k+1}$ of $d$-tree when it adds a level in the bottom. More precisely,
the $\Delta_{j-1}$ of $(d-r)$-tree (in the partition of $\Delta_{k}$ of
$d$-tree) becomes to a $\Delta_{j}$ of $(d-r)$-tree and $r(d-r)^{j-1}$ many
$\Delta_{0}$ for all $1\leq j\leq k+1$ (in the partition of $\Delta_{k+1}$ of
$d$-tree). Then, by the assumption that $\Delta_{k}$ of $d$-tree can be
divided into a $\Delta_{k}$ of $(d-r)$-tree and $rd^{k-j}$ many $\Delta_{k-j}$
of $(d-r)$-tree for all $1\leq j\leq k$, we conclude that the $\Delta_{k+1}$
of $d$-tree can be divided into a $\Delta_{k+1}$ of $(d-r)$-tree, $rd^{k-j}$
many $\Delta_{k+1-j}$ of $(d-r)$-tree for all $1\leq j\leq k+1$ and
$r(d-r)^{k}+\sum_{j=1}^{k}r(rd^{k-j})(d-r)^{j-1}=rd^{k}$ many $\Delta_{0}$ in
which

$\left|\mathcal{P}\left(\Delta_{k+1},E^{r}\times_{i=1}^{d-r}X_{i}\right)\right|=\left|\mathcal{P}\left(\Delta_{k+1},\times_{i=1}^{d-r}X_{i}\right)\right|\prod_{j=1}^{k+1}\left|\mathcal{P}\left(\Delta_{j-1},\times_{i=1}^{d-r}X_{i}\right)\right|^{rd^{k+1-j}}.$

The proof of induction is complete. See Figure 3 for the induction step when
$n=1$.

Figure 3. The induction step for $d$-tree’s $\Delta_{1}$ to $d$-tree’s
$\Delta_{2}$.

Then we have the following two cases: (1) When $r=d$, we have

$\displaystyle
h\left(E^{d}\right)=\lim_{n\to\infty}\frac{\log\left|\mathcal{P}\left(\Delta_{n},E^{d}\right)\right|}{\left|\Delta_{n}\right|}=\lim_{n\to\infty}\frac{\log
2^{\left|\Delta_{n}\right|}}{\left|\Delta_{n}\right|}=\log 2.$

(2) When $1\leq r<d$, since

$0\leq\lim_{n\to\infty}\frac{\log\left|\mathcal{P}\left(\Delta_{n},\times_{i=1}^{d-r}X_{i}\right)\right|}{1+d+\cdots+d^{n}}\leq\lim_{n\to\infty}\frac{\log
2^{1+(d-r)+\cdots+(d-r)^{n}}}{1+d+\cdots+d^{n}}=0.$

We have

$\displaystyle h\left(E^{r}\times_{i=1}^{d-r}X_{i}\right)$
$\displaystyle=\lim_{n\to\infty}\frac{\log\left|\mathcal{P}\left(\Delta_{n},E^{r}\times_{i=1}^{d-r}X_{i}\right)\right|}{\left|\Delta_{n}\right|}$
$\displaystyle=\lim_{n\to\infty}\frac{\log\left|\mathcal{P}\left(\Delta_{n},\times_{i=1}^{d-r}X_{i}\right)\right|\prod_{j=1}^{n}\left|\mathcal{P}\left(\Delta_{j-1},\times_{i=1}^{d-r}X_{i}\right)\right|^{rd^{n-j}}}{1+d+\cdots+d^{n}}$
$\displaystyle=\lim_{n\to\infty}\frac{\sum_{j=1}^{n}rd^{n-j}\log\left|\mathcal{P}\left(\Delta_{j-1},\times_{i=1}^{d-r}X_{i}\right)\right|}{1+d+\cdots+d^{n}}+\lim_{n\to\infty}\frac{\log\left|\mathcal{P}\left(\Delta_{n},\times_{i=1}^{d-r}X_{i}\right)\right|}{1+d+\cdots+d^{n}}$
$\displaystyle=\lim_{n\to\infty}\frac{\sum_{j=1}^{n}rd^{-j}\log\left|\mathcal{P}\left(\Delta_{j-1},\times_{i=1}^{d-r}X_{i}\right)\right|}{1+d^{-1}+\cdots+d^{-n}}$
$\displaystyle=r(d-1)\sum_{j=1}^{\infty}\frac{\log\left|\mathcal{P}\left(\Delta_{j-1},\times_{i=1}^{d-r}X_{i}\right)\right|}{d^{j+1}}.$

The proof is complete. ∎

The set $\times_{i=1}^{d-r}X_{i}$ is called _nonempty_ if there exists a
global pattern. For $k\in\mathbb{N}$, a _$k$ -permutation matrix_ is a $k$ by
$k$ binary matrix that has exactly one entry of 1 in each row and each column
and 0 elsewhere.

######  Theorem 3.2.

Let $\mathcal{A}=\\{0,...,k-1\\}$. For $d\geq 2$, $1\leq r\leq d$ and
$\times_{i=1}^{d-r}X_{i}\subseteq\mathcal{A}^{\mathcal{T}_{d-r}}$, we have the
following assertions.

$\inf h(E^{r}\times_{i=1}^{d-r}X_{i})=\frac{r(d-1)\log k}{d^{2}}$, where the
infimum is taken over all nonempty $\times_{i=1}^{d-r}X_{i}$.

$\inf h(E^{r}\times_{i=1}^{d-r}X_{i})=\frac{r\log k}{d}$, where the infimum is
taken over all subshift $\times_{i=1}^{d-r}X_{i}$.

If $X_{1},...,X_{d-r}$ are irreducible SFTs and $\times_{i=1}^{d-r}X_{i}$ is a
subshift, then $h(E^{r}\times_{i=1}^{d-r}X_{i})=\frac{r\log k}{d}$ if and only
if $X_{1},...,X_{d-r}$ are determined by $k$-permutation matrices.

###### Proof.

Since $|\mathcal{P}(\Delta_{0},\times_{i=1}^{d-r}X_{i})|=k$ and
$|\mathcal{P}(\Delta_{n},\times_{i=1}^{d-r}X_{i})|\geq 1$ for all $n\geq 1$,
then the proof is completed by the Theorem 3.1 (2).

It is clear that if $\times_{i=1}^{d-r}X_{i}$ is a subshift, then
$|\mathcal{P}(\Delta_{n},\times_{i=1}^{d-r}X_{i})|\geq k$ for all $n\geq 0$.
Let $X_{1}=\cdots=X_{d-r}=\\{0^{\infty},...,(k-1)^{\infty}\\}$ be an subshift
that contains only $k$ points, then $\times_{i=1}^{d-r}X_{i}$ is also an
subshift that contains only $k$ points. Thus,
$|\mathcal{P}(\Delta_{n},\times_{i=1}^{d-r}X_{i})|=k$ for all $n\geq 0$, then
the proof is completed by the Theorem 3.1 (2).

For $d=2$ and $r=1$, if $A$ is a transition matrix of $X_{1}$, then
$|A^{n}|=k$ for all $n\geq 1$ if and only if $A$ is a permutation matrix or
union of permutation matrices with one sink. Then the irreducibility of $A$
implies $A$ is a $k$-permutation matrix. For $1<r<d$, it is easy to see that
if $X_{1},...,X_{d-r}$ are $k$-permutation matrices, then
$|\mathcal{P}(\Delta_{n},\times_{i=1}^{d-r}X_{i})|=k$ for all $n\geq 0$.
Conversely, if $|\mathcal{P}(\Delta_{n},\times_{i=1}^{d-r}X_{i})|=k$ for all
$n\geq 0$, then for the root of $(d-r)$-tree labels $i\in\\{0,...,k-1\\}$, we
have $\mathcal{P}(\Delta_{0},\times_{i=1}^{d-r}X_{i})=\\{0,...,k-1\\}$. This
implies that if two global patterns $t_{1}$ and $t_{2}$ on $(d-r)$-tree with
the same root symbol, then $t_{1}=t_{2}$. This implies that the symbol $i$ in
each $X_{1},...,X_{d-r}$ goes to unique $j\in\\{0,1,...,k-1\\}$. Then the
irreducibility implies that $X_{r+1},...,X_{d}$ are determined by
$k$-permutation matrices. The proof is complete. ∎

###  3.2. Application: surface entropy

Let $F_{n,m}=[1,n]\times[1,m]\subseteq\mathbb{N}^{2}$ and $X$ be a shift on
$\mathbb{N}^{2}$, the _surface entropy_ of $X$ with eccentricity $\alpha$ is

$h_{s}(X,\alpha)=\sup_{\\{(x_{n},y_{n})\\}\in\Gamma_{\alpha}}\limsup\limits_{n\rightarrow\infty}S_{X}(x_{n},y_{n})\text{,}$

where

$S_{X}(x_{n},y_{n})=\frac{\log\left|\mathcal{P}(F_{x_{n},y_{n}},X)\right|-x_{n}y_{n}h(X)}{x_{n}+y_{n}}\text{,}$

and
$\Gamma_{\alpha}=\\{\\{(x_{n},y_{n})\\}\in\left(\mathbb{N}^{2}\right)^{\mathbb{N}}:\frac{y_{n}}{x_{n}}\rightarrow\alpha$
and $x_{n}\rightarrow\infty\\}$. The main purpose of this study is to
understand the ‘linear term’ of the complexity function
$\log\left|\mathcal{P}(F_{x_{n},y_{n}},X)\right|$. This concept was first
introduced by Pace [24], wherein the author obtains the rigorous formula for
$\mathbb{Z}$ SFTs and a lot of interesting properties for $\mathbb{Z}^{2}$
SFTs. Some related results for the surface entropy may also be found in [9,
22]. Below we introduce the concept of the $k$-multiplicative integer system
($k$-MIS) on $\mathbb{N}^{d}$, and the main aim of this section is to combine
the results of the surface entropy for $k$-MIS and the full axial extension to
derive the surface entropy for the complexity function
$\log\mathcal{P}(\Delta_{n},E^{d-1}\times\Omega)$.

Let $\Omega\subseteq\mathcal{A}^{\mathbb{N}}$ be a $\mathbb{N}$ subshift and
$\mathbf{p}_{1},\ldots,\mathbf{p}_{k-1}\in\mathbb{N}^{d}$, $d\geq 1$. The
$k$_-multiplicative integer system_ with respect to $\Omega$ is defined as

(4)
$X_{\Omega}^{\mathbf{p}_{1},\ldots,\mathbf{p}_{k-1}}=\\{(x_{\mathbf{i}})_{\mathbf{i}\in\mathbb{N}^{d}}\in\mathcal{A}^{\mathbb{N}^{d}}:x_{\mathbf{i}}x_{\mathbf{ip}_{1}}\cdots
x_{\mathbf{ip}_{k-1}}\in\Omega_{k}\text{
}\forall\mathbf{i}\in\mathbb{N}^{d}\\}\text{,}$

where $\Omega_{k}$ denotes the set of admissible blocks of $\Omega$ with
length $1<k\in\mathbb{N}$. The
$X_{\Omega}^{\mathbf{p}_{1},\ldots,\mathbf{p}_{k-1}}$ in (4) is the
multidimensional version of the multiplicative shift $X_{\Omega}^{p}$ defined
by [14, 17].

$X_{\Omega}^{p}=\\{(x_{k})_{k=1}^{\infty}\in\mathcal{A}^{\mathbb{N}}:(x_{ip^{l}})_{l=0}^{\infty}\in\Omega\text{
for all }i\text{, }p\nmid i\\}\text{.}$

The $k$-MIS or multiplicative shifts have been studied in depth because they
have a close relationship with multifractal analysis of the multiple ergodic
theory (cf. [13]). We refer the reader to [13, 1] for more details on the
multiple ergodic theory and find the complete bibliography. In [8], the
authors consider the surface entropy of $2$-MISs on $\mathbb{N}^{d},d\geq 1$.

######  Theorem 3.3 (Theorem 3.3 with $d=1$ [8]).

For a sequence $\\{a_{n}\\}_{n=1}^{\infty}$ with
$\lim_{n\to\infty}a_{n}=\infty$, we have

$\log\left|\mathcal{P}\left(\mathbb{Z}_{x_{n}},X^{p}_{\Omega}\right)\right|-x_{n}h=\left(1-\frac{1}{p}\right)^{2}\sum_{i=r_{n}+1}^{\infty}\frac{x_{n}}{p^{i-1}}\log|\Omega_{i}|+o\left(a_{n}x_{n}\right),$

where $r_{n}=\left\lfloor\frac{\log x_{n}}{\log p}\right\rfloor$.

If $x_{n}=p^{n}\pm k$ and $1\leq k\leq p$, then we have

$\log\left|\mathcal{P}\left(\mathbb{Z}_{x_{n}},X^{p}_{\Omega}\right)\right|-x_{n}h=O\left(n\right).$

######  Theorem 3.4.

We have

$h\left(E^{d-1}\times
X\right)=(d-1)^{2}\sum_{i=1}^{\infty}\frac{\log\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|}{d^{i+1}}.$

For $d\geq 2$, let $\Omega$ be a subshift, we have

$\log\left|\mathcal{P}\left(\Delta_{n},E^{d-1}\times\Omega\right)\right|=\left|\Delta_{n}\right|h(E^{d-1}\times\Omega)+(d-1)^{2}\sum_{i=r_{n}+1}^{\infty}\frac{|\Delta_{n}|}{d^{i-1}}\log|\mathcal{P}(\mathbb{Z}_{i},\Omega)|+o\left(a_{n}|\Delta_{n}|\right),$

where $r_{n}=\left\lfloor\frac{\log|\Delta_{n}|}{\log d}\right\rfloor$ and
$\\{a_{n}\\}_{n=1}^{\infty}$ is a sequence with
$\lim_{n\to\infty}a_{n}=\infty$.

In particular, when $d=2$, we have

$\log\left|\mathcal{P}\left(\Delta_{n},E\times\Omega\right)\right|=\left|\Delta_{n}\right|h(E\times\Omega)+O(n).$

###### Proof.

By Theorem 3.1 with $r=d-1$, we have

$h\left(E^{d-1}\times
X\right)=(d-1)^{2}\sum_{i=1}^{\infty}\frac{\log\left|\mathcal{P}(\Delta_{i-1},X)\right|}{d^{i+1}}.$

Since $\mathcal{P}(\Delta_{i-1},X)=\mathcal{P}(\mathbb{Z}_{i},X)$ for all
$i\geq 1$, the proof is thus complete.

When $d\geq 2$, the proof is complete by taking $x_{n}=|\Delta_{n}|$ and $p=d$
in Theorem 3.3 (1). When $d=2$, since $|\Delta_{n}|=2^{n+1}-1$, the proof is
complete by applying Theorem 3.3 (2). ∎

##  4\. General trees

In this section, we study the entropy of the full axial extension on general
trees. Let $d\geq 2$, the $d$-tree $\mathcal{T}_{d}=\langle
f_{1},...,f_{d}\rangle$ is a semigroup which is generated by $d$ generators
$f_{1},...,f_{d}$. For any $g\in\mathcal{T}_{d}$ and a binary matrix
$M=[M_{i,j}]_{d\times d}$, we say $g$ is _admissible_ with respect to $M$ if
$g=f_{a_{1}}\cdots f_{a_{k}},k\geq 1,a_{1},...,a_{k}\in\\{1,...,d\\}$
satisfies $M_{a_{j},a_{j+1}}=1$ for all $1\leq j\leq k-1$. Define the _Markov-
Cayley tree_ $\mathcal{T}$ with the adjacency matrix $M=[M_{i,j}]_{d\times d}$
is the set $\left\\{g\in\mathcal{T}_{d}:g\mbox{ is admissible w.r.t.
}M\right\\}$. The _golden-mean tree_ $\mathcal{G}$ is defined by the adjacency
matrix $M=\left[\begin{array}[]{cc}1&1\\\ 1&0\end{array}\right]$. That is,
$\mathcal{G}=\left\\{g\in\mathcal{T}_{2}:g\mbox{ is admissible w.r.t.
}\left[\begin{array}[]{cc}1&1\\\ 1&0\end{array}\right]\right\\}$.

The following result demonstrates that the values $h(E\times X)$ and
$h(X\times E)$ are not general coincident.

######  Theorem 4.1 (Golden-mean tree).

If $X$ is a subshift, then the entropy of $E\times X$ and $X\times E$ on
$\mathcal{G}$ are equal to
$\frac{\log\left|\mathcal{P}(\mathbb{Z}_{1},X)\right|}{\rho^{3}}+\frac{\log\left|\mathcal{P}(\mathbb{Z}_{2},X)\right|}{\rho^{2}}$
and
$\sum_{i=1}^{\infty}\frac{\log\left|\mathcal{P}(\mathbb{Z}_{i},X)\right|}{\rho^{i+3}}$
respectively, where $\rho=\frac{1+\sqrt{5}}{2}$.

###### Proof.

For the entropy of $E\times X$, we claim that for $n\geq 2$,
$\Delta_{n}^{\mathcal{G}}$ can be divided into $a_{n}$ many $\mathbb{Z}_{1}$
lattices and $\frac{\sum_{i=1}^{n+1}a_{i}-a_{n}}{2}$ many $\mathbb{Z}_{2}$
lattices in which

$\left|\mathcal{P}\left(\Delta^{\mathcal{G}}_{n},E\times
X\right)\right|=\left|\mathcal{P}(\mathbb{Z}_{1},X)\right|^{a_{n}}\left|\mathcal{P}\left(\mathbb{Z}_{2},X\right)\right|^{\frac{\sum_{i=1}^{n+1}a_{i}-a_{n}}{2}},$

where $\Delta^{\mathcal{G}}_{n}=\\{g\in\mathcal{G}:|g|\leq n\\}$ is the
$n$-block of the golden-mean tree and $a_{n+2}=a_{n+1}+a_{n}$ with $a_{1}=1$
and $a_{2}=2$. Indeed, due to the rule of $E\times X$, we divide the
$\Delta^{\mathcal{G}}_{n}$ into $|\mathcal{T}_{n-1}^{\mathcal{G}}|$ many
$\mathbb{Z}_{1}$ lattices and all the other lattices of the partition of
$\Delta^{\mathcal{G}}_{n}$ are $\mathbb{Z}_{2}$ in which

$\left|\mathcal{P}\left(\Delta^{\mathcal{G}}_{n},E\times
X\right)\right|=\left|\mathcal{P}(\mathbb{Z}_{1},X)\right|^{|\mathcal{T}_{n-1}^{\mathcal{G}}|}\left|\mathcal{P}\left(\mathbb{Z}_{2},X\right)\right|^{\frac{\left|\Delta_{n}^{\mathcal{G}}\right|-\left|\mathcal{T}_{n-1}^{\mathcal{G}}\right|}{2}},$

where $\mathcal{T}_{n}^{\mathcal{G}}=\\{g\in\mathcal{G}:|g|=n\\}$. The proof
of claim is completed by
$\left|\Delta_{n}^{\mathcal{G}}\right|=\sum_{i=1}^{n+1}a_{i}$ and
$\left|\mathcal{T}_{n-1}^{\mathcal{G}}\right|=a_{n}$. By the construction of
the partitions based on $X\times E$ (See the left hand side of Figure 4 for
the partitions of $\Delta_{1}^{\mathcal{G}}$, $\Delta_{2}^{\mathcal{G}}$ and
$\Delta_{3}^{\mathcal{G}}$ based on $E\times X$.), thus we have

Figure 4. The left (resp. right) hand side illustrate the partition of
$\Delta_{1}^{\mathcal{G}}$, $\Delta_{2}^{\mathcal{G}}$ and
$\Delta_{3}^{\mathcal{G}}$ based on $E\times X$ (resp. $X\times E$).

$\displaystyle h\left(E\times X\right)$
$\displaystyle=\lim_{n\to\infty}\frac{\log\left|\mathcal{P}\left(\Delta^{\mathcal{G}}_{n},E\times
X\right)\right|}{\left|\Delta^{\mathcal{G}}_{n}\right|}$
$\displaystyle=\lim_{n\to\infty}\frac{\log\left|\mathcal{P}(\mathbb{Z}_{1},X)\right|^{a_{n}}\left|\mathcal{P}\left(\mathbb{Z}_{2},X\right)\right|^{\frac{\sum_{i=1}^{n+1}a_{i}-a_{n}}{2}}}{a_{1}+a_{2}+\cdots+a_{n+1}}$
$\displaystyle=\lim_{n\to\infty}\frac{a_{n}\log\left|\mathcal{P}(\mathbb{Z}_{1},X)\right|+\frac{1}{2}\left(\sum_{i=1}^{n+1}a_{i}-a_{n}\right)\log\left|\mathcal{P}\left(\mathbb{Z}_{2},X\right)\right|}{a_{1}+a_{2}+\cdots+a_{n+1}}$
$\displaystyle=\lim_{n\to\infty}\frac{a_{n}\log\left|\mathcal{P}(\mathbb{Z}_{1},X)\right|+\frac{1}{2}\left(a_{n+3}-2-a_{n}\right)\log\left|\mathcal{P}\left(\mathbb{Z}_{2},X\right)\right|}{a_{n+3}-2}$
$\displaystyle=\lim_{n\to\infty}\frac{a_{n}\log\left|\mathcal{P}(\mathbb{Z}_{1},X)\right|+\left(a_{n+1}-1\right)\log\left|\mathcal{P}\left(\mathbb{Z}_{2},X\right)\right|}{a_{n+3}-2}$
$\displaystyle=\frac{1}{\rho^{3}}\log\left|\mathcal{P}(\mathbb{Z}_{1},X)\right|+\frac{1}{\rho^{2}}\log\left|\mathcal{P}\left(\mathbb{Z}_{2},X\right)\right|,$

where $\rho=\frac{1+\sqrt{5}}{2}$.

For the entropy of $X\times E$, we claim that for $n\geq 2$,
$\Delta_{n}^{\mathcal{G}}$ can be divided into a $\mathbb{Z}_{n+1}$ lattice, a
$\mathbb{Z}_{n}$ lattice and $a_{n-i}$ many $\mathbb{Z}_{i}$ lattices for all
$1\leq i\leq n-1$ in which

$\left|\mathcal{P}\left(\Delta^{\mathcal{G}}_{n},X\times
E\right)\right|=\left|\mathcal{P}\left(\mathbb{Z}_{n+1},X\right)\right|\left|\mathcal{P}\left(\mathbb{Z}_{n},X\right)\right|\prod_{i=1}^{n-1}\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|^{a_{n-i}}.$

Indeed, due to the rule of $E\times X$, for $n=2$, we divide the
$\Delta_{2}^{\mathcal{G}}$ into a $\mathbb{Z}_{3}$ lattice, a $\mathbb{Z}_{2}$
lattice and $|\mathcal{T}_{0}^{\mathcal{G}}|=a_{1}=1$ many $\mathbb{Z}_{1}$
lattices in which

$\left|\mathcal{P}\left(\Delta^{\mathcal{G}}_{2},X\times
E\right)\right|=\left|\mathcal{P}\left(\mathbb{Z}_{3},X\right)\right|\left|\mathcal{P}\left(\mathbb{Z}_{2},X\right)\right|\left|\mathcal{P}\left(\mathbb{Z}_{1},X\right)\right|^{a_{1}}.$

Assume the claim holds for $n=k$, that is, we divide
$\Delta_{k}^{\mathcal{G}}$ into a $\mathbb{Z}_{k+1}$ lattice, a
$\mathbb{Z}_{k}$ lattice and $a_{k-i}$ many $\mathbb{Z}_{i}$ lattices for all
$1\leq i\leq k-1$ in which

$\left|\mathcal{P}\left(\Delta^{\mathcal{G}}_{k},X\times
E\right)\right|=\left|\mathcal{P}\left(\mathbb{Z}_{k+1},X\right)\right|\left|\mathcal{P}\left(\mathbb{Z}_{k},X\right)\right|\prod_{i=1}^{k-1}\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|^{a_{k-i}}.$

Since the $\mathbb{Z}_{i}$ of the partition of $\Delta_{k}^{\mathcal{G}}$ can
be extended to the $\mathbb{Z}_{i+1}$ of the partition of
$\Delta_{k+1}^{\mathcal{G}}$ for all $1\leq i\leq k+1$, and the number of
$\mathbb{Z}_{1}$ of the partition of $\Delta_{j}^{\mathcal{G}}$ is $a_{j-2}$
for all $j\geq 2$. Then by assumption of claim, the
$\Delta_{k+1}^{\mathcal{G}}$ can be divided into a $\mathbb{Z}_{k+2}$ lattice,
a $\mathbb{Z}_{k+1}$ lattice and $a_{k+1-i}$ many $\mathbb{Z}_{i}$ lattices
for all $1\leq i\leq k$ in which

$\left|\mathcal{P}\left(\Delta^{\mathcal{G}}_{k},X\times
E\right)\right|=\left|\mathcal{P}\left(\mathbb{Z}_{k+2},X\right)\right|\left|\mathcal{P}\left(\mathbb{Z}_{k+1},X\right)\right|\prod_{i=1}^{k}\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|^{a_{k+1-i}}.$

The proof of claim is complete. By the construction of the partitions based on
$X\times E$ (See the right hand side of Figure 4 for the partitions of
$\Delta_{1}^{\mathcal{G}}$, $\Delta_{2}^{\mathcal{G}}$ and
$\Delta_{3}^{\mathcal{G}}$ based on $X\times E$.), thus we have

$\displaystyle h\left(X\times E\right)$
$\displaystyle=\lim_{n\to\infty}\frac{\log\left|\mathcal{P}\left(\Delta^{\mathcal{G}}_{n},X\times
E\right)\right|}{\left|\Delta^{\mathcal{G}}_{n}\right|}$
$\displaystyle=\lim_{n\to\infty}\frac{\log\left|\mathcal{P}\left(\mathbb{Z}_{n+1},X\right)\right|\left|\mathcal{P}\left(\mathbb{Z}_{n},X\right)\right|\prod_{i=1}^{n-1}\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|^{a_{n-i}}}{a_{1}+a_{2}+\cdots+a_{n+1}}$
$\displaystyle=\lim_{n\to\infty}\frac{\sum_{i=1}^{n-1}a_{n-i}\log\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|}{a_{n+3}-2}$
$\displaystyle=\sum_{i=1}^{\infty}\frac{\log\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|}{\rho^{i+3}}.$

The proof is complete. ∎

######  Remark 4.2.

We remark that if $X_{A}$ is a full shift with $A=\left[\begin{matrix}1&1\\\
1&1\end{matrix}\right]$, then
$\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|=2^{i}$ for all $i\geq
1$. This implies $h(E\times X)=h(X\times E)=h(X)=\log 2$. However, the
equalities do not hold in general. If we let $X=X_{A}$ be an SFT with
$A=\left[\begin{matrix}1&0\\\ 0&1\end{matrix}\right]$, then
$\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|=2$ for all $i\geq 1$.
This implies $h(E\times X)=(\frac{1}{\rho^{3}}+\frac{1}{\rho^{2}})\log 2$,
$h(X\times E)=\frac{1}{\rho^{2}}\log 2$ and $h(X)=0$.

In the following , we consider $h(E\times X)$ and $h(X\times E)$ on the
Markov-Cayley trees.

######  Theorem 4.3 (General trees).

If the Markov-Cayley tree $\mathcal{T}$ is a subtree of $\mathcal{T}_{2}$
satisfies $\gamma>1$ where
$\gamma=\lim_{n\to\infty}\frac{|\mathcal{T}_{n+1}|}{|\mathcal{T}_{n}|}$ and
$\mathcal{T}_{n}=\\{g\in\mathcal{T}:|g|=n\\}$, then $h(E\times X)>h(X)$ and
$h(X\times E)>h(X)$ when $X$ is not a full shift.

There is a Markov-Cayley tree $\mathcal{T}$ that satisfies $\gamma=1$ such
that $h(E\times X)=h(X)$ for all $X=X_{A}$ that is an SFT with primitive
transition matrix $A$.

###### Proof.

Since $\gamma>1$ and $\mathcal{T}$ is a subtree of $\mathcal{T}_{2}$, the
ratio of 2 branching
$p_{n}=\frac{|\\{g\in\mathcal{T}_{n}:gf_{1},gf_{2}\in\mathcal{T}_{n+1}\\}|}{|\mathcal{T}_{n}|}$
has a limit $p=\lim_{n\to\infty}p_{n}$ that is bounded away from zero. Due to
the rule of $E\times X$, the $\Delta_{n}$ of $\mathcal{T}$ can be divided into
$a_{n;i}$ many $\mathbb{Z}_{i}$, $1\leq i\leq n$ in which

$|\mathcal{P}(\Delta_{n}^{\mathcal{T}},E\times
X)|=\prod_{i=1}^{n}|\mathcal{P}(\mathbb{Z}_{i},X)|^{a_{n;i}},$

where
$a_{n;1}\geq|\\{g\in\mathcal{T}_{n-1}:gf_{1},gf_{2}\in\mathcal{T}_{n}\\}|$.

Thus,

$\displaystyle h(E\times X)$
$\displaystyle=\limsup_{n\to\infty}\frac{\log|\mathcal{P}(\Delta_{n}^{\mathcal{T}},E\times
X)|}{|\Delta_{n}^{\mathcal{T}}|}$
$\displaystyle=\limsup_{n\to\infty}\frac{\log\prod_{i=1}^{n}|\mathcal{P}(\mathbb{Z}_{i},X)|^{a_{n;i}}}{|\Delta_{n}^{\mathcal{T}}|}$
$\displaystyle=\limsup_{n\to\infty}\sum_{i=1}^{n}\frac{\log|\mathcal{P}(\mathbb{Z}_{i},X)|^{a_{n;i}}}{|\Delta_{n}^{\mathcal{T}}|}.$

Since $\log|\mathcal{P}(\mathbb{Z}_{n},X)|\geq nh(X)$ for all $n\geq 1$, we
have

$\displaystyle h(E\times
X)\geq\limsup_{n\to\infty}\sum_{i=1}^{n}\frac{a_{n;i}nh(X)}{|\Delta_{n}^{\mathcal{T}}|}=h(X).$

Since $X$ is not a full shift, it is clear that
$\log|\mathcal{P}(\mathbb{Z}_{1},X)|>h(X)$. Then, it is enought to show that
$\lim_{n\to\infty}\frac{a_{n;1}}{|\Delta_{n}^{\mathcal{T}}|}>0$ for proving
$h(E\times X)>h(X)$. Indeed, it is equivalent to show the following limit
exists and positive. We now claim

$\lim_{n\to\infty}\frac{|\mathcal{T}_{n}|}{\sum_{i=1}^{n}|\mathcal{T}_{i}|}p_{n}=\frac{\gamma-1}{\gamma}p.$

For any $0<\epsilon<\frac{1}{2\gamma}$, since
$\frac{1}{\gamma}=\lim_{n\to\infty}\frac{|\mathcal{T}_{n}|}{|\mathcal{T}_{n+1}|}$,
there is an $m\in\mathbb{N}$ such that
$\frac{1}{\gamma}-\epsilon<\frac{|\mathcal{T}_{n}|}{|\mathcal{T}_{n+1}|}<\frac{1}{\gamma}+\epsilon$
for all $n\geq m$. Then for $n\geq m$, we have

$\displaystyle\frac{1}{m(\frac{1}{\gamma}+\epsilon)^{n-m}+\frac{\gamma}{\gamma-1-\gamma\epsilon}}\leq\frac{1}{m\frac{|\mathcal{T}_{m}|}{|\mathcal{T}_{n}|}+\sum_{i=0}^{\infty}(\frac{1}{\gamma}+\epsilon)^{i}}$
$\displaystyle\leq\frac{|\mathcal{T}_{n}|}{\sum_{i=1}^{n}|\mathcal{T}_{i}|}=\frac{1}{\sum_{i=1}^{m-1}\frac{|\mathcal{T}_{i}|}{|\mathcal{T}_{n}|}+\sum_{i=m}^{n}\frac{|\mathcal{T}_{i}|}{|\mathcal{T}_{n}|}}$
$\displaystyle\leq\frac{1}{\sum_{i=1}^{m-1}\frac{|\mathcal{T}_{i}|}{|\mathcal{T}_{n}|}+\sum_{i=0}^{n-m}(\frac{1}{\gamma}-\epsilon)^{i}}\leq\frac{1}{\sum_{i=0}^{n-m}(\frac{1}{\gamma}-\epsilon)^{i}}.$

Taking $n\to\infty$, we have

$\frac{\gamma-1-\gamma\epsilon}{\gamma}\leq\lim_{n\to\infty}\frac{|\mathcal{T}_{n}|}{\sum_{i=1}^{n}|\mathcal{T}_{i}|}\leq\frac{\gamma-1+\gamma\epsilon}{\gamma}.$

Since $\epsilon$ is arbitrary, we obtain
$\lim_{n\to\infty}\frac{|\mathcal{T}_{n}|}{\sum_{i=1}^{n}|\mathcal{T}_{i}|}=\frac{\gamma-1}{\gamma}$.
The proof of claim is complete.

Therefore, we have $h(E\times X)>h(X)$. The proof of $h(X\times E)>h(X)$ is
similar and thus we omit it.

Let $\mathcal{G}_{1}$ be a Markov-Cayley tree with the adjacency matrix
$M=\left[\begin{matrix}1&1\\\ 0&1\end{matrix}\right]$. Due to the rule of
$E\times X$, for $n\geq 2$, the $\Delta_{n}^{\mathcal{G}_{1}}$ can be divided
into a $\mathbb{Z}_{1}$, a $\mathbb{Z}_{2}$, … and a $\mathbb{Z}_{n+1}$ in
which

$\left|\mathcal{P}\left(\Delta^{\mathcal{G}_{1}}_{n},E\times
X\right)\right|=\prod_{i=1}^{n+1}\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|.$

See Figure 5 for the partitions of $\Delta_{1}^{\mathcal{G}_{1}}$,
$\Delta_{2}^{\mathcal{G}_{1}}$ and $\Delta_{3}^{\mathcal{G}_{1}}$ based on
$E\times X$.

Figure 5. The partitions of $\Delta_{1}^{\mathcal{G}_{1}}$,
$\Delta_{2}^{\mathcal{G}_{1}}$ and $\Delta_{3}^{\mathcal{G}_{1}}$ based on
$E\times X$.

Thus, we have the entropy of $E\times X$ on $\mathcal{G}_{1}$ is equal to

$\displaystyle h(E\times X)$
$\displaystyle=\lim_{n\to\infty}\frac{\log\left|\mathcal{P}\left(\Delta^{\mathcal{G}_{1}}_{n},E\times
X\right)\right|}{|\Delta^{\mathcal{G}_{1}}_{n}|}=\lim_{n\to\infty}\frac{\log\prod_{i=1}^{n+1}\left|\mathcal{P}\left(\mathbb{Z}_{i},X\right)\right|}{1+2+\cdots+(n+1)}=\lim_{n\to\infty}\frac{\sum_{i=1}^{n+1}\log\left|A^{i}\right|}{\frac{(n+1)(n+2)}{2}}.$

Assume $A$ is a primitive matrix and let $X=X_{A}$, there are constants
$c_{1}$ and $c_{2}$ such that $c_{1}\lambda_{A}^{i}\leq|A^{i}|\leq
c_{2}\lambda_{A}^{i}$ for all $i\geq 1$. Then we have

$\displaystyle\log\lambda_{A}=\lim_{n\to\infty}\frac{\sum_{i=1}^{n+1}\log
c_{1}\lambda_{A}^{i}}{\frac{(n+1)(n+2)}{2}}\leq h(E\times X)$
$\displaystyle\leq\lim_{n\to\infty}\frac{\sum_{i=1}^{n+1}\log
c_{2}\lambda_{A}^{i}}{\frac{(n+1)(n+2)}{2}}=\log\lambda_{A}.$

The proof is complete by $h(X)=\log\lambda_{A}$. ∎

## References

  * [1] I. Assani and K. Presser, _A survey of the return times theorem_ , Ergodic Theory and Dynamical Systems. Proceedings of the Ergodic Theory workshops at University of North Carolina at Chapel Hill, 2011–2012, 2014, p. 126.
  * [2] N. Aubrun and M.-P. Beal, _Tree-shifts of finite type_ , Theoretical Computer Science 459 (2012), 16–25.
  * [3] J.-C. Ban and C.-H. Chang, _Tree-shifts: Irreducibility, mixing, and chaos of tree-shifts_ , Trans. Am. Math. Soc. 369 (2017), 8389–8407.
  * [4] by same author, _Tree-shifts: The entropy of tree-shifts of finite type_ , Nonlinearity 30 (2017), 2785–2804.
  * [5] J.-C. Ban, C.-H. Chang, W.-G. Hu, and Y.-L. Wu, _On structure of topological entropy for tree-shift of finite type_ , Journal of Differential Equations 292 (2021), 325–353.
  * [6] by same author, _Topological entropy for shifts of finite type over $\mathbb{Z}$ and trees_, Theoretical Computer Science 930 (2022), 24–32.
  * [7] J.-C. Ban, C.-H. Chang, Y.-L. Wu, and Y.-Y. Wu, _Stem and topological entropy on cayley trees_ , Mathematical Physics, Analysis and Geometry (2022), no. 25.
  * [8] J.-C. Ban, W.-G. Hu, and G.-Y. Lai, _Boundary complexity and surface entropy of 2-multiplicative integer systems on $\mathbb{N}^{d}$_, arXiv preprint arXiv:2210.09115 (2022).
  * [9] A. Callard and P. Vanier, _Computational characterization of surface entropies for $\mathbb{Z}^{2}$ subshifts of finite type_, Leibniz International Proceedings in Informatics (LIPIcs), vol. 198, 2021.
  * [10] T. Ceccherini-Silberstein and M. Coornaert, _Cellular automata and groups_ , Springer Science & Business Media, 2010.
  * [11] N. Chandgotia and B. Marcus, _Mixing properties for hom-shifts and the distance between walks on associated graphs_ , Pacific J. Math. 294 (2018), 41–69.
  * [12] A. Desai, _Subsystem entropy for $\mathbb{Z}^{d}$ sofic shifts_, Indagationes Mathematicae 17 (2006), no. 3, 353–359.
  * [13] A.-H. Fan, _Some aspects of multifractal analysis_ , Geometry and Analysis of Fractals, Springer, 2014, pp. 115–145.
  * [14] A.-H. Fan, L.-M. Liao, and J.-H. Ma, _Level sets of multiple ergodic averages_ , Monatshefte für Mathematik 168 (2012), no. 1, 17–26.
  * [15] M. Hochman and T. Meyerovitch, _A characterization of the entropies of multidimensional shifts of finite type_ , Ann. of Math. 171 (2010), 2011–2038.
  * [16] A. Johnson, S. Kass, and K. Madden, _Projectional entropy in higher dimensional shifts of finite type_ , Complex Systems 17 (2007), no. 3, 243.
  * [17] R. Kenyon, Y. Peres, and B. Solomyak, _Hausdorff dimension for fractals invariant under multiplicative integers_ , Ergodic Theory and Dynamical Systems 32 (2012), no. 5, 1567–1584.
  * [18] S. Kolyada and L. Snoha, _Some aspects of topological transitivity–a survey_ , Grazer Math. Ber. 334 (1997), 3–35.
  * [19] D. Lind, _The entropies of topological Markov shifts and a related class of algebraic integers_ , Ergodic Theory and Dynamical Systems 4 (1984), no. 2, 283–300.
  * [20] D. Lind and B. Marcus, _An introduction to symbolic dynamics and coding_ , Cambridge University Press, Cambridge, 1995.
  * [21] E. Louidor, B. Marcus, and R. Pavlov, _Independence entropy of $\mathbb{Z}^{d}$ shift spaces_, Acta Applicandae Mathematicae 126 (2013), no. 1, 297–317.
  * [22] T. Meyerovitch, _Growth-type invariants for $\mathbb{Z}^{d}$ subshifts of finite type and classes arithmetical of real numbers_, Inventiones Mathematicae 184 (2011), no. 3, 567–589.
  * [23] T. Meyerovitch and R. Pavlov, _On independence and entropy for high-dimensional isotropic subshifts_ , Proceedings of the London Mathematical Society 109 (2014), no. 4, 921–945.
  * [24] D. Pace, _Surface entropy of shifts of finite type_ , Ph.D. thesis, University of Denver, 2018.
  * [25] K. Petersen and I. Salama, _Tree shift topological entropy_ , Theoret. Comput. Sci. 743 (2018), 64–71.
  * [26] by same author, _Entropy on regular trees_ , Discrete & Continuous Dynamical Systems 40 (2020), no. 7, 4453.
  * [27] by same author, _Asymptotic pressure on some self-similar trees_ , Stochastics and Dynamics (2021).
  * [28] M. Schraudner, _Projectional entropy and the electrical wire shift_ , Discrete & Continuous Dynamical Systems 26 (2010), no. 1, 333.