1. Introduction
In topological dynamics, the study of chaotic behaviour of a dynamical system is often based on some properties of continuous functions. One of the frequently studied properties of such functions in the theory of topological dynamical systems is the entropy of a continuous function
$f:X\rightarrow X$
on a compact metric space X, which serves as a measure of the complexity of the dynamical system. This often leads to studying the entropy of the shift map
$\sigma $
on the inverse limit
$\varprojlim (X,f)$
. More precisely, suppose X is a compact metric space. If
$f:X \to X$
is a continuous function, the inverse limit space generated by f is the subspace of
$\prod _{i={ 0}}^{\infty } X $
equipped with the usual product topology, given by
$$ \begin{align*} \varprojlim(X,f):=\bigg\{({ x_0},x_{1},x_{2},x_{3},\ldots) \in \prod_{i={ 0}}^{\infty} X \mid \text{for each non-negative integer } i, x_{i}= f(x_{i+1})\bigg\}, \end{align*} $$
also abbreviated as
$\varprojlim f$
. The map f on X induces a natural homeomorphism
$\sigma $
on
$\varprojlim f$
, called the shift map, defined by
$$ \begin{align*}\sigma({ x_0},x_{1},x_{2},x_3, x_4,\ldots)=({ x_1}, x_{2},x_{3},x_4,\ldots)\end{align*} $$
for each
$({ x_0},x_{1},x_{2},x_3, x_4,\ldots )$
in
$\varprojlim f$
. Note that the shift map
$\sigma $
is the inverse function to the map defined by
$({ x_0},x_1,x_2,\ldots )\mapsto (f(x_0),x_0,x_1,x_2,\ldots )$
.
To study such inverse limits
$\varprojlim f$
and shift maps
$\sigma :\varprojlim f\rightarrow \varprojlim f$
, the study of backward orbits of points of dynamical systems
$(X,f)$
is also required; note that the inverse limit
$\varprojlim f$
is the space of all backward orbits in
$(X,f)$
. Such backward orbits of points are actually forward orbits of points in the dynamical system
$(X,f^{-1})$
if
$f^{-1}$
is well defined. However, usually,
$f^{-1}$
is not a well-defined function; therefore, a more general tool is needed to study these properties. Note that for a continuous function
$f:X\rightarrow X$
, the set
$$ \begin{align*} \Gamma(f)^{-1}=\{(y,x) \in X\times X \mid y=f(x)\} \end{align*} $$
is a closed relation on X that describes best the dynamics of
$(X,f)$
in the backward direction when
$f^{-1}$
is not well defined. So, generalizing topological dynamical systems
$(X,f)$
to topological dynamical systems
$(X,{ F})$
with closed relations F on X by making the identification
$(X,f)=(X,\Gamma (f))$
is only natural. So, what we are engaged in is the replacement of a map
$f:X\rightarrow X$
by a closed relation F on X.
Recently, many such generalizations of dynamical systems were introduced and studied (see [Reference Banič, Erceg, Gril Rogina and Kennedy1, Reference Banič, Erceg and Kennedy2, Reference Cordeiro and Pacfico7, Reference Kelly and Woods12, Reference Kennedy and Nall14–Reference Metzger, Arnoldo Morales Rojas and Thieullen19, Reference Romn-Flores22], where more references may be found). However, there is not much known of such dynamical systems and, therefore, there are many properties of such set-valued dynamical systems that are yet to be studied. In [Reference Banič, Erceg and Kennedy2], the notion of topological entropy
$h(f)$
of continuous functions
$f:X\rightarrow X$
on compact metric spaces X was generalized to the notion of topological entropy
${\mathrm {ent}}({ F})$
of closed relations F on compact metric spaces X. In this paper, we continue our research from [Reference Banič, Erceg and Kennedy2]. We introduce the notions of returns and well-aligned sets for closed relations on compact metric spaces, and then use them to obtain non-trivial sufficient conditions for such relations to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy. We also show that, unlike topological entropy for closed relations on compact metric spaces in general, in the case of finite relations, positive entropy is equivalent to the shift map on the Mahavier product being Li–Yorke chaotic as well as equivalent to DC-2 distributional chaos for the shift map, as well as equivalent to F having a
$(k,\varepsilon )$
-return.
When constructing a closed relation F on a compact metric space X, a standard topological dynamical system
$(X_F^+,\sigma )$
, where
$X_F^+$
is the Mahavier product of the relation F and
$\sigma $
is the shift map on
$X_F^+$
, is constructed at the same time. This is a new way of constructing topological dynamical systems with interesting properties. Therefore, it is only natural to study the properties of the relation F that imply interesting topological or dynamical properties of the dynamical system
$(X_F^+,\sigma )$
. Here, we study the properties of the closed relation F on X (such as returns and well-aligned sets) that imply that the shift map
$\sigma $
on
$X_F^+$
has positive entropy. Then, for example, one can apply these results to obtain mappings on compact metric spaces with non-zero entropy. Oprocha constructed in [Reference Oprocha21] a transitive zero-entropy homeomorphism on the Lelek fan. He asked then if there existed also a transitive non-zero entropy homeomorphism on the Lelek fan [Reference Oprocha21, Question 3.11]. In [Reference Banič, Erceg and Kennedy3], Banič, Erceg and Kennedy constructed a transitive homeomorphism on the Lelek fan. The construction was based on producing a closed relation F on
$X=[0,1]$
in such a way that the Mahavier product
$X_F^+$
is a Lelek fan and the shift map
$\sigma $
on it is transitive. In [Reference Banič, Erceg and Kennedy2], the authors showed that the entropy of the relation F from [Reference Banič, Erceg and Kennedy3] has positive entropy. In Example 4.12, we use the introduced theory of returns to shorten and simplify that proof from [Reference Banič, Erceg and Kennedy2] and then we use Theorem 3.16 to show that this implies that the shift map
$\sigma $
on
$X_F^+$
from [Reference Banič, Erceg and Kennedy3] has non-zero entropy. This gives an answer to Oprocha’s question [Reference Oprocha21, Question 3.11] in the affirmative. The same technique of closed relations on compact metric spaces was used later in [Reference Banič, Erceg, Kennedy, Mouron and Nall4] to produce a transitive homeomorphism on the Cantor fan. The notions of returns can be used to give simple proofs that a transitive homeomorphism on the Cantor fan from [Reference Banič, Erceg, Kennedy, Mouron and Nall4] also has positive entropy. See Examples 4.10 and 4.11.
We proceed as follows. In §2, basic definitions and notation that are needed later in the paper are given and presented. In §3, the topological entropy for closed relations is defined and in addition, basic results from [Reference Banič, Erceg and Kennedy2] are presented. In §4, our first main result as well as illustrative examples and corollaries are given and proved. In our last section, §5, we restrict ourselves to finite relations on compact metric spaces. Here, our second main result, a characterization of finite relations with non-zero entropy, is presented and proved.
2. Definitions and notation
First, we define some properties from the continuum theory and the theory of inverse limits that are used later in the paper.
Definition 2.1. Let X be a non-empty compact metric space. We always use
$\rho $
to denote the metric on X.
Definition 2.2. Suppose X is a non-empty compact metric space. If for each non-negative integer n,
$f_n:X \to X$
is a continuous function, the inverse limit space generated by
$(f_n)$
is the subspace of
$\prod _{i={ 0}}^{\infty } X $
equipped with the usual product topology, given by
$$ \begin{align*} \varprojlim(X,f_n):=\bigg\{({ x_0},x_{1},x_{2},x_{3},\ldots ) \in \prod_{i={ 0}}^{\infty}\! X \mid \text{for each non-negative integer } i, x_{i}= f_i(x_{i+1})\bigg\}. \end{align*} $$
Definition 2.3. A continuum is a non-empty connected compact metric space. A continuum is degenerate if it consists of only a single point. Otherwise it is non-degenerate. A subcontinuum is a subspace of a continuum which itself is a continuum.
Next, we define chainable continua (using inverse limits); see [Reference Nadler20, §XII] for more details.
Definition 2.4. A continuum X is chainable if there is a sequence
$(f_n)$
of continuous surjections
$f_n:[0,1]\rightarrow [0,1]$
such that X is homeomorphic to
$\varprojlim ([0,1],f_n)$
.
Definition 2.5. A continuum X is decomposable if there are proper subcontinua A and B of X (
$A,B\neq X$
) such that
$X=A\cup B$
. A continuum is indecomposable if it is not decomposable. A continuum is hereditarily indecomposable if each of its subcontinua is indecomposable.
Definition 2.6. A pseudoarc is any non-degenerate hereditarily indecomposable chainable continuum.
Bing showed in [Reference Bing5] that any two pseudoarcs are homeomorphic. Next, we present basic definitions and well-known results about closed relations and Mahavier products.
Definition 2.7. We use:
-
(1)
$\mathbb N$
to denote the set of positive integers; -
(2)
$\mathbb N_m$
to denote the set
$\{1,2,3,\ldots ,m\}$
for each positive integer m; -
(3)
${\mathbb Z}_+$
to denote the set of non-negative integers; and -
(4)
${\mathbb Z}$
to denote the set of integers.
Let m be a positive integer. For any set X, we use:
-
(1)
$X^m$
to denote the Cartesian product
$\prod _{i=0}^{m-1}X$
; -
(2)
$X^{{\mathbb Z}_+}$
to denote the Cartesian product
$\prod _{i=0}^{\infty }X$
.
For each non-negative integer i and for each
${\mathbf {x}}=(x_0,x_1,x_2,\ldots )\in X^{{\mathbb Z}_+}$
( or
${\mathbf {x}}=(x_0,x_1,x_2,\ldots ,x_{m-1})\in X^{m}$
), we use
$\pi _i({\mathbf {x}})$
or
${\mathbf {x}}(i)$
or
${\mathbf {x}}_i$
to denote the ith coordinate
$x_i$
of the point
${\mathbf {x}}$
. For all non-negative integers k and
$\ell $
such that
$k\leq \ell $
, we use
$[k,\ell ]$
to denote the set
$\{k, k+1,k+2,\ldots ,\ell \}$
and
$\pi _{[k,\ell ]}:\prod _{i=0}^{\infty }X\rightarrow \prod _{i=k}^{\ell }X$
to denote the projection that is defined by
$ \pi _{[k,\ell ]}(x_0,x_1,x_2,\ldots ,x_k,x_{k+1},\ldots , x_{\ell },x_{\ell +1},\ldots )=(x_k,x_{k+1},x_{k+2},\ldots , x_{\ell }). $
For
$k=\ell $
, we use
$\pi _k$
to denote the projection
$\pi _{[k,k]}$
. For a non-empty compact metric space X, we use
$p_1:X\times X\rightarrow X$
and
$p_2:X\times X\rightarrow X$
to denote the standard projections defined by
$p_1(s,t)=s$
and
$p_2(s,t)=t$
for all
$(s,t)\in X\times X$
.
Definition 2.8. Let X and Y be metric spaces, and let
$f:X\rightarrow Y$
be a function. We use
$\Gamma (f)=\{(x,y)\in X\times Y \mid y=f(x)\}$
to denote the graph of the function f.
Definition 2.9. Let X be a non-empty compact metric space and let
${F}\subseteq X\times X$
be a relation on X. If F is closed in
$X\times X$
, then we say that F is a closed relation on X.
Definition 2.10. Let X be a set and let F be a relation on X. Then we define
${F}^{-1}=\{(y,x)\in X\times X \mid (x,y)\in { F}\}$
to be the inverse relation of the relation F on X.
Definition 2.11. Let X be a non-empty compact metric space and let F be a closed relation on X. Then we call
$$ \begin{align*} X_F^m=\bigg\{({ x_0},x_1,x_2,x_3,\ldots ,{ x_m})\in \prod_{i={ 0}}^{{ m}}\!X \mid \text{for each } i\in \{{ 0,1,2},\ldots ,m{ -1}\}, (x_{i},x_{i+1})\in { F}\bigg\} \end{align*} $$
for each positive integer m, the mth Mahavier product of F, and we call
$$ \begin{align*} X_F^+=\bigg\{({ x_0},x_1,x_2,x_3,\ldots )\in \prod_{i={ 0}}^{\infty}X \mid \text{for each { non-negative} integer } i, (x_{i},x_{i+1})\in { F}\bigg\} \end{align*} $$
the infinite Mahavier product of F.
In our previous papers, we have also used
$\star _{i={ 0}}^{m{ -1}}{ }{ F}$
to denote the mth Mahavier product and
$\star _{i=0}^{\infty }{ }{ F}$
to denote the infinite Mahavier product of F.
Definition 2.12. Let X be a non-empty compact metric space and let F be a closed relation on X. The function
$\sigma : { X_F^+} \rightarrow { X_F^+}$
, defined by
$$ \begin{align*} \sigma ({ x_0,x_1,x_2,x_3},\ldots)=({ x_1,x_2,x_3},\ldots) \end{align*} $$
for each
$({ x_0,x_1,x_2,x_3},\ldots )\in { X_F^+}$
, is called the shift map on
$X_F^+$
.
Definition 2.13. We use
$\Sigma _2$
to denote the set
$$ \begin{align*} \Sigma_2= \{ {\mathbf{s}}:{ \mathbb{Z}_+} \rightarrow \{0,1\} \} \end{align*} $$
and for each non-negative integer j, we use
$\Sigma _2^j$
to denote the set
$$ \begin{align*} \Sigma_2^j = \{ {\mathbf{s}}:{ [0,j]}\rightarrow \{0,1\} \}. \end{align*} $$
Finally, we define the operation
$\star $
on Mahavier products.
Definition 2.14. Let X be a non-empty compact metric space and let F be a closed relation on X. Also, let
$m,n\in { {\mathbb Z}_+}$
and let
$(x_{ 0},x_{ 1}, x_{ 2},\ldots ,x_{ n}) \in { X_F^n}$
and
$(y_{ 0},y_{ 1},y_{ 2}, \ldots , y_{ m} )\in { X_F^m}$
be such that
$x_{ n}=y_{ 0}$
. Then we define
$(x_{ 0}, x_{ 1}, x_{ 2},\ldots , x_{ n})\star (y_{ 0}, y_{ 1}, y_{ 2}, \ldots , y_{ m})$
by
$$ \begin{align*} (x_{ 0}, x_{ 1}, x_{ 2},\ldots , x_{ n})\star (y_{ 0}, y_{ 1},y_{ 2}, \ldots , y_{ m}) = (x_{ 0}, x_{ 1},x_{ 2},\ldots , x_{ n}, y_{ 1}, y_{ 2},\ldots , y_{ m}). \end{align*} $$
Observation 2.1. Let X be a non-empty compact metric space and let F be a closed relation on X. Also, let
$m,n\in { {\mathbb Z}_+}$
, and let
$(x_{ 0},x_{ 1}, x_{ 2},\ldots ,x_{ n}) \in { X_F^n}$
and
$(y_{ 0},y_{ 1},y_{ 2}, \ldots , y_{ m} )\in { X_F^m}$
be such that
$x_{ n}=y_{ 0}$
. Then
$(x_{ 0}, x_{ 1}, x_{ 2},\ldots , x_{ n})\star (y_{ 0}, y_{ 1},y_{ 2}, \ldots , y_{ m}) \in { X_F^{n+m}}$
.
3. Topological entropy of closed relations on compact metric spaces
In this section, we summarize the generalization of topological entropy to closed relations on a compact metric space introduced in [Reference Banič, Erceg and Kennedy2], where the entropy of a closed relation F on X is defined using Mahavier products
$X_{F^{-1}}^m$
. To simplify the notion of entropy of closed relations and to avoid any possible confusion when dealing with F and
$F^{-1}$
at the same time, we define here the entropy of F using Mahavier products
$X_{F}^m$
. This new definition is equivalent to the definition of entropy from [Reference Banič, Erceg and Kennedy2].
Definition 3.1. Let X be a non-empty compact metric space and let
${\mathcal S}$
be a family of subsets of X. We use
$|{\mathcal S}|$
to denote the cardinality of
${\mathcal S}$
.
Definition 3.2. Let X be a non-empty compact metric space and let
${\mathcal S}$
be a family of subsets of X. For each positive integer n, we use
${\mathcal S}^n$
to denote the family
$$ \begin{align*} {\mathcal S}^n=\{S_{ 0}\times S_{ 1}\times S_{ 2}\times \cdots \times S_{ n-1} \mid S_{ 0},S_{ 1},S_{ 2},\ldots,S_{ n-1}\in {\mathcal S}\}. \end{align*} $$
We call the elements
$S_{ 0}\times S_{ 1}\times S_{ 2}\times \cdots \times S_{ n-1}$
of
${\mathcal S}$
the n-boxes (generated by the family
${\mathcal S}$
).
Definition 3.3. Let X be a non-empty compact metric space and let
${\mathcal U}$
be an open cover for X. We use
$N({\mathcal U})$
to denote
$$ \begin{align*} N({\mathcal U})=\min \{|\mathcal V| \mid \mathcal V \text{ is a finite subset of } {\mathcal U} \text{ that covers } X\}. \end{align*} $$
Definition 3.4. Let X be a non-empty compact metric space, let n be a positive integer, let K be a closed subset of the product
${X^n}$
and let
${\mathcal U}$
be a non-empty family of open subsets of
${X^n}$
such that
$K\subseteq \bigcup {\mathcal U}$
. We use
$N(K,{\mathcal U})$
to denote
$$ \begin{align*} N(K,{\mathcal U})=\min\bigg\{|\mathcal V| \mid \mathcal V \text{ is a non-empty subfamily of } {\mathcal U} \text{ such that } K\subseteq \bigcup \mathcal V\bigg\}. \end{align*} $$
Observation 3.1. Let X be a non-empty compact metric space, let K be a closed subset of the product
$X^n$
and let
${\mathcal U}$
be a non-empty family of open subsets of
$X^n$
such that
$K\subseteq \bigcup {\mathcal U}$
. Note that even in the case when
$K=\emptyset $
,
$N(K,{\mathcal U})$
is a positive integer since
$|\mathcal V|\geq 1$
for each non-empty subfamily
$\mathcal V$
of
${\mathcal U}$
. Also, note that for each positive integer m and for each open cover
$\alpha $
for X,
$$ \begin{align*} 1\leq N({ X_F^m},\alpha^{m+1})\leq (N(\alpha ))^{m+1}, \end{align*} $$
and that for some positive integer m,
$X_F^m$
may be empty. Note that even in this case,
$$ \begin{align*} 1\leq N({ X_F^m},\alpha^{m+1})\leq (N(\alpha ))^{m+1}. \end{align*} $$
Observation 3.2. Let X be a non-empty compact metric space, let F be a closed relation on X such that
$p_2(F)\subseteq p_1(F)$
and let
$\alpha $
be an open cover for X. Note that for each positive integer m,
$$ \begin{align*} N({ X_F^m},\alpha^{m+1})\leq N(X_F^{m+1},\alpha^{m+2}). \end{align*} $$
Lemma 3.3. Let X be a non-empty compact metric space, let F be a closed relation on X and let
$\alpha $
be an open cover for X. Then
$$ \begin{align*} N(X_F^{m+n},\alpha^{m+n+1})\leq N(X_F^m,\alpha^{m+1})\cdot N(X_F^n,\alpha^{n+1}) \end{align*} $$
for all positive integers m and n.
Proof. Let m and n be positive integers, and let:
-
(1)
$N(X_F^m,\alpha ^{m+1})=k_m$
and let
$\gamma =\{\Gamma _0,\Gamma _1,\Gamma _2,\ldots ,\Gamma _{k_{m}-1}\}$
be a subfamily of
$\alpha ^{m+1}$
such that
$X_F^{m}\subseteq \Gamma _0\cup \Gamma _1\cup \Gamma _2\cup \cdots \cup \Gamma _{k_{m}-1}$
; and -
(2)
$N(X_F^n,\alpha ^{n+1})=k_n$
and let
$\delta =\{\Delta _0,\Delta _1,\Delta _2,\ldots ,\Delta _{k_{n}-1}\}$
be a subfamily of
$\alpha ^{n+1}$
such that
$X_F^{n}\subseteq \Delta _0\cup \Delta _1\cup \Delta _2\cup \cdots \cup \Delta _{k_{n}-1}$
.
Observe that
$$ \begin{align*}X_F^m \times X_F^n \subseteq \bigcup_{i=0}^{k_m-1}\bigcup_{j=0}^{k_n-1} {\Gamma_i \times \Delta_j}. \end{align*} $$
Let
$\pi :X_F^m \times X_F^n\rightarrow X_F^m \times X_F^{n-1}$
be the projection which omits the first coordinate in
$X^n_F$
:
$$ \begin{align*} \pi((x_0,x_1,x_2,\ldots,x_{m}),(y_{0},y_{1}, y_2,\ldots ,y_{n}))=((x_0,x_1,x_2,\ldots,x_{m}),(y_{1}, y_2,\ldots ,y_{n})) \end{align*} $$
for each
$((x_0,x_1,x_2,\ldots ,x_{m}),(y_{0},y_{1}, y_2,\ldots ,y_{n}))\in X_F^m \times X_F^n$
. Also, let
$$ \begin{align*} Z = \{({\mathbf{x}}, {\mathbf{y}}) \in X_F^m \times X_F^n \mid {\mathbf{x}}_m = {\mathbf{y}}_0\}. \end{align*} $$
Then
$\pi (Z)$
is homeomorphic to
$X_F^{m+n}$
. Furthermore, note that for each
$i\in \{0,1,2,\ldots , k_m-1\}$
and for each
$j\in \{0,1,2,\ldots ,k_n-1\}$
,
$$ \begin{align*} \pi(\Gamma_i\times \Delta_j) \subseteq \alpha^{m+n+1} \end{align*} $$
and the result follows.
We use the following lemma in the proof of Theorem 3.5.
Lemma 3.4. Let
$(a_m)$
be a sequence in
$\mathbb R$
such that:
-
(1) for each positive integer m,
$a_m\geq 0$
; and -
(2) for all positive integers m and n,
$a_{m+n}\leq a_m+a_n$
.
Then the limit
$\lim _{m\to \infty }{a_m}/{m}$
exists and
$$ \begin{align*} \lim_{m\to \infty}\frac{a_m}{m}=\inf\bigg\{\frac{a_m}{m} \mid m \text{ is a positive integer}\bigg\}. \end{align*} $$
Proof. The proof can be found in [Reference Walters24, Theorem 4.9, p. 87].
Theorem 3.5. Let X be a non-empty compact metric space, let F be a closed relation on X and let
$\alpha $
be an open cover for X. Then the limit
$$ \begin{align*} \lim_{m\to \infty}\frac{\log N({ X_F^m},\alpha^{m+1})}{m} \end{align*} $$
exists.
Proof. For each positive integer m, let
$$ \begin{align*} a_m=\log N(X_F^m,\alpha^{m+1}). \end{align*} $$
By Observation 3.1, for each positive integer m,
$N(X_F^m,\alpha ^{m+1})$
is a positive integer. It follows that for each positive integer m,
$ a_m\geq 0$
. Next, we prove that the above limit exists. By Lemma 3.4, it suffices to show that for all positive integers m and n,
$a_{m+n}\leq a_m+a_n$
. It follows (using Lemma 3.3) that for all positive integers m and n,
$$ \begin{align*} a_{m+n}&=\log N(X_F^{m+n},\alpha^{m+n+1})\leq \log (N(X_F^m,\alpha^{m+1})\cdot N(X_F^n,\alpha^{n+1})) \\ &=\log N(X_F^m,\alpha^{m+1})+\log N(X_F^n,\alpha^{n+1})=a_m+a_n.\\[-32pt] \end{align*} $$
Definition 3.5. Let X be a non-empty compact metric space, let F be a closed relation on X and let
$\alpha $
be an open cover for X. We define the entropy of F with respect to the open cover
$\alpha $
by
$$ \begin{align*} {\mathrm{ent}}({ F},\alpha )=\lim_{m\to \infty}\frac{\log N({ X_F^m},\alpha^{m+1})}{m}. \end{align*} $$
Observation 3.6. Let X be a non-empty compact metric space, let F be a closed relation on X and let
$\alpha $
be an open cover for X. Since X is non-empty, it follows that
$\alpha $
is non-empty. Therefore,
$\log N(X_F^m,\alpha ^{m+1})$
is well defined and is a non-negative number (even in the case when
$X_F^m=\emptyset $
for some positive integer m), since
$N(X_F^m,\alpha ^{m+1})$
is defined in Definition 3.4 as
$$ \begin{align*} N(X_F^m,\alpha^{m+1})=\min\bigg\{|\mathcal V| \mid \mathcal V \textit{ is a non-empty subfamily of } \alpha^{m+1} \text{ such that } X_F^m\subseteq\! \bigcup\! \mathcal V\bigg\}. \end{align*} $$
Therefore, the entropy
${\mathrm {ent}}({ F},\alpha )$
with respect to the open cover
$\alpha $
is always well defined – also in the case when
$X_F^m=\emptyset $
for some positive integer m. Note that if for some positive integer
$m_0$
,
$X_F^{m_0}=\emptyset $
, then for each
$m\geq m_0$
, also
$X_F^{m}=\emptyset $
. It follows that in this case,
$N({ X_F^m},\alpha ^{m+1})=1$
and, therefore,
$$ \begin{align*} {\mathrm{ent}}({ F},\alpha )=\lim_{m\to \infty}\frac{\log N({ X_F^m},\alpha^{m+1})}{m}=\lim_{m\to \infty}\frac{\log1}{m}=0. \end{align*} $$
Definition 3.6. Let X be a metric space and let
${\mathcal S}$
and
${\mathcal T}$
be families of subsets of X. We say that the family
${\mathcal S}$
refines the family
${\mathcal T}$
if for each
$S\in {\mathcal S}$
, there is
$T\in {\mathcal T}$
such that
$S\subseteq T$
. The notation
${\mathcal T}\leq {\mathcal S}$
means that the family
${\mathcal S}$
refines the family
${\mathcal T}$
.
Proposition 3.7. [Reference Banič, Erceg and Kennedy2, Proposition 1]
Let X be a non-empty compact metric space and let F be a closed relation on X. For all non-empty open covers
$\alpha $
and
$\beta $
,
$$ \begin{align*} \alpha\leq \beta \Longrightarrow {\mathrm{ent}}({ F},\alpha )\leq {\mathrm{ent}}({ F},\beta ). \end{align*} $$
Proof. Let
$\alpha $
and
$\beta $
be any open covers for X such that
$\alpha \leq \beta $
. Then
$\alpha ^n\leq \beta ^n$
for each positive integer n. Let m be a positive integer, let
$k=N(X_F^m,\beta ^{m+1})$
and let
$\{B_0,B_1,B_2,\ldots ,B_{k-1}\}\subseteq \beta ^{m+1} $
be such that
$X_F^m\subseteq B_0\cup B_1\cup B_2\cup \cdots \cup B_{k-1}$
. For each
$i\in \{0,1,2,\ldots ,k-1\}$
, let
$A_i\in \alpha $
such that
$B_i\subseteq A_i$
. Therefore,
$X_F^m\subseteq A_0\cup A_1\cup A_2\cup \cdots \cup A_{k-1}$
and
$N(X_F^m,\alpha ^{m+1})\leq N(X_F^m,\beta ^{m+1})$
follows. Therefore,
${\mathrm {ent}}(F,\alpha )\leq {\mathrm {ent}}(F,\beta )$
.
Proposition 3.8. [Reference Banič, Erceg and Kennedy2, Proposition 2]
Let X be a non-empty compact metric space and let
$\alpha $
be a non-empty open cover for X. For all closed relations F and G on X,
$$ \begin{align*} { F}\subseteq G \Longrightarrow {\mathrm{ent}}({ F},\alpha )\leq {\mathrm{ent}}(G,\alpha ). \end{align*} $$
Proof. The proposition follows from the fact that
$N(X_F^m,\alpha ^{m+1})\leq N(X_G^m,\alpha ^{m+1}) $
for each positive integer m.
Definition 3.7. Let X be a non-empty compact metric space, let F be a closed relation on X and let
$E=\{{\mathrm {ent}}({ F},\alpha ) \mid \alpha \text { is an open cover for } X\}$
. We define the entropy of F by
$$ \begin{align*} {\mathrm{ent}}({ F})=\begin{cases} \sup (E)\text{;} & E \text{ is bounded in } \mathbb R, \\ \infty\text{;} & E \text{ is not bounded in } \mathbb R. \end{cases} \end{align*} $$
Observation 3.9. Let X be a non-empty compact metric space and let F be a closed relation on X. Note that if for some positive integer
$m_0$
,
$X_F^{m_0}=\emptyset $
, then, as seen in Observation 3.6, for each open cover
$\alpha $
for X,
$ {\mathrm {ent}}({ F},\alpha )=0. $
Therefore, in this case,
$ E=\{{\mathrm {ent}}({ F},\alpha ) \mid \alpha \text { is an open cover for } X\}=\{0\} $
and since E is bounded in
$\mathbb R$
, it follows that
$ {\mathrm {ent}}(F)=\sup (E)=\sup \{0\}=0. $
The following theorems summarize the basic properties of
${\mathrm {ent}}({ F})$
.
Theorem 3.10. Let X be a non-empty compact metric space. For all closed relations F and G on X,
$$ \begin{align*} { F}\subseteq G \Longrightarrow {\mathrm{ent}}({ F})\leq {\mathrm{ent}}(G). \end{align*} $$
Proof. The theorem follows directly from Proposition 3.8.
Theorem 3.11. Let X be a non-empty compact metric space and let F be a closed relation on X. Then
$$ \begin{align*}{\mathrm{ent}}({ F}^{-1}) = {\mathrm{ent}}({ F}). \end{align*} $$
Proof. Let T be the homeomorphism on
$X^{m+1}$
which reverses the order of the coordinates. Then
$T(X_F^m) =X_{F^{-1}}^m$
and
$T(X_{F^{-1}}^m) =X_{F}^m$
. Note that for any subset U of
$\alpha ^{m+1}$
which covers
$X_F^m$
, the set
$T(U)$
is a subset of
$\alpha ^{m+1}$
which covers
$X_{F^{-1}}^m$
, and that for any subset U of
$\alpha ^{m+1}$
which covers
$X_{F^{-1}}^m$
, the set
$T(U)$
is a subset of
$\alpha ^{m+1}$
which covers
$X_{F}^m$
. The result follows.
In Theorem 3.12, we show that the entropy of closed relations F on a non-empty compact metric space X is a generalization of the well-known topological entropy of continuous functions
$f:X\rightarrow X$
. Before stating and proving the theorem, we give the following definitions.
Definition 3.8. Let X be a set, let
$f:X\rightarrow X$
be a function and let
${\mathcal S}$
be a family of subsets of X. Then we define
$f^{-1}({\mathcal S})=\{f^{-1}(S) \mid S\in {\mathcal S}\}$
.
Definition 3.9. Let X be a set and let
$\mathcal A_0$
,
$\mathcal A_1$
,
$\mathcal A_2 ,\ldots , \mathcal A_{m-1}$
be families of subsets of X. Then we define
$$ \begin{align*} \vee_{i=0}^{m-1}\mathcal A_i=\{A_0\cap A_1\cap A_2\cap \cdots \cap A_{m-1} \mid \text{for each } i\in \{0,1,2,\ldots ,m-1\}, A_i\in \mathcal A_i\}. \end{align*} $$
Definition 3.10. Let X be a non-empty compact metric space and let
$f:X\rightarrow X$
be a continuous function. For any open cover
$\alpha $
for X, we define
$$ \begin{align*} h(f,\alpha)=\lim_{m\to\infty}\frac{\log(N(\vee_{i=0}^{m}f^{-i}(\alpha)))}{m}. \end{align*} $$
Definition 3.11. Let X be a non-empty compact metric space and let
$f:X\rightarrow X$
be a continuous function. Also, let
$E=\{h(f,\alpha ) \mid \alpha \text { is an open cover for } X\}$
. Then we define
$$ \begin{align*} h(f)=\begin{cases} \sup (E)\text{;} & E \text{ is bounded in } \mathbb R, \\ \infty\text{;} & E \text{ is not bounded in } \mathbb R, \end{cases} \end{align*} $$
to be the entropy of the function f.
The following theorem shows that the topological entropy of a continuous function
$f:X\rightarrow X$
on compact metric spaces X is just a special case of the entropy of closed relations F on X. Therefore, the concept
${\mathrm {ent}}(F)$
is a generalization of the concept of
$h(f)$
.
Theorem 3.12. Let X be a non-empty compact metric space and let
$f:X\rightarrow X$
be a continuous function. Then
$$ \begin{align*} {\mathrm{ent}}(\Gamma(f))=h(f). \end{align*} $$
Proof. Observe that for each
$x\in X$
and for all
$A_0, A_1, A_2, \ldots , A_n\in \alpha $
,
$(x, f(x), f^2(x)\ldots , f^n(x)) \in A_0 \times A_1\times A_2\times \cdots \times A_n$
if and only if
$x \in \bigcap _{i=0}^n f^{-i}(A_i)$
. So, a collection of elements of
$\alpha ^{n+1}$
covers
$X_{\Gamma (f)}^n$
if and only if the corresponding elements of
$\vee _{i=0}^{n} f^{-i}(\alpha )$
cover X. Therefore, for any open cover
$\alpha $
for X and for any positive integer m,
$$ \begin{align*} N(X_{\Gamma(f)}^m,\alpha^{m+1})=N(\vee_{i=0}^{m}f^{-i}(\alpha)). \end{align*} $$
Finally, we prove that
${\mathrm {ent}}(\Gamma (f))=h(f)$
:
$$ \begin{align*} {\mathrm{ent}}(\Gamma(f))&=\sup\{{\mathrm{ent}}(\Gamma(f),\alpha) \mid \alpha \text{ is an open cover for } X\}\\ &=\sup\bigg\{\lim_{m\to\infty}\frac{\log N(X_{\Gamma(f)}^m,\alpha^{m+1})}{m} \mid \alpha \text{ is an open cover for } X\bigg\}\\ &=\sup\bigg\{\lim_{m\to\infty}\frac{\log N(\vee_{i=0}^{m}f^{-i}(\alpha))}{m} \mid \alpha \text{ is an open cover for } X\bigg\}\\ &=\sup\{h(f,\alpha) \mid \alpha \text{ is an open cover for } X\}\\ &=h(f). \end{align*} $$
This completes the proof.
For a more thorough discussion of topological entropy
$h(f)$
, see [Reference Walters24]. In Theorem 3.16, we present a relationship between
${\mathrm {ent}}(F)$
and
$h(\sigma )$
. We use the following notation in its proof.
Definition 3.12. Let X be a non-empty compact metric space, let F be a closed relation on X and let
${\mathcal U}$
be an open cover for
$X^{{\mathbb Z}_+}$
. Then we use
${\mathcal U}\cap X_F^+$
to denote
$$ \begin{align*} {\mathcal U}\cap X_F^+=\{U\cap X_F^+ \mid U\in {\mathcal U}\}. \end{align*} $$
Next, let m be a positive integer and let
$\mathcal V$
be an open cover for
$X^m$
. Then we use
$\mathcal V\times X^{{\mathbb Z}_+}$
to denote
$$ \begin{align*} \mathcal V\times X^{{\mathbb Z}_+}=\{V\times X^{{\mathbb Z}_+} \mid V\in \mathcal V\}, \end{align*} $$
which is an open cover for
$X^{{\mathbb Z}_+}$
.
Lemma 3.13. Let X be a non-empty compact metric space, let F be a closed relation on X such that
$p_2(F)\subseteq p_1(F)$
and let
$\sigma $
be the shift map on
${X_F^+}$
. Then
$$ \begin{align*} {\mathrm{ent}}(F,\alpha)=h(\sigma, (\alpha^{m+1}\times X^{{\mathbb Z}_{+}})\cap X_F^+) \end{align*} $$
for any open cover
$\alpha $
for X and for any positive integer m.
Proof. First, we define
$\sigma _X:X^{{\mathbb Z}_+}\rightarrow X^{{\mathbb Z}_+}$
to be the function defined by
$$ \begin{align*} \sigma_X(x_0,x_1,x_2,\ldots)=(x_1,x_2,x_3,\ldots) \end{align*} $$
for each
$(x_0,x_1,x_2,\ldots )\in X^{{\mathbb Z}_+}$
. Note that
$(\sigma _X)|_{X_F^{+}}=\sigma $
. Let
$\alpha $
be an open cover for X and let m be a positive integer. First, note that for each positive integer k and for any
$V\in \alpha ^{m+1}$
,
$$ \begin{align*} \sigma_X^{-k}(V\times X^{{\mathbb Z}_+})=X^k\times V\times X^{{\mathbb Z}_+}. \end{align*} $$
It follows that
$\sigma ^{-k}(\alpha ^{m+1} \times X^{{\mathbb Z}_+}) = X^k \times \alpha ^{m+1} \times X^{{\mathbb Z}_+}$
and, therefore,
$$ \begin{align*} N(X_F^+,\vee_{i=0}^{k}\sigma_X^{-i}(\alpha^{m+1}\times X^{{\mathbb Z}_+}))=N(X_F^+,\alpha^{m+k+1}\times X^{{\mathbb Z}_+}). \end{align*} $$
Let
$\pi :X_F^+ \rightarrow X_{F}^{m+k}$
be the projection to the first
$m + k + 1$
coordinates. Note that
$\pi $
is surjective since
$p_2(F)\subseteq p_1(F)$
. Then
$$ \begin{align*} N(X_F^+,\alpha^{m+k+1}\times X^{{\mathbb Z}_+})=N(X_F^{m+k},\alpha^{m+k+1}) \end{align*} $$
follows since for each
${\mathbf {x}}\in X_F^+$
and for each
$V\in \alpha ^{m+k+1}$
,
$$ \begin{align*} {\mathbf{x}}\in V\times X^{{\mathbb Z}_+} =\pi^{-1}(V) \Longleftrightarrow\, \pi({\mathbf{x}})\in V. \end{align*} $$
Note also that for all positive integers m and k,
$$ \begin{align*} &N(\vee_{i=0}^{k}\sigma^{-i}((\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+))=N(\vee_{i=0}^{k}(\sigma_X^{-i}(\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap \sigma^{-i}(X_F^+)))\\ &\quad=N(\vee_{i=0}^{k}(\sigma_X^{-i}(\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+))=N(X_F^+,\vee_{i=0}^{k}(\sigma_X^{-i}(\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+)) \\ &\quad=N(X_F^+,\vee_{i=0}^{k}\sigma_X^{-i}(\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+)=N(X_F^+,\vee_{i=0}^{k}\sigma_X^{-i}(\alpha^{m+1}\times X^{{\mathbb Z}_+})). \end{align*} $$
It follows that for all positive integers m and k,
$$ \begin{align*} N(X_F^{m+k},\alpha^{m+k+1})=N(\vee_{i=0}^{k}\sigma^{-i}((\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+)). \end{align*} $$
Next, we prove that
$$ \begin{align*} \lim_{k\to \infty}\frac{\log N(X_F^{m+k},\alpha^{m+k+1})}{k}=\lim_{k\to \infty}\frac{\log N(X_F^{k},\alpha^{k+1})}{k}. \end{align*} $$
For each positive integer n, let
$a_n=\log N(X_F^{n},\alpha ^{n+1})$
. Then
$a_k=\log N(X_F^{k},\alpha ^{k+1})$
and
$a_{m+k}=\log N(X_F^{m+k},\alpha ^{m+k+1})$
for all positive integers m and k. It follows from Observation 3.2 that
$a_k\leq a_{m+k}$
and from Lemma 3.3 that
$a_{m+k}\leq a_{m}+a_k$
for all positive integers m and k. Therefore,
$a_k\leq a_{m+k}\leq a_{m}+a_k$
, and
${a_k}/{k} \leq {a_{m+k}}/{k} \leq {a_{m}}/{k} + {a_k}/{k}$
follows for all positive integers m and k. By Theorem 3.5, the limit
$\lim _{k\to \infty }({a_k}/{k})$
exists, therefore, for each positive integer m,
$$ \begin{align*} \lim_{k\to\infty}\frac{a_k}{k}\leq \lim_{k\to\infty}\frac{a_{m+k}}{k}\leq \lim_{k\to\infty}\frac{a_{m}}{k}+\lim_{k\to\infty}\frac{a_{k}}{k}=\lim_{k\to\infty}\frac{a_{k}}{k}. \end{align*} $$
Thus, for each positive integer m,
$\lim _{k\to \infty }{a_{m+k}}/{k}=\lim _{k\to \infty }{a_k}/{k}$
and
$$ \begin{align*} \lim_{k\to \infty}\frac{\log N(X_F^{m+k},\alpha^{m+k+1})}{k}=\lim_{k\to \infty}\frac{\log N(X_F^{k},\alpha^{k+1})}{k} \end{align*} $$
follows for each positive integer m. Therefore, for any positive integer m, we get that
$$ \begin{align*} h(\sigma,(\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+)&=\lim_{k\to \infty}\frac{\log N(\vee_{i=0}^{k}\sigma^{-i}((\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+))}{k}\\ &=\lim_{k\to \infty}\frac{\log N(X_F^{m+k},\alpha^{m+k+1})}{k}\\ &=\lim_{k\to \infty}\frac{\log N(X_F^{k},\alpha^{k+1})}{k}\\ &={\mathrm{ent}}(F,\alpha).\\[-34pt] \end{align*} $$
Lemma 3.14. Let X be a non-empty compact metric space, let
${F}$
be a closed relation on X and let
$\sigma $
be the shift map on
${X_F^+}$
. Then
$$ \begin{align*} h(\sigma)=\sup\{h(\sigma,(\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+) \mid \alpha \text{ is an open cover for } X, m\ \textit{is a positive integer}\} \end{align*} $$
Proof. Let
$A=\{h(\sigma ,(\alpha ^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+) \mid \alpha \text { is an open cover for } X, m \text { is a positive } \mathrm {integer}\}$
and let
$B=\{h(\sigma ,{\mathcal U}) \mid {\mathcal U} \text { is an open cover for } X_F^+\}$
. Note that
$h(\sigma )=\sup (B)$
. We distinguish the following possible cases.
-
• A is not bounded in
$\mathbb R$
. Then
$\sup (A)=\infty $
and since
$A\subseteq B$
, it follows that
$\sup (B)=\infty $
. Therefore, in this case,
$\sup (A)=\sup (B)$
. -
• A is bounded in
$\mathbb R$
. Since
$A\subseteq B$
, it follows that
$\sup (A)\leq \sup (B)$
. To show that
$\sup (A)\geq \sup (B)$
, we show that for every open cover
${\mathcal U}$
for
$X_F^+$
, there are an open cover
$\alpha $
for X and a positive integer m such that
$ h(\sigma ,{\mathcal U})\leq h(\sigma ,(\alpha ^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+)$
. Let
${\mathcal U}$
be any open cover for
$X_F^+$
. To show that there are an open cover
$\alpha $
for X and a positive integer m such that
$h(\sigma ,{\mathcal U})\leq h(\sigma ,(\alpha ^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+)$
, we show that there are an open cover
$\alpha $
for X and a positive integer m such that
${\mathcal U}\leq (\alpha ^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+$
. Since
$X_F^+$
is a compact metric space, it follows that
${\mathcal U}$
has a finite subcover. Let
$U_0,U_1,U_2,\ldots ,U_k\in {\mathcal U}$
be such that
$X_F^+=U_0\cup U_1\cup U_2 \cup \cdots \cup U_k$
. Let
$\unicode{x3bb} $
be a Lebesgue number (a positive number
$\unicode{x3bb} $
is a Lebesgue number of an open cover
${\mathcal U}$
of X if for each set
$A\subseteq X$
with diameter less than
$\unicode{x3bb} $
, there is U in
${\mathcal U}$
such that
$A\subseteq U$
) of the open cover
$\{U_0,U_1,U_2,\ldots ,U_k\}$
for
$X_F^+$
, let
$\alpha $
be an open cover for X and m a positive integer such that for each
$A\in \alpha $
,
${\mathop {\mathrm {diam}}\nolimits }(A)<\unicode{x3bb} $
and
${{\mathop {\mathrm {diam}}\nolimits }(X_F^+)}/{2^m}<\unicode{x3bb} $
. It follows that for each element
$U\in (\alpha ^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+$
,
${\mathop {\mathrm {diam}}\nolimits }(U)<\unicode{x3bb} $
. Therefore, for each
$U\in (\alpha ^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+$
, there is an
$i\in \{0,1,2,\ldots , k\}$
such that
$U\subseteq B_i$
. Thus,
$\{U_0,U_1,U_2,\ldots ,U_k\}\leq (\alpha ^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+$
. Since
$\{U_0,U_1,U_2,\ldots ,U_k\}\subseteq {\mathcal U}$
, it follows that also
${\mathcal U} \leq (\alpha ^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+$
.
Lemma 3.15. Let X be a non-empty compact metric space, let F be a closed relation on X such that
$p_2(F)\subseteq p_1(F)$
and let
$\sigma $
be the shift map on
${X_F^+}$
. Then
$$ \begin{align*} {\mathrm{ent}}(F)=\sup\{h(\sigma,(\alpha^{m+1}\times X^{{\mathbb Z}_+})\!\cap\! X_F^+) \mid \alpha \text{ is an open cover for } X, m \text{ is a positive integer}\}. \end{align*} $$
Proof. It follows from Lemma 3.13 that for each positive integer m,
$$ \begin{align*} &\sup\{{\mathrm{ent}}(F,\alpha) \mid \alpha \text{ is an open cover for } X\}= \\ &\sup\{h(\sigma,(\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+) \mid \alpha \text{ is an open cover for } X\}. \end{align*} $$
For each positive integer m, let
$A_m=\{h(\sigma ,(\alpha ^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+) \mid \alpha \text { is an open cover } \text {for } X\}$
, let
$A{\kern2pt}={\kern2pt}\{h(\sigma ,(\alpha ^{m+1}\times X^{{\mathbb Z}_+}){\kern2pt}\cap{\kern2pt} X_F^+) \mid \alpha \text { is an open cover for } X, m \text { is a positive } \text {integer}\}$
and let
$B=\{{\mathrm {ent}}(F,\alpha ) \mid \alpha \text { is an open cover for } X\}$
. It follows from
$A=\bigcup _{i=1}^{\infty }A_i$
and from
$\sup (A_m)=\sup B$
for each positive integer m that
$\sup (A)= \sup (B)$
.
Theorem 3.16. Let X be a non-empty compact metric space, let F be a closed relation on X such that
$p_2(F)\subseteq p_1(F)$
and let
$\sigma $
be the shift map on
${X_F^+}$
. Then
$$ \begin{align*} {\mathrm{ent}}({ F})=h(\sigma). \end{align*} $$
Proof. The theorem follows from Lemmas 3.14 and 3.15:
$$ \begin{align*} &{\mathrm{ent}}({ F})\\&\quad=\sup\{h(\sigma,(\alpha^{m+1}\times X^{{\mathbb Z}_+})\cap X_F^+) \mid \alpha \text{ is an open cover for } X, m \text{ is a positive integer}\}\\ &\quad= h(\sigma).\\[-33pt] \end{align*} $$
The following well-known result is a corollary of Theorem 3.16.
Corollary 3.17. Let X be a non-empty compact metric space, let
$f:X\rightarrow X$
be a continuous function and let
$\sigma : \varprojlim (X,f) \rightarrow \varprojlim (X,f) $
be the shift function. Then
$$ \begin{align*} h(f)=h(\sigma). \end{align*} $$
We conclude this section by stating some open problems. Let X be a non-empty compact metric space and let F be a closed relation on X. If there is a positive integer m such that
$X_F^m=\emptyset $
, then, obviously,
${\mathrm {ent}}(F)=0$
. Note that there is a positive integer m such that
$X_F^m=\emptyset $
if and only if
$\bigcap _{n=0}^{\infty }F^n(X)=\emptyset $
, where
$F^n(x)=\{y\in X \mid \text {there is } {\mathbf {x}}\in X_F^n\text { such that } \pi _0({\mathbf {x}})=x \text { and } \pi _{n}({\mathbf {x}})=y\}$
and
$ F^n(X)=\bigcup _{x\in X}F^n(x)$
. This gives rise to the following problems that came up during a discussion on the topic with our anonymous referee.
Problem 3.18. Let X be a non-empty compact metric space and let F be a closed relation on X. Is it true that
$$ \begin{align*} {\mathrm{ent}}(F)={\mathrm{ent}}\bigg(F\cap \bigg(\bigg(\bigcap_{n=0}^{\infty}F^n(X)\bigg)\times \bigg(\bigcap_{n=0}^{\infty}F^n(X)\bigg)\bigg)\bigg)? \end{align*} $$
Note that for any non-empty closed relation F on a compact metric space,
$p_2(F)\subseteq p_1(F)$
is a stronger condition than
$\bigcap _{n=0}^{\infty }F^n(X)\neq \emptyset $
. So, it is natural to state the following problem.
Problem 3.19. Let X be a non-empty compact metric space, let F be a closed relation on X such that
$\bigcap _{n=0}^{\infty }F^n(X)\neq \emptyset $
and let
$\sigma $
be the shift map on
${X_F^+}$
. Is it true that also in this case,
$$ \begin{align*} {\mathrm{ent}}({ F})=h(\sigma)? \end{align*} $$
An affirmative solution to the problem gives a nice generalization of Theorem 3.16.
4. Returns
In this section, we present returns for closed relations on compact metric spaces and use them to obtain non-trivial sufficient conditions for a relation on a compact metric space to have non-zero entropy. First, we introduce the notion of a
$(k,\varepsilon )$
-return on a set.
Definition 4.1. Let X be a non-empty compact metric space, let A be a non-empty subset of X, let F be a closed relation on X, let k be a positive integer and let
$\varepsilon>0$
. We say that
${F}$
has a
$(k,\varepsilon )$
-return on A if for each
$a\in A$
, there are positive integers j and
$j'$
such that
${0}<j'\leq j \leq k$
and points
$ { {\mathbf {x}},{\mathbf {y}}\in X_F^{j}}$
such that:
-
(1)
$ { {\mathbf {x}}(0)={\mathbf {y}}(0)=a}$
; -
(2)
${\mathbf {x}}(j),{\mathbf {y}}(j)\in A$
; and -
(3)
$\rho ( { {\mathbf {x}}(j'),{\mathbf {y}}(j')}) \geq \varepsilon $
.
We say that
${F}$
has a
$(k,\varepsilon )$
-return if there is a non-empty set
${B}\subseteq X$
such that F has a
$(k,\varepsilon )$
-return on
${B}$
.
In Construction 4.1, we give detailed instructions on how to construct an
$(F,A,k,\varepsilon )$
- return function for any element
$a\in A$
(which is defined later in Definition 4.2).
Construction 4.1. Let X be a non-empty compact metric space, let
$A\subseteq X$
, let F be a closed relation on X, let k be a positive integer and let
$\varepsilon>0$
. Suppose that F has a
$(k,\varepsilon )$
-return on A.
-
• Step 1. For each
$a \in A$
, we choose and fix positive integers
$j'(a)$
and
$j(a)$
such that
$0 < j'(a) \leq j(a)\leq k$
and points
${\mathbf {x}}(a,0),{\mathbf {x}}(a,1) \in X_F^{j(a)}$
such that:-
(1)
$a={\mathbf {x}}(a,0)_0 ={\mathbf {x}}(a,1)_0$
; -
(2)
$\rho ({\mathbf {x}}(a, 0)_{j'(a)}, {\mathbf {x}}(a, 1)_{j'(a)}) \geq \varepsilon $
; -
(3)
${\mathbf {x}}(a, 0)_{j(a)}, {\mathbf {x}}(a, 1)_{j(a)} \in A$
.
Then we define
$$ \begin{align*} \delta( a, 0) = {\mathbf{x}}(a, 0)_{j(a)} \quad\mathrm{and}\quad \delta(a, 1) = {\mathbf{x}}(a, 1)_{j(a)}. \end{align*} $$
-
-
• Step 2. Choose and fix
$a^{*}\in A$
. We inductively define functions by
$$ \begin{align*} \Delta : \Sigma_2 \rightarrow A^{{\mathbb Z}_+} \quad\mathrm{and}\quad J :\Sigma_2 \rightarrow {\mathbb Z}_+^{{\mathbb Z}_+} \end{align*} $$
$\Delta ({\mathbf {s}})_0 = a^{*}$
and for each non-negative integer n, we define and
$$ \begin{align*} \Delta({\mathbf{s}})_{n+1} = \delta(\Delta({\mathbf{s}})_n, {\mathbf{s}}_n) \end{align*} $$
$J({\mathbf {s}})_0 = 0$
, and for each non-negative integer n, we define for any sequence
$$ \begin{align*} J({\mathbf{s}})_{n+1} = J({\mathbf{s}})_n + j(\Delta({\mathbf{s}})_n) \end{align*} $$
${\mathbf {s}}\in \Sigma _2$
.
-
• Step 3. We define the function
$\Psi : \Sigma _2 \rightarrow X_F^+$
by for each
$$ \begin{align*} \Psi ({\mathbf{s}})_{J ({\mathbf{s}})_n} = \Delta ({\mathbf{s}})_n \end{align*} $$
${\mathbf {s}}\in \Sigma _2$
and for each positive integer n, and for each
$$ \begin{align*} \Psi ({\mathbf{s}})_{J ({\mathbf{s}})_n+i} = {\mathbf{x}}(\Delta ({\mathbf{s}})_n, {\mathbf{s}}_n)_i \end{align*} $$
${\mathbf {s}}\in \Sigma _2$
, for each positive integer n and for each
$i \in [0, j(\Delta ({\mathbf {s}})_n)]$
.
Definition 4.2. Let X be a non-empty compact metric space, let
$A\subseteq X$
, let F be a closed relation on X, let k be a positive integer, let
$\varepsilon>0$
such that F has a
$(k,\varepsilon )$
-return on A and let
$a^{*}\in A$
. We call such a function
$\Psi : \Sigma _2 \rightarrow X_F^+$
from Construction 4.1 a
$({ F},A,k,\varepsilon )$
-return function for
${a^{*}}$
.
The following observation follows directly from Construction 4.1.
Observation 4.2. Let X be a non-empty compact metric space, let F be a closed relation on X, let k be a positive integer, let
$\varepsilon>0$
, let
$A\subseteq X$
be such that F has a
$(k,\varepsilon )$
-return on A and let
$a^{*}\in A$
. Also, let
$\Psi $
be a
$({ F},A,k,\varepsilon )$
-return function for
${a^{*}}$
.
Note that for all sequences
${\mathbf {s}},{\mathbf {t}}\in \Sigma _2$
and for each positive integer n:
-
(1)
$J({\mathbf {s}})_{n+1} \leq kn$
; -
(2) if
${\mathbf {s}}_i ={\mathbf {t}}_i$
for each
$i\in [0,n]$
, then
$\Delta ({\mathbf {s}})_i =\Delta ({\mathbf {t}})_i$
and
$J({\mathbf {s}})_i =J({\mathbf {t}})_i$
for each
$i\in [0,n+1]$
; -
(3)
$\Psi ({\mathbf {s}})_0 = a^{*}$
; -
(4) for
$i = 0$
, and for
$$ \begin{align*} \Psi({\mathbf{s}})_{J ({\mathbf{s}})_n +i} = {\mathbf{x}}(\Delta({\mathbf{s}})_n , s_n )_0 = \Delta({\mathbf{s}})_n = \Psi({\mathbf{s}})_{J({\mathbf{s}})_n}, \end{align*} $$
$i = j(\Delta ({\mathbf {s}})_n)$
, it holds that
$J({\mathbf {s}})_n +i = J ({\mathbf {s}})_{n+1}$
, and follows.
$$ \begin{align*} \Psi ({\mathbf{s}})_{J({\mathbf{s}})_n+i} = {\mathbf{x}}(\Delta({\mathbf{s}})_n,{\mathbf{s}}_n)_{j(\Delta({\mathbf{s}})_n)} = \Delta({\mathbf{s}})_{n+1} = \Psi({\mathbf{s}})_{J({\mathbf{s}})_{n+1}} \end{align*} $$
Lemma 4.3. Let X be a non-empty compact metric space, let F be a closed relation on X, let k and m be positive integers, let
$\varepsilon>0$
, let
$A\subseteq X$
be such that F has a
$(k,\varepsilon )$
-return on A, and let
$a^{*}\in A$
. Also, let
$\Psi $
be a
$({ F},A,k,\varepsilon )$
-return function for
${a^{*}}$
, let
$j'= j'(\Delta ({\mathbf {s}})_m)$
and let
${\mathbf {s}},{\mathbf {t}} \in \Sigma _2$
be any sequences of
$0$
s and
$1$
s such that:
-
• for any
$i\in [0,m-1]$
,
${\mathbf {s}}_i={\mathbf {t}}_i$
and -
•
${\mathbf {s}}_m\neq {\mathbf {t}}_m$
.
Then:
-
(1) for each
$i \in [0,J({\mathbf {s}})_m]$
,
$$ \begin{align*} \Psi({\mathbf{s}})_i = \Psi({\mathbf{t}})_i; \end{align*} $$
-
(2) for each
$i \in [0, j(\Delta ({\mathbf {s}})_m)] $
,
$$ \begin{align*} \Psi({\mathbf{s}})_{J({\mathbf{s}})_m+i} = {\mathbf{x}}(\Delta({\mathbf{s}})_m, {\mathbf{s}}_m)_i \quad\mathrm{and}\quad \Psi({\mathbf{t}})_{J({\mathbf{s}})_m+i} = {\mathbf{x}}(\Delta({\mathbf{s}})_m, {\mathbf{t}}_m)_i; \end{align*} $$
-
(3)
$\rho (\Psi ({\mathbf {s}})_{J({\mathbf {s}})_m+j'} , \Psi ({\mathbf {t}})_{J({\mathbf {s}})_m+j'} ) \geq \varepsilon $
and
$$ \begin{align*} J(s)_m + j' \leq J(s)_m + j(\Delta(s)_m) = J(s)_{m+1} \leq mk. \end{align*} $$
Proof. The lemma follows directly from the construction of the function
$\Psi $
and from Observation 4.2.
Corollary 4.4. Let X be a non-empty compact metric space, let F be a closed relation on X, let k and m be positive integers, let
$\varepsilon>0$
, let
$A\subseteq X$
be such that F has a
$(k,\varepsilon )$
-return on A, and let
$a^{*}\in A$
. Also, let
$\Psi $
be a
$({ F},A,k,\varepsilon )$
-return function for
${a^{*}}$
, and let
${\mathbf {s}},{\mathbf {t}} \in \Sigma _2$
be any sequences of
$0$
s and
$1$
s. Then the following hold.
-
(1) If there is a non-negative integer
$i\leq m$
such that
${\mathbf {t}}_i \neq {\mathbf {s}}_i$
, then there is a positive integer
$j<mk$
such that
$$ \begin{align*} \rho(\Psi({\mathbf{s}})_j , \Psi({\mathbf{t}})_j ) \geq \varepsilon. \end{align*} $$
-
(2) Let
$\alpha $
be an open cover for X consisting of sets with diameter less than
$\varepsilon $
and let
$A^{*}\in \alpha $
be such that
$a^{*}\in A^{*}$
. Then for any
$U\in A^{*}\times \alpha ^{mk-1}$
,
$$ \begin{align*} \{\pi_{[0,mk]}(\Psi({\mathbf{s}})),\pi_{[0,mk]}(\Psi({\mathbf{t}}))\}\not \subseteq U. \end{align*} $$
-
(3)
$N(\pi _{[0,mk]}(\Psi (\Sigma _2)),A^{*}\times \alpha ^{mk-1})\geq 2^{m-1}$
.
Proof. Item (1) follows directly from Lemma 4.3, item (2) follows from item (1) and item (3) follows from item (2).
Theorem 4.5 is our first main result of the paper. It says that for a closed relation F on a compact metric space X, the existence of a
$(k,\varepsilon )$
-return on a subset of X implies that the entropy of G is non-zero.
Theorem 4.5. Let X be a non-empty compact metric space, let
$A\subseteq X$
, let F be a closed relation on X, let k be a positive integer and let
$\varepsilon>0$
such that F has a
$(k,\varepsilon )$
-return on A. Then
$$ \begin{align*} {\mathrm{ent}}({ F})\geq \frac{\log (2)}{k}. \end{align*} $$
Proof. Let
$a^{*}\in A$
and let
$\Psi $
be a
$({ F},A,k,\varepsilon )$
-return function for
${a^{*}}$
. Let
$\alpha $
be an open cover for X such that for any
$ { A}\in \alpha $
, the diameter of
$ { A}$
is less than
$\varepsilon $
. It follows from Corollary 4.4 that for each positive integer m,
$N({ X_F^{mk-1}},\alpha ^{mk})\geq 2^{m{ -1}}$
. Thus,
$$ \begin{align*} {\mathrm{ent}}({ F}) \!\geq\! \lim_{m\rightarrow \infty} \frac{\log( N({ X_F^{mk-1}},\alpha^ {mk{ }}))}{mk { -1}}\geq \lim_{m\rightarrow \infty } \frac{ \log(2^{m{ -1}}) }{mk { -1}}= \lim_{m\rightarrow \infty} \frac {{ (}m{ -1)}\log(2)}{mk { -1}}= \frac {\log(2)}{ k}.\\[-20pt] \end{align*} $$
Note that it follows from the following observation that there are closed relations F on compact metric spaces X such that
${\mathrm {ent}}({ F})\neq 0$
and for each non-empty
$A\subseteq X$
, F has no
$(k,\varepsilon )$
-return on A. So, sufficient conditions from Theorem 4.5 are not necessary conditions for non-zero entropy of F. Thus, we do not have a characterization of non-zero entropy.
Observation 4.6. If F is a graph of a continuous function from X to X, then for any positive integer k and for any
$\varepsilon>0$
, F does not have a
$(k, \varepsilon )$
-return since for each positive integer k and for all
${\mathbf {x}},{\mathbf {y}}\in X_F^+$
,
${\mathbf {x}}(0) = {\mathbf {y}}(0)$
implies that
${\mathbf {x}}(j) = {\mathbf {y}}(j)$
for all
$j\leq k$
.
In particular, if
$\varphi $
is a homeomorphism on X with non-zero entropy, then
$\Gamma (\varphi )$
is a closed relation on X with non-zero entropy that admits no
$(k, \varepsilon )$
-returns. An example of such a space X and a homeomorphism
$\varphi $
on X is the Cantor set
$\{0, 1\}^{{\mathbb Z}}$
and the shift homeomorphism
$\sigma $
on it. Also, note that such a space X may be connected. For example, let X be the pseudo-arc. It was proved by Kennedy in [Reference Kennedy13] that there is a homeomorphism
$\varphi $
on X such that
$h(\varphi )\neq 0$
.
Theorem 4.7. Let X be a non-empty compact metric space, let
$A\subseteq X$
, let F be a closed relation on X, let k be a positive integer and let
$\varepsilon>0$
such that
${ }$
has a
$(k,\varepsilon )$
-return on A. Then there is a countable subset
$A'\subseteq A$
such that F has a
$(k,\varepsilon )$
-return on
$A'$
.
Proof. Let
$a^{*}\in A$
and let
$\Psi $
be a
$({ F},A,k,\varepsilon )$
-return function for
${a^{*}}$
. While
$\Sigma _2$
is uncountable, for each positive integer m, the set
$\{\Delta ({\mathbf {s}})_i : {\mathbf {s}} \in \Sigma _2, i \leq m\}$
is finite and so, the set
$\{\Delta ({\mathbf {s}})_i : {\mathbf {s}} \in \Sigma _2, i \in {\mathbb Z}_+\}$
is a countable set
$A'$
such that F has a
$(k,\varepsilon )$
-return on
$A'$
.
Observation 4.8. For any integer
$N \geq 2$
, one can also define that F has a
$(k,\varepsilon ,N)$
-return on A if for every
$a\in A$
, there exist
$j\in \{2,3,4,\ldots ,k\}$
and the set
$$ \begin{align*} \{{\mathbf{x}}^\ell \mid \ell\in\{ 0,\ldots,N - 1\}\} \subseteq X_F^j \end{align*} $$
such that for each
$\ell \in \{ 0,1,2,\ldots ,N - 1\}$
,
${\mathbf {x}}^{\ell }(0) = a$
and
${\mathbf {x}}^{\ell }(j) \in A$
, and for all
$\ell ,\ell '\in \{ 0,1,2,\ldots ,N - 1\}$
such that
$\ell \neq \ell '$
, there exists a positive integer
$j' \leq j$
such that
$\rho ({\mathbf {x}}^{\ell }(j'), {\mathbf {x}}^{\ell '} (j')) \geq \varepsilon $
. One can then embed
$\Sigma _N$
into
$X_F^+$
and obtain that the entropy of the relation F is at least
${\log (N)}/{k}$
.
Next, we give some examples showing how easy it is to prove, using the notion of returns, that the entropy of a function is positive. First, we show how the notion of returns can be used to give a simple proof that the tent map has non-zero entropy.
Example 4.9. Let
$I=[0,1]$
and let
$f:[0,1]\rightarrow [0,1]$
be defined by
$$ \begin{align*}f(x)= \begin{cases} 2x &\text{for } x\leq \tfrac{1}{2},\\ 2-2x &\text{for } x> \tfrac{1}{2}, \end{cases}\end{align*} $$
see Figure 1.
The graph of the function f from Example 4.9.
Let
$F=\Gamma (f)^{-1}$
. It is easy to check that for each
$x\in [0,1]$
, both
$(x,\tfrac 12x , \tfrac 14 x)$
and
$(x, 1- \tfrac 12x, \tfrac 12+\tfrac 14 x )$
are elements of
${ I_F^2}$
. Since
$|\tfrac 14x - ( \tfrac 12+\tfrac 14 x)|=\tfrac 12>\tfrac 13$
,
$\Gamma (f)^{-1}$
has a
$(3,\tfrac 13)$
-return on
$A=[0,1]$
. It follows from Theorem 4.5 that
$h(f)={\mathrm {ent}}({ F})>0$
.
The following example presents how returns may be used to produce a homeomorphism on a Cantor fan with positive entropy (a continuum X is a Cantor fan if X is homeomorphic to the continuum
$\bigcup _{c\in C}{S}_c$
, where
$C\subseteq [0,1]$
is the standard Cantor set and for each
$c\in C$
,
${S}_c$
is the straight line segment in the plane from
$(0,0)$
to
$(c,1)$
).
Example 4.10. Let
$X=[-1,1]$
and let
$F=\Gamma (f_1)\cup \Gamma (f_2)$
, where
$f_1:[-1,1]\rightarrow [-1,1]$
and
$f_2:[-1,1]\rightarrow [-1,1]$
are defined by
$$ \begin{align*} f_1(x)=-x\quad \mathrm{and}\quad f_2(x)=x \end{align*} $$
for each
$x\in X$
. By [Reference Ingram11, Example 2.7], the Mahavier product
$X_F^+$
is a Cantor fan. Note that
$A=\{1\}$
is a
$(3,2)$
-return with sequences
$(1,-1,1)$
and
$(1,1,1)$
. So, the entropy of the shift map
$\sigma _F$
on
$X_F^+$
is non-zero. The same argument as in [Reference Banič, Erceg, Kennedy, Mouron and Nall4, Example 4.15] shows that the inverse limit
$\varprojlim (X_F^+,\sigma _F)$
is also a Cantor fan. Therefore, the shift map
$\sigma $
on
$\varprojlim (X_F^+,\sigma _F)$
is a homeomorphism on a Cantor fan with positive entropy.
The relation F from Example 4.11.
In the following example, the same simple argument is used to produce a transitive homeomorphism on a Cantor fan with positive entropy. (The function
$f:X\rightarrow X$
is transitive if for all non-empty open sets U and V in X, there is a non-negative integer n such that
$f^n(U)\cap V\neq \emptyset $
.) Note that the homeomorphism from Example 4.10 is not transitive.
Example 4.11. Let
$X=[-1,1]$
and let
$F=\Gamma (f_1)\cup \Gamma (f_2)$
, where
$f_1:[-1,1]\rightarrow [-1,1]$
and
$f_2:[-1,1]\rightarrow [-1,1]$
are defined by
$$ \begin{align*} f_1(x)=-x\quad\mathrm{and}\quad f_2(x)=\begin{cases} x^{{1}/{3}}\text{;} & x\in [-1,0],\\ x^2\text{;} & x\in [0,1] \end{cases} \end{align*} $$
for each
$x\in X$
; see Figure 2. By [Reference Banič, Erceg, Kennedy, Mouron and Nall4, Example 4.15], the Mahavier product
$X_F^+$
is a Cantor fan and the shift map
$\sigma _F$
on
$X_F^+$
is transitive. Note that
$A=\{1\}$
is a
$(3,2)$
-return with sequences
$(1,-1,1)$
and
$(1,1,1)$
. So, the entropy of the shift map
$\sigma _F$
on
$X_F^+$
is positive. It follows from [Reference Banič, Erceg, Kennedy, Mouron and Nall4, Example 4.15] that the inverse limit
$\varprojlim (X_F^+,\sigma _F)$
is also a Cantor fan. Therefore, the shift map
$\sigma $
on
$\varprojlim (X_F^+,\sigma _F)$
is a transitive homeomorphism on a Cantor fan with positive entropy.
The closed relation in the following example was shown in [Reference Banič, Erceg and Kennedy2] to produce a dynamical system with positive entropy that does not have periodic points or finitely generated Cantor sets. (Let X be a compact metric space and let F be a closed relation on X. We say that F has finitely generated Cantor sets, if there is a Cantor set C in
$X_F^+$
such that for some finite collection
$G\subseteq F$
,
$X_G^+=C$
.) The notion of well-aligned sets was introduced in [Reference Banič, Erceg and Kennedy2] as a way to generalize the proof that the relation in Example 4.12 produces a dynamical system with positive entropy. While the existence of well-aligned sets may be hard to verify in many different settings, the use of returns introduced in this paper serves a different purpose in that it can greatly simplify proofs of positive entropy in this setting of shift maps on Mahavier products as well as in more traditional settings as in Example 4.9. However, it is especially useful when there are no finitely generated Cantor sets. Our examples also tend to illustrate that it is much easier to verify the existence of a return than the existence of well-aligned sets.
Example 4.12. Let
$I=[0,1]$
. In [Reference Banič, Erceg and Kennedy2], it was shown that if
$0<a<1$
and
$0<b<1$
, and
$$ \begin{align*} { F}=\bigg\{ (x,y) \in [0,1] \times [0,1] \mid y={ \frac{1}{a}} x \text{ or } y={ b}x\bigg\}, \end{align*} $$
see Figure 3, then
${\mathrm {ent}}({ F})\neq 0$
.
The relation F from Example 4.12.
We present a much streamlined proof of this fact by showing that if
$0<a\leq b<1$
, then F has a
$(k,\varepsilon )$
-return on the set
$A=[ab,a]$
. For the case
$0 < b \leq a <1$
, a similar argument shows that
${F}^{-1}$
has a
$(k,\varepsilon )$
-return on
$A=[ab,b]$
.
So assume
$0<a\leq b<1$
. Note if
$b>a$
, then there is an
$m\in \mathbb {N}$
such that
$b^{m-1}>a$
and
$b^m \leq a$
, and thus
$ab { <} b^m \leq a$
. So if
$y\in (a,b]$
, then there is an
$m_y\in \mathbb {N}$
such that
$m_y \leq m$
and
$b^{m_y-1} y>a $
and
$b^{m_y} y \leq a$
, and thus
$ab \leq b^{m_y} y\leq a$
.
We show that if
$k=m+1$
and
$\varepsilon = {({1}/{a} - b)ab}/{2}$
, then F has a
$(k,\varepsilon )$
-return on
$[ab,a]$
.
Let
$x\in [ab,a]$
. If
$({b}/{a})x \leq a$
, then
$({b}/{a})x \geq ({b}/{a})ab =b\cdot b \geq ab$
and
$(x,bx) \in { F}$
with
$bx < x { \leq } a $
, so
$(bx, ({b}/{a}) x) \in { F}$
. It follows that
$(x, bx, ({b}/{a})x) \in { I_F^2}$
. It is easy to see that also
$(x, ({1}/{a})x , ({b}/{a}) x )\in { I_F^2}$
. Observe that
$x\in [ab,a]$
,
$({b}/{a}) x \in [ab,a]$
and
$|({1}/{a})x-bx|=( ({1}/{a})-b)x\geq (({1}/{a})-b)ab>\varepsilon $
. So, for
$({b}/{a})x \leq a$
, F has an
$(m+1,\varepsilon )$
-return on
$[ab,a]$
.
If
$({b}/{a}) x> a$
, then
$({b}/{a}) x \leq ({b}/{a}) a =b$
. So, there is a positive integer
$m_x$
such that
$m_x\leq m$
and
$ab\leq b^{m_x} ({b}/{a})x \leq a$
. Now
$$ \begin{align*} \bigg(x, bx, \frac{b}{a} x, b \frac{b}{a} x, b^2 \frac{b}{a} x, \ldots , b^{m_x} \frac{b}{a} x\bigg) \in { I_F^{m_x+2}} \end{align*} $$
and
$$ \begin{align*} \bigg(x, \frac{1}{a} x, \frac{b}{a} x, b \frac{b}{a} x, b^2 \frac{b}{a} x, \ldots , b^{m_x} \frac{b}{a} x\bigg) \in { I_F^{m_x+2}}. \end{align*} $$
Again observe that
$x\in [ab,a]$
,
$b^{m_x} ({b}/{a}) x \in [ab,a]$
and
$|({1}/{a})x-bx|=(({1}/{a})-b) x\geq (({1}/{a})-b)ab>\varepsilon $
. So, for
$({b}/{a})x> a$
, F has an
$(m+1,\varepsilon )$
-return on
$[ab,a]$
. So, by Theorem 4.5, it follows that
${\mathrm {ent}}({ F})\neq 0$
.
Observation 4.13. Let F be the closed relation on
$I=[0,1]$
from Example 4.12 for
$a=\tfrac 12$
and
$b=\tfrac 13$
. Note that it follows from Theorem 3.16 that the shift mapping
$\sigma _F$
on
$I_F^+$
has non-zero entropy and that it follows from [Reference Banič, Erceg and Kennedy3, Theorem 4.3] that
$\sigma _F$
is transitive. It follows from [Reference Banič, Erceg and Kennedy3, Theorem 5.17] that the inverse limit
$\varprojlim (I_F^+,\sigma _F)$
is a Lelek fan (let X be a Cantor fan and let Y be a subcontinuum of X. We say that Y is a Lelek fan if the end-points of Y are dense in Y) and from [Reference Banič, Erceg and Kennedy3, Observation 5.3] that the shift homeomorphism
$\sigma $
on
$\varprojlim (I_F^+,\sigma _F)$
is transitive. Therefore, the shift homeomorphism
$\sigma $
on
$\varprojlim (I_F^+,\sigma _F)$
is a transitive homeomorphism on a Lelek fan with non-zero entropy. This answers the question [Reference Oprocha21, Question 3.11] by Oprocha in the affirmative.
We conclude the section by stating and proving various corollaries to Theorem 4.5.
Corollary 4.14. Let X be a non-empty compact metric space, let F be a closed relation on X and let k be a positive integer. If there are two sets J and K in
${X_F^k}$
with
$\rho (J,K)>0$
(where
$\rho (J,K)=\inf \{\rho (x,y) \mid x\in J, y\in K\}$
) and such that
$\pi _k(J)\cup \pi _k(K)\subseteq \pi _{ 0}(J)\cap \pi _{ 0}(K)$
, then
${F}$
has a
$(k,\rho (J,K))$
-return. In particular, F has non-zero entropy.
Proof. Let
$\varepsilon =\rho (J,K)$
. We show that F has a
$(k,\varepsilon )$
-return on X. Since
$\rho (J,K)>0$
, if
$({ x_0},x_1, x_2, \ldots ,x_k) \in J $
and
$({ y_0},y_1, y_2, \ldots ,y_k) \in K $
with
$x_{ 0}=y_{ 0}$
, then there is an integer
$i\in \{{ 1,2,3},\ldots k \}$
such that
$\rho (x_i, y_i)>\varepsilon $
. Let
$x\in \pi _{ 0}(J)\cap \pi _{ 0}(K)$
. Then there are elements
$({ x_0},x_1, x_2, \ldots ,x_k) \in J $
and
$({ y_0},y_1, y_2, \ldots ,y_k) \in K $
with
$x=x_{ 0}=y_{ 0}$
and since
$\pi _k(J)\cup \pi _k(K)\subseteq \pi _{ 0}(J)\cap \pi _{ 0}(K)$
, we have
$\{x_k,y_k\} \subseteq \pi _{ 0}(J)\cap \pi _{ 0}(K)$
. So F has a
$(k,\varepsilon )$
-return on
$\pi _{ 0}(J)\cap \pi _{ 0}(K)$
. By Theorem 4.5,
${\mathrm {ent}}({ F})\neq 0$
.
Corollary 4.15. Let X be a non-empty compact metric space and let F be a closed relation on X. Also, let
$k_x$
and
$k_y$
be two positive integers, let
${\mathbf {x}}\in { X_F^{k_x}}$
and
${\mathbf {y}} \in { X_F^{k_y}}$
be such that
$ { {\mathbf {x}}(k_x)}={ {\mathbf {x}}(0)}={ {\mathbf {y}}(0)}={ {\mathbf {y}}(k_y)}$
, and let j be a positive integer such that
${0}<j\leq \min \{ k_x , k_y \}$
and
${ {\mathbf {x}}(j)}\neq { {\mathbf {y}}(j)}$
. Then there are a positive integer k and an
$\varepsilon>0$
such that F has a
$(k,\varepsilon )$
-return. In particular, F has non-zero entropy.
Proof. We show that there is a set
$A\subseteq X$
, a positive integer k and an
$\varepsilon>0$
such that F has a
$(k,\varepsilon )$
-return on A. Let
${\mathbf {s}}={\mathbf {x}} \star {\mathbf {y}}$
and
${\mathbf {t}}={\mathbf {y}} \star {\mathbf {x}}$
. Then
${\mathbf {s}},{\mathbf {t}}\in { X_F^{k_x+k_y}}$
such that
${ {\mathbf {s}}(0)}={ {\mathbf {t}}(0)}={ {\mathbf {s}}(k_x+k_y)}={ {\mathbf {t}}(k_x+k_y)}$
and
${ {\mathbf {s}}(j)}\neq { {\mathbf {t}}(j)}$
. Therefore, for
$A=\{ { {\mathbf {s}}(0)} \}$
, F has a
$(k_x+k_y,\varepsilon )$
-return on A, where
$\varepsilon =\tfrac 12 \rho ({ {\mathbf {s}}(j)},{ {\mathbf {t}}(j)})$
. By Theorem 4.5,
${\mathrm {ent}}({ F})\neq 0$
.
In the last corollary (Corollary 4.17) to Theorem 4.5, so-called well-aligned sets are used to detect non-zero entropy. They form a more visual or geometric apparatus for spotting non-zero entropy. Before stating and proving the corollary, we give the following definitions to describe this apparatus.
Definition 4.3. Let X be a non-empty compact metric space, let
${F}$
be a closed relation on X, and let L and R be non-empty subsets of F. We say that the sets L and R are well aligned in F, if:
-
(1)
$p_2(L)\cap p_2(R)\neq \emptyset $
; -
(2) there is
$\varepsilon>0$
such that for all
$t\in p_2(L)\cap p_2(R)$
, there are
$\ell \in p_1(p_2^{-1}(t)\cap L)$
and
$r\in p_1(p_2^{-1}(t)\cap R)$
such that
$\rho (r,\ell )\geq \varepsilon $
; -
(3)
$p_1(L)\cup p_1(R)\subseteq p_2(L\cup R)$
; -
(4) there is a positive integer N such that for each
$t\in p_2(L\cup R)$
, there are a positive integer
$i_0\leq N$
and a point
$(a_{ 0},a_{ 1},a_{ 2},\ldots ,a_{ i_0-1},a_{ i_0})\in { X_{F^{-1}}^{i_0}} $
such that:-
(a)
$a_{ 0}=t$
and -
(b)
$a_{ i_0}\in p_2(L)\cap p_2(R)$
;
-
see Figure 4.
The sets L and R from Definition 4.3.
Example 4.16. Let
${F}=\{(0,1),(0,\tfrac 34),(\tfrac 34,0),(1,0)\}$
be a closed relation on
$[0,1]$
. It is easy to see that
${\mathrm {ent}}({ F})\neq 0$
. Let
$L=\{(\tfrac 34,0)\}$
,
$R=\{(1,0),(0,1),(0,\tfrac 34)\}$
. Then the sets L and R are well aligned in F.
Definition 4.4. Let X be a non-empty compact metric space, let F be a closed relation on X. We say that the relation
${F}$
is well aligned if there are
$L,R\subseteq { F}$
such that the sets L and R are well aligned in F.
Corollary 4.17. Let X be a non-empty compact metric space and let F be a closed relation on X. If F or
${F}^{-1}$
is a well-aligned relation, then
${\mathrm {ent}}({ F})\neq 0$
.
Proof. Suppose that F is a well-aligned relation. Let L and R be non-empty closed subsets of F such that L and R are well aligned in F; that is:
-
(1)
$p_2(L)\cap p_2(R)\neq \emptyset $
; -
(2) there is
$\varepsilon>0$
such that for all
$t\in p_2(L)\cap p_2(R)$
, there are
$\ell \in p_1(p_2^{-1}(t)\cap L)$
and
$r\in p_1(p_2^{-1}(t)\cap R)$
such that
$\rho (r,\ell )\geq \varepsilon $
; -
(3)
$p_1(L)\cup p_1(R)\subseteq p_2(L\cup R)$
; -
(4) there is a positive integer N such that for each
$t\in p_2(L\cup R)$
, there are a positive integer
$i_0\leq N$
and a point
$(a_{ 0},a_{ 1},a_{ 2},\ldots ,a_{ i_0-1},a_{ i_0})\in X_F^{i_0}$
such that:-
(a)
$a_{ 0}=t$
and -
(b)
$a_{ i_0}\in p_2(L)\cap p_2(R)$
.
-
Choose and fix such a positive integer N and
$\varepsilon>0$
. Let
$A=p_2(L\cup R)$
. Then
${ F^{-1}}$
has an
$(N, {\varepsilon }/{2})$
-return on A. Thus, by Theorem 4.5,
${\mathrm {ent}}({ F^{-1}})\neq 0$
. It follows that
${\mathrm {ent}}(F)\neq 0$
. If
${F}^{-1}$
is a well-aligned relation, the proof is similar.
5. Finite relations
Finite relations on compact metric spaces X can have positive entropy. This may happen even in the case where X is not finite. We show in this final section that, unlike topological entropy for closed relations on compact metric spaces in general, in the case of finite relations, positive entropy is equivalent to the shift map on the Mahavier product being Li–Yorke chaotic as well as equivalent to DC-2 distributional chaos for the shift map, as well as equivalent to G having a
$(k,\varepsilon )$
-return. Before stating and proving our theorems, we present the following definitions.
Definition 5.1. Let X be a non-empty compact metric space, let
$f:X\rightarrow X$
be a continuous function and let
$x,y\in X$
such that
$x\neq y$
. The set
$\{x,y\}$
is called a Li–Yorke pair for f if
$$ \begin{align*} \liminf(\rho(f^n(x),f^n(y)))=0 \quad\mathrm{and}\quad \limsup(\rho(f^n(x),f^n(y)))>0. \end{align*} $$
Definition 5.2. Let X be a non-empty compact metric space, let
$f:X\rightarrow X$
be a continuous function and let
$S\subseteq X$
. We say that the set S is a scrambled set or a Li–Yorke set in
$(X,f)$
if for all
$x,y\in S$
,
$$ \begin{align*} x\neq y \Longrightarrow \{x,y\} \quad\text{is a Li--Yorke pair for } f. \end{align*} $$
Definition 5.3. Let X be a non-empty compact metric space and let
$f:X\rightarrow X$
be a continuous function. The dynamical system
$(X,f)$
is called Li–Yorke chaotic if X contains an uncountable scrambled set.
The following is a well-known result.
Theorem 5.1. Let X be a non-empty compact metric space and let
$f:X\rightarrow X$
be a continuous function. If
$h(f)>0$
, then the dynamical system
$(X,f)$
is Li–Yorke chaotic.
Proof. See the proof of [Reference Blanchard, Glasner, Kolyada and Maass6, 2 from Corollary 2.4, p. 10].
See [Reference Blanchard, Glasner, Kolyada and Maass6] for more references and information about Li–Yorke chaotic topological dynamical systems.
The approach used in the following theorem was suggested by our referee. Let
$X=\{0,1,2,\ldots ,m-1\}$
be a finite set and F be a relation on X. We define the zero-one
$m\times m$
-matrix
$M_F$
on
$(X,F)$
as follows: for all
$x,y\in X$
,
$M_{xy}=1$
if
$(x,y) \in F$
and
$M_{xy}=0$
if
$(x,y) \not \in F$
. Observe that for all
$x,y\in X$
and for each positive integer N,
$M_{xy}^N$
counts the number of distinct paths of length N from x to y, that is,
$M_{xy}^N$
is the cardinality of the set
$\{{\mathbf {x}}\in X_F^N \mid {\mathbf {x}}(0)=x \text { and } {\mathbf {x}}(N)=y\}$
.
Theorem 5.2. Let
$X=\{0,1,2,\ldots ,m-1\}$
be a finite set and F be a relation on X. If there are
$x\in X$
and a positive integer k such that
$M_{xx}^k>1$
, then
${\mathrm {ent}}(F)\geq ({\log N})/{k}$
.
Proof. Let
$x\in X$
and let k be a positive integer such that
$M_{xx}^k>1$
. Let
$N=M_{xx}^k $
. Then F has
$(k,1,N)$
-return on
$\{x\}$
(see Observation 4.8 for the definition of
$(k,\varepsilon ,N)$
-returns) and so, the entropy of the relation F is at least
$({\log N})/{k}$
.
For any point
$x\in X$
, we say that x is recurrent in
$(X,F)$
if
$M_{xx}^k\geq 1$
for some positive integer k. Using this, one can easily prove the following theorem.
Theorem 5.3. Let
$X=\{0,1,2,\ldots ,m-1\}$
be a finite set and F be a relation on X. The following statements are equivalent.
-
(1) There are
$x\in X$
and a positive integer k such that
$M_{xx}^k>1$
. -
(2)
${\mathrm {ent}}(F)\neq 0$
.
Proof. Let
$x\in X$
and let k be a positive integer such that
$M_{xx}^k>1$
. It follows from Theorem 5.2 that
${\mathrm {ent}}(F)\neq 0$
. This proves the implication from statement (1) to (2). To prove the implication from statement (2) to (1), suppose that for every
$x\in X$
and for every positive integer k,
$M_{xx}^k\leq 1$
. First, we show that for each recurrent point x in
$(X,F)$
, there is exactly one element
$y\in X$
such that
$(x,y)\in F$
. To show this, let
$x\in X$
be a recurrent point in
$(X,F)$
. Also, let k be a positive integer such that
$M_{xx}^k\geq 1$
. It follows from our assumption that
$M_{xx}^k=1$
. Therefore, there is exactly one element
$y\in X$
such that
$(x,y)\in F$
.
For each recurrent point x in
$(X,F)$
, we denote by
$f(x)$
the point in X such that
$(x,f(x))\in F$
. Note that for each recurrent point x in
$(X,F)$
, the point
$f(x)$
is also a recurrent point in
$(X,F)$
. It follows that for any recurrent point x in
$(X,F)$
and for any positive integer m, there is exactly one point
${\mathbf {x}}\in X_F^m$
such that
${\mathbf {x}}(0)=x$
.
Next, we examine non-recurrent points in
$(X,F)$
. For any non-recurrent point x in
$(X,F)$
, either:
-
(1) there is a positive integer m such that for each
${\mathbf {x}}\in X_F^m$
,
${\mathbf {x}}(0)\neq x$
; or -
(2) there are a positive integer m and a point
${\mathbf {x}}\in X_F^m$
such that
${\mathbf {x}}(0) = x$
and
${\mathbf {x}}(m)$
is a recurrent point.
Let R be the set of recurrent points in
$(X,F)$
and let
$$ \begin{align*} A=\{x\in X\setminus R \mid \text{there is a positive integer } m \text{ such that for each } {\mathbf{x}}\in X_F^m, {\mathbf{x}}(0)\neq x \} \end{align*} $$
and
$$ \begin{align*} B&=\{x \in X\setminus R \mid \text{there are a positive integer } m \text{ and }\\ &\qquad{\mathbf{x}} \in X_F^m \text{ such that } {\mathbf{x}}(0)= x \text{ and } {\mathbf{x}}(m)\in R\}. \end{align*} $$
For each
$x\in A$
, let
$m_x$
be a positive integer such that for each
${\mathbf {x}}\in X_F^{m_x}$
,
${\mathbf {x}}(0)\neq x$
, and let
$m_A=\max \{m_x \mid x\in A\}$
. Let
$n=|X\setminus R|$
. Note that for each
$x\in B$
, for each positive integer
$m>n$
and for each
${\mathbf {x}}\in X_F^m$
,
$$ \begin{align*} {\mathbf{x}}(0)=x \Longrightarrow {\mathbf{x}}(m)\in R. \end{align*} $$
For each
$x\in B$
, let
$\ell _x=|\{{\mathbf {x}}\in X_F^{n+1} \mid {\mathbf {x}}(0)=x\}|$
. Then
$|\{{\mathbf {x}}\in X_F^{m} \mid {\mathbf {x}}(0)=x\}|=\ell _x$
for each positive integer
$m>n$
. Let
$m_0=\max \{n+1,m_A\}$
. It follows that for each positive integer
$m>m_0$
,
$|X_F^m|\leq |R|+\sum _{x\in B}\ell _x$
. Therefore, for any open cover
$\alpha $
for X and for any positive integer m,
$$ \begin{align*} m\geq m_0 \Longrightarrow N(X_F^m,\alpha^{m+1})\leq |R|+\sum_{x\in B}\ell_x. \end{align*} $$
Let
$\alpha $
be any open cover for X. Then
$$ \begin{align*} {\mathrm{ent}}(F,\alpha)=\lim_{m\to \infty}\frac{\log N(X_F^m,\alpha^{m+1})}{m}\leq \lim_{m\to \infty}\frac{\log (|R|+\sum_{x\in B}\ell_x)}{m}=0 \end{align*} $$
and it follows that
${\mathrm {ent}}(F)=0$
.
Lemma 5.4. Let X be a non-empty compact metric space and let F be a non-empty closed relation on X. If for each positive integer k, for each
$\varepsilon>0$
and for each
$a\in X$
, F has no
$(k,\varepsilon )$
returns on
$\{a\}$
, then the following hold.
-
(1) For each
$a \in X$
and for each positive integer m, there is at most one point
${\mathbf {x}} \in X_F^m$
such that
${\mathbf {x}}(0) = {\mathbf {x}}(m) = a$
. -
(2) For each positive integer m, for each
$a\in X$
and for each
${\mathbf {x}} \in X_F^m$
such that
${\mathbf {x}}(0) = {\mathbf {x}}(m) = a$
, we use
$\overline {{\mathbf {x}}} \in X_F^+$
to denote the periodic point that is defined by:-
(a)
$\pi _{[0,m]}(\overline {{\mathbf {x}}}) ={\mathbf {x}}$
; and -
(b)
$\overline {{\mathbf {x}}}(i+m) = \overline {{\mathbf {x}}}(i)$
for all
$i \in \mathbf Z_+$
.
Then for each positive integer m, for each
$a\in X$
, for each
${\mathbf {x}} \in X_F^m$
such that
${\mathbf {x}}(0) = {\mathbf {x}}(m) = a$
and for each
${\mathbf {y}}\in X_F^+$
, the following holds:
$$ \begin{align*} {\mathbf{y}}(i) = a \quad\text{for infinitely many non-negative integers } i \Longrightarrow {\mathbf{y}}=\overline{{\mathbf{x}}}. \end{align*} $$
-
Proof. Suppose that for each positive integer k, for each
$\varepsilon>0$
and for each
$a\in X$
, F has no
$(k,\varepsilon )$
returns on
$\{a\}$
. The first part of the lemma is obvious. To prove the second part of Lemma 5.4, let m be a positive integer, let
$a\in X$
and let
${\mathbf {x}} \in X_F^m$
such that
${\mathbf {x}}(0) = {\mathbf {x}}(m)~=~a$
. Also, let
${\mathbf {y}}\in X_F^+$
be such that
$ {\mathbf {y}}(i) = a$
for infinitely many non-negative integers i. To show that
${\mathbf {y}}=\overline {{\mathbf {x}}}$
, suppose that
${\mathbf {y}}\neq \overline {{\mathbf {x}}}$
. Then there is a positive integer i such that
${\mathbf {y}}(i)\neq \overline {{\mathbf {x}}}(i)$
. Choose and fix such a positive integer i and let
$m_1>i$
and
$m_2>i$
be positive integers such that
${\mathbf {y}}(m_1)=\overline {{\mathbf {x}}}(m_2)=a$
. Then
$$ \begin{align*} \pi_{[0,m_1]}({\mathbf{y}})\star \pi_{[0,m_2]}(\overline{{\mathbf{x}}}), \pi_{[0,m_2]}(\overline{{\mathbf{x}}})\star \pi_{[0,m_1]}({\mathbf{y}})\in X_F^{m_1+m_2} \end{align*} $$
are such points that:
-
(1)
$(\pi _{[0,m_1]}({\mathbf {y}})\star \pi _{[0,m_2]}(\overline {{\mathbf {x}}}))(0)=(\pi _{[0,m_1]}({\mathbf {y}})\star \pi _{[0,m_2]}(\overline {{\mathbf {x}}}))(m_1+m_2)=a$
and
$(\pi _{[0,m_2]} (\overline {{\mathbf {x}}})\star \pi _{[0,m_1]}({\mathbf {y}}))(0)=(\pi _{[0,m_2]}(\overline {{\mathbf {x}}})\star \pi _{[0,m_1]}({\mathbf {y}}))(m_1+m_2)=a$
; and -
(2)
$(\pi _{[0,m_1]}({\mathbf {y}})\star \pi _{[0,m_2]}(\overline {{\mathbf {x}}}))(i)\neq (\pi _{[0,m_2]}(\overline {{\mathbf {x}}})\star \pi _{[0,m_1]}({\mathbf {y}}))(i)$
.
It follows that F has an
$(m_1+m_2, ({d({\mathbf {y}}(i), \overline {{\mathbf {x}}}(i))})/{2})$
-return on
$\{a\}$
, which is a contradiction.
Lemma 5.5. Let X be a non-empty compact metric space and let F be a finite subset of
$X\times X$
. If
${\mathrm {ent}}({ F})\neq 0$
, then
${X_F^{+}}$
is uncountable.
Proof. Assume that
${\mathrm {ent}}({ F})\neq 0$
. First, let
$Z=p_1(F)\cup p_2(F)$
. Then Z is finite and F is a relation on Z, so we can use Theorem 5.3 for
$(Z,F)$
. Let
$x\in X$
and let k be a positive integer such that
$M_{xx}^k>1$
. It follows from Theorem 5.3 that such a point x and such a positive integer k do exist. Let
$n=M_{xx}^k$
and let
${\mathbf {x}}_0,{\mathbf {x}}_1,{\mathbf {x}}_2,\ldots ,{\mathbf {x}}_{n-1}\in Z_F^k$
be n distinct points such that for each
$i\in \{0,1,2,\ldots ,n-1\}$
,
${\mathbf {x}}_{i}(0)=x$
and
${\mathbf {x}}_{i}(k)=x$
. Next, let
$ Y=\{{\mathbf {x}}_{i}(j) \mid i\in \{0,1,2,\ldots ,n-1\}, j\in \{0,1,2,\ldots ,k\}\}$
and let
$$ \begin{align*} G=\{({\mathbf{x}}_{i}(j),{\mathbf{x}}_{i}(j+1)) \mid i\in \{0,1,2,\ldots,n-1\}, j\in \{0,1,2,\ldots,k-1\} \}. \end{align*} $$
Note that
$Y\subseteq X$
and that
$G\subseteq F$
is a relation on Y such that
$p_2(G)\subseteq p_1(G)$
. Also note that
${\mathrm {ent}}(G)\neq 0$
since G has a
$(k,\varepsilon )$
-return on Y. By Theorem 3.16, the topological entropy of the shift map
$\sigma $
on
${ Y_G^{+}}$
is non-zero. Therefore, by Theorem 5.1, the dynamical system
$({ Y_G^{+}},\sigma )$
is Li–Yorke chaotic, which implies that there is an uncountable scrambled set in
${ Y_G^{+}}$
. Therefore,
${ Y_G^{+}}$
is uncountable. Since
$Y_G^{+}\subseteq X_F^{+}$
, it follows that
$X_F^{+}$
is uncountable.
Theorem 5.6. Let X be a non-empty compact metric space and let F be a finite subset of
$X \times X$
. The following statements are equivalent.
-
(1)
${\mathrm {ent}}({ F})\neq 0$
. -
(2) There are a set
$A\subseteq X$
, a positive integer k and an
$\varepsilon>0$
such that F has a
$(k,\varepsilon )$
-return on A. -
(3) There are:
-
(a) positive integers
$k_x$
and
$k_y$
; -
(b) points
${\mathbf {x}}\in { X_F^{k_x}}$
and
${\mathbf {y}} \in { X_F^{k_y}}$
such that
$ { {\mathbf {x}}(k_x)}={ {\mathbf {x}}(0)}={ {\mathbf {y}}(0)}={ {\mathbf {y}}(k_y)}$
; and -
(c) a positive integer j such that
${0}<j\leq \min \{ k_x , k_y \} \text { and } { {\mathbf {x}}(j)}\neq { {\mathbf {y}}(j)}$
.
-
-
(4)
${X_F^{+}}$
is uncountable.
Proof. The implication from statement (1) to (4) is Lemma 5.5, the implication from statement (2) to (1) is Theorem 4.5 and the implication from statement (3) to (1) is Corollary 4.15.
Now we prove the implication from statement (4) to (1). Assume
${\mathrm {ent}}({ F})=0$
. By Theorem 4.5, there do not exist a non-empty set
$A\subseteq X$
, a positive integer k and an
$\varepsilon>0$
such that F has a
$(k,\varepsilon )$
return on A. According to Lemma 5.4, for each
$a\in p_1(G) $
, there is at most one point in
${X_F^{+}}$
with first coordinate a and in which a occurs as a coordinate infinitely many times. Let B be the set of all points in
${X_F^{+}}$
whose first coordinate occurs infinitely many times as a coordinate. Since F is finite, B is finite. Also, for each element
${\mathbf {x}}\in { X_F^{+}}$
, some coordinate must be repeated infinitely many times, so there is a non-negative integer k such that
$\sigma ^k({\mathbf {x}})\in B$
. Since
$\sigma ^{-1}({\mathbf {x}})$
is finite for each
${\mathbf {x}}\in { X_F^{+}}$
and since
${X_F^{+}}= \bigcup _{k=0}^\infty \sigma ^{-k}(B)$
, it follows that
${X_F^{+}}$
is countably infinite or finite.
We prove the implication from statement (1) to (2) similarly as the implication from statement (4) to (1): if we assume that there does not exist a set
$A\subseteq X$
, an integer k and an
$\varepsilon>0$
such that F has a
$(k,\varepsilon )$
-return on A, then again Lemma 5.4 can be used as above in the proof of the implication from statement (4) to (1) to show that
${X_F^{+}}$
is countable, and thus, by Lemma 5.5, it follows that
${\mathrm {ent}}({ F})=0$
.
Finally, we prove the implication from statement (2) to (3). Let
$A\subseteq X$
, let k be a positive integer and let
$\varepsilon>0$
such that F has a
$(k,\varepsilon )$
-return on A. Also, let
$a^*\in A$
and let
$\Psi : \Sigma _2 \rightarrow X_F^+$
be an
$(F,A,k,\varepsilon )$
-return function for
${a^{*}}$
. Since F is finite, there are at least two different sequences
${\mathbf {s}}, {\mathbf {t}}\in \Sigma _2$
of
$0$
s and
$1$
s for which there are strictly increasing sequences
$(i_n)$
and
$(j_n)$
of positive integers such that:
-
•
$i_{ 0}=1$
and
$j_{ 0}=1$
; -
• for each positive integer
$\ell $
,
${\Psi ({\mathbf {s}})_{i_{\ell }},\Psi ({\mathbf {t}})_{j_{\ell }}}\in A $
; and -
• for all positive integers
$\ell _1$
and
$\ell _2$
,
${\Psi ({\mathbf {s}})_{i_{\ell _1}}}={ \Psi ({\mathbf {s}})_{i_{\ell _2}}}={ \Psi ({\mathbf {t}})_{j_{\ell _1}}}={ \Psi ({\mathbf {t}})_{j_{\ell _2}}}$
.
Since
${\mathbf {s}}\neq {\mathbf {t}}$
, it follows that there is a positive integer m such that
${\Psi ({\mathbf {s}})_m}\neq { \Psi ({\mathbf {t}})_m}$
. Fix such a positive integer m. Then, let:
-
•
$\ell _1$
and
$\ell _2$
be such positive integers that
$i_{\ell _1}>m$
and
$j_{\ell _2}>m$
and let
$$ \begin{align*} k_x=i_{\ell_1} \quad\mathrm{and}\quad k_y=j_{\ell_2}; \end{align*} $$
-
•
${\mathbf {x}}=\pi _{[1,k_x]}({ \Psi ({\mathbf {s}})})$
and
${\mathbf {y}} = \pi _{[1,k_y]}({ \Psi ({\mathbf {t}})})$
; and -
•
$j=m$
.
Note that for constructed
$k_x$
,
$k_y$
,
${\mathbf {x}}$
,
${\mathbf {y}}$
and j, statement (3) follows. This completes the proof.
The following is a corollary of Theorem 5.6.
Corollary 5.7. Let X be a non-empty compact metric space and let F be a finite subset of
$X \times X$
. If
${\mathrm {ent}}({ F})\neq 0$
, then there are elements
${\mathbf {s}}_1$
and
${\mathbf {s}}_2$
of F such that
$$ \begin{align*} p_1({\mathbf{s}}_1)=p_1({\mathbf{s}}_2) \quad\mathrm{and}\quad p_2({\mathbf{s}}_1) \neq p_2 ({\mathbf{s}}_2), \end{align*} $$
and there are elements
${\mathbf {t}}_1$
and
${\mathbf {t}}_2$
of F such that
$$ \begin{align*} p_2({\mathbf{t}}_1)=p_2({\mathbf{t}}_2) \quad\mathrm{and}\quad p_1({\mathbf{t}}_1)\neq p_1({\mathbf{t}}_2). \end{align*} $$
Proof. Let
${\mathrm {ent}}(F)\neq 0$
. By Theorem 5.6, there are:
-
(1) positive integers
$k_x$
and
$k_y$
; -
(2) points
${\mathbf {x}}\in { X_F^{k_x}}$
and
${\mathbf {y}} \in { X_F^{k_y}}$
such that
$ { {\mathbf {x}}(k_x)}={ {\mathbf {x}}(0)}={ {\mathbf {y}}(0)}={ {\mathbf {y}}(k_y)}$
; and -
(3) a positive integer j such that
$0<j\leq \min \{ k_x , k_y \} \text { and } { {\mathbf {x}}(j)}\neq { {\mathbf {y}}(j)}$
.
Fix such
$k_x$
,
$k_y$
,
${\mathbf {x}}$
and
${\mathbf {y}}$
, and let
$j_1$
be the smallest among all positive integers
$j\in \{{ 1,2,3},\ldots , k_x\}$
such that
${ {\mathbf {x}}(j)}\neq { {\mathbf {y}}(j)}$
. Let
$$ \begin{align*} {\mathbf{s}}_1=({ {\mathbf{x}}(j_1-1)},{ {\mathbf{x}}(j_1)}) \quad\mathrm{and}\quad {\mathbf{s}}_2=({ {\mathbf{y}}(j_1-1)},{ {\mathbf{y}}(j_1))}. \end{align*} $$
Then
$p_1({\mathbf {s}}_1)=p_1({\mathbf {s}}_2)$
and
$p_2({\mathbf {s}}_1) \neq p_2 ({\mathbf {s}}_2)$
. This proves the first part of the claim. Next, we prove the second part of the claim. Suppose that there are no elements
${\mathbf {t}}_1$
and
${\mathbf {t}}_2$
in F such that
$ p_2({\mathbf {t}}_1)=p_2({\mathbf {t}}_2)$
and
$p_1({\mathbf {t}}_1)\neq p_1({\mathbf {t}}_2)$
. Then F is a graph of a single-valued function
$f:p_2({ F})\rightarrow p_1({ F})$
. Note that
$\varphi :p_2({ F})\rightarrow { X_{F^{-1}}}$
, defined by
$\varphi (x)=(x,f(x),f^2(x),f^3(x),\ldots )$
for any
$x\in p_2({ F})$
, is a homeomorphism. Since F is finite, it follows that
$p_2({ F})$
is finite. Therefore,
${X_{F^{-1}}^{+}}$
is finite, which is a contradiction since by Theorem 5.6,
${X_{F^{-1}}^{+}}$
is uncountable (since
${\mathrm {ent}}(F^{-1})$
is also non-zero). This completes the proof.
Definition 5.4. Let X be a non-empty compact metric space, let
$f:X\rightarrow X$
be a continuous function and let
$x,y\in X$
such that
$x\neq y$
. The set
$\{x,y\}$
is called a DC2-pair for f if
$$ \begin{align*} \liminf\bigg(\frac{1}{n}\cdot \sum_{i=1}^{n} \rho(f^i(x),f^i(y))\bigg)=0 \quad\mathrm{and}\quad \limsup\bigg(\frac{1}{n}\cdot \sum_{i=1}^{n} \rho(f^i(x),f^i(y))\bigg)>0. \end{align*} $$
Definition 5.5. Let X be a non-empty compact metric space, let
$f:X\rightarrow X$
be a continuous function and let
$S\subseteq X$
. We say that the set S is a DC2-scrambled set in
$(X,f)$
if for all
$x,y\in S$
,
$$ \begin{align*} x\neq y \Longrightarrow \{x,y\} \quad\text{is a DC2-pair for } f. \end{align*} $$
Definition 5.6. Let X be a non-empty compact metric space and let
$f:X\rightarrow X$
be a continuous function. The dynamical system
$(X,f)$
is called DC2-chaotic if X contains an uncountable DC2-scrambled set.
Observation 5.8. Let X be a non-empty compact metric space and let
$f:X\rightarrow X$
be a continuous function. If
$(X,f)$
is DC2-chaotic, then
$(X,f)$
is Li–Yorke chaotic. See [Reference Doleželová-Hantáková, Roth and Roth8] for more information.
Putting these facts together, we see that the case where F is a finite subset of
$X\times X$
produces a dynamical system
$({ X_F^{+}},\sigma )$
where several forms of chaos are equivalent. See the following corollary. Note that it is a known fact that, in general, Li–Yorke chaos does not imply positive entropy, see [Reference Smítal23] for more information.
Corollary 5.9. Suppose X is a non-empty compact metric space and F is a finite subset of
$X \times X$
. The following statements are equivalent.
-
(1)
${\mathrm {ent}}({ F})\neq 0$
. -
(2)
$({ X_F^{+}},\sigma )$
is Li–Yorke chaotic. -
(3)
$({ X_F^{+}},\sigma )$
has a DC2-scrambled Cantor set.
Proof. First, we show the implication from statement (1) to (3). Suppose that
${\mathrm {ent}}({ F})\neq 0$
. A similar argument as in the first part of the proof of Lemma 5.5 gives that
$h(\sigma )\neq 0$
, where
$\sigma $
is the shift map on
${X_F^{+}}$
. By [Reference Downarowicz9, Theorem 1.1, p. 138]
$({ X_F^{+}},\sigma )$
is DC2-chaotic and it follows from [Reference Downarowicz9, Remark 3, p. 148] that
$({ X_F^{+}},\sigma )$
has a DC2-scrambled Cantor set.
The implication from statement (3) to (2) follows from Observation 5.8.
Finally, we prove the implication from statement (2) to (1). Suppose that
$({ X_F^{+}},\sigma )$
is Li–Yorke chaotic. Then
${X_F^{+}}$
contains an uncountable scrambled set. It follows that
${X_F^{+}}$
is uncountable. By Theorem 5.6,
${\mathrm {ent}}({ F})\neq 0$
.
The following examples, where a countable closed subset
${F}$
of
$[0,1]\times [0,1]$
, such that:
-
(1)
${\mathrm {ent}}({ F})=0$
; and -
(2)
${X_F^{+}}$
is uncountable,
is presented, together with two problems, is a good place to finish the paper.
Example 5.10. Let
$X=[0,1]$
and let
$ { F}=\{(0,0)\}\cup \{ ({1}/{2^i}, {1}/{2^{i+1}}) \mid i\in \{0,1,2,3,\ldots \} \}{\kern1pt}\cup{\kern1pt} \{ ( {1}/{2^i}, {1}/{2^{i+1}}) \mid i\in \{0,1,2, 3,\ldots \}\} {\kern1pt}\cup{\kern1pt} \{ ({1}/{2^i},{1}/{2^{i+2}}) \mid i\in \{0,1,2,3,\ldots \} \}$
. Then
${\mathrm {ent}}({ F})=0$
since
${F} \subseteq \{ (x,y) \in [0,1]\times [0,1] \mid y\leq x \}$
(see [Reference Erceg and Kennedy10]), and
${X_F^{+}}$
is uncountable since each coordinate a of an element of
${X_F^{+}}$
can be followed by either
$\tfrac 12 a$
or
$\tfrac 14 a$
.
Problem 5.11. Let X be a non-empty compact metric space and let F be a countable closed relation on X such that
${\mathrm {ent}}({ F})\neq 0$
. Is it true that either F or
${F}^{-1}$
has a
$(k,\varepsilon )$
-return for some positive integer k and some
$\varepsilon>0$
?
Problem 5.12. Let
$X=[0,1]$
and let
$f:X\rightarrow X$
be a continuous function such that
$h(f)\neq 0$
. Is it true that
$\Gamma (f)$
has a
$(k,\varepsilon )$
-return for some positive integer k and some
$\varepsilon>0$
?
Acknowledgements
The authors thank the anonymous referee for careful reading. Their comments helped us to improve our paper significantly. This work is supported in part by the Slovenian Research Agency (research projects J1-4632, BI-US/22-24-086 and BI-US/22-24-094, and research program P1-0285).
