SUFFICIENT CONDITIONS FOR NON-ZERO ENTROPY AND FINITE RELATIONS IZTOK BANI ˇC RENE GRIL ROGINA JUDY KENNEDY V AN NALL

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SUFFICIENT CONDITIONS FOR NON-ZERO ENTROPY AND FINITE

RELATIONS

IZTOK BANI ˇ

C, RENE GRIL ROGINA, JUDY KENNEDY, VAN NALL

Abstract. We introduce the notions of returns, dispersions and well-aligned sets for closed

relations on compact metric spaces and then we use them to obtain non-trivial suﬃcient

conditions for such a relation to have non-zero entropy. In addition, we give a characteri-

zation of ﬁnite relations with non-zero entropy in terms of Li-Yorke and DC2-chaos.

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 con-

tinuous function f:X→Xon 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 σon the inverse limit lim

←−−(X,f). More precisely, suppose Xis a compact metric space.

If f:X→Xis a continuous function, the inverse limit space generated by fis

lim

←−−(X,f) :=n(x1,x2,x3,...)∈

∞

i=1

X|for each positive integer i,xi=f(xi+1)o,

also abbreviated as lim

←−− f. The map fon Xinduces a natural homeomorphism σon lim

←−− f,

called the shift map, deﬁned by

σ(x1,x2,x3,x4,...)=(x2,x3,x4,...)

for each (x1,x2,x3,x4,...) in lim

←−− f. It is a well-known result that the entropy of fis then

equal to the entropy of σ[7, Proposition 5.2].

To study such inverse limits lim

←−− fand shift maps σ: lim

←−− f→lim

←−− f, the study of back-

ward orbits of points of dynamical systems (X,f) is also required; note that the inverse

limit lim

←−− fis 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−1is well-deﬁned.

But usually, f−1is not a well-deﬁned function, therefore, a more general tool is needed to

study these properties. Note that for a continuous function f:X→X, the set

Γ(f)−1={(y,x)∈X×X|y=f(x)}

is a closed relation on Xthat describes best the dynamics of (X,f) in the backward direc-

tion when f−1is not well-deﬁned. So, generalizing topological dynamical systems (X,f)

to topological dynamical systems (X,G) with closed relations Gon Xby making the iden-

tiﬁcation (X,f)=(X,Γ(f)) is only natural.

2020 Mathematics Subject Classiﬁcation. Primary: 37B40, 37B45, 37E05; Secondary: 54C08, 54E45,

54F15, 54F17.

Key words and phrases. Entropy, closed relations, ﬁnite relations.

This work was supported by the Slovenian Research Agency under the research program P1-0285.

arXiv:2210.01880v1 [math.DS] 4 Oct 2022

2 IZTOK BANI ˇ

C, RENE GRIL ROGINA, JUDY KENNEDY, VAN NALL

Recently, many such generalizations of dynamical systems were introduced and studied

(see [4,5,8,15,17,18,19,20,22,23,25], where more references may be found). However,

there is not much known of such dynamical systems and therefore, there are many proper-

ties of such set-valued dynamical systems that are yet to be studied. In [4], the notion of

topological entropy h(f) of continuous functions f:X→Xon compact metric spaces X

was generalized to the notion of topological entropy ent(G) of closed relations Gon com-

pact metric spaces X. In this paper, we continue our research from [4]. We introduce the

notions of returns, dispersions 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 Ghaving a (k, ε)-return.

We proceed as follows. In Section 2, basic deﬁnitions and notation that are needed

later in the paper are given and presented. In Section 3, the topological entropy for closed

relations is deﬁned and in addition, basic results from [4] are presented. In Section 4, our

ﬁrst main result as well as many illustrative examples and corollaries are given and proved.

In our last section, 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 will be used later in the paper.

Deﬁnition 2.1. Let Xbe a compact metric space. We always use ρto denote the metric on

Deﬁnition 2.2. Suppose Xis a compact metric space. If for each positive integer n,fn:

X→Xis a continuous function, the inverse limit space generated by (fn)is

lim

←−−(X,fn) :=n(x1,x2,x3,...)∈

∞

i=1

X|for each positive integer i,xi=fi(xi+1)o.

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 also a continuum.

Next, we deﬁne chainable continua (using inverse limits); see [24, Section XII] for more

details.

Deﬁnition 2.4. A continuum Xis chainable if there is a sequence ( fn) of continuous sur-

jections fn: [0,1] →[0,1] such that Xis homeomorphic to lim

←−−([0,1],fn)∞

n=1.

Deﬁnition 2.5. A continuum Xis decomposable, if there are proper subcontinua Aand Bof

X(A,B,X) such that X=A∪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.

SUFFICIENT CONDITIONS FOR NON-ZERO ENTROPY AND FINITE RELATIONS 3

Bing showed in [2] that any two pseudoarcs are homeomorphic.

Next, we present basic deﬁnitions and well-known results about closed relations and

Mahavier products.

Deﬁnition 2.7. Let Xand Ybe metric spaces, and let f:X→Ybe a function. We use

Γ(f)={(x,y)∈X×Y|y=f(x)}to denote the graph of the function f .

Deﬁnition 2.8. Let Xbe a compact metric space and let G⊆X×Xbe a relation on X. If G

is closed in X×X, then we say that Gis a closed relation on X.

Deﬁnition 2.9. Let Xbe a set and let Gbe a relation on X. Then we deﬁne G−1={(y,x)∈

X×X|(x,y)∈G}to be the inverse relation of the relation G on X.

Deﬁnition 2.10. Let Xbe a compact metric space and let Gbe a closed relation on X. Then

we call

i=1G−1=n(x1,x2,x3,...,xm+1)∈

m+1

i=1

X|for each i∈ {1,2,3,...,m},(xi+1,xi)∈Go

for each positive integer m,the m-th Mahavier product of G−1, and

?∞

i=1G−1=n(x1,x2,x3,...)∈

∞

i=1

X|for each positive integer i,(xi+1,xi)∈Go

the inﬁnite Mahavier product of G−1.

Deﬁnition 2.11. Let Xbe a compact metric space and let Gbe a closed relation on X. The

function

σ:?∞

n=1G−1→?∞

n=1G−1,

deﬁned by

σ(x1,x2,x3,x4,...)=(x2,x3,x4,...)

for each (x1,x2,x3,x4,...)∈?∞

n=1G−1, is called the shift map on ?∞

n=1G−1.

3. Topological entropy of closed relations on compact metric spaces

In this section we will summarize the generalization of topological entropy to closed

relations on a compact metric space introduced in [4].

Deﬁnition 3.1. Let Xbe a compact metric space and let Sbe a family of subsets of X. We

use |S| to denote the cardinality of S.

Deﬁnition 3.2. Let Xbe a compact metric space and let Sbe a family of subsets of X. For

each positive integer n, we use Snto denote the family

Sn={S1×S2×S3×...×Sn|S1,S2,S3,...,Sn∈ S}.

We call the elements S1×S2×S3×...×Snof Sthe n-boxes (generated by the family S).

Deﬁnition 3.3. Let Xbe a compact metric space and let Ube a non-empty open cover for

X. We use N(U)to denote

N(U)=min{|V| | V is a non-empty ﬁnite subcover of U}.

Deﬁnition 3.4. Let Xbe a compact metric space, let Kbe a closed subset of the product

i=1X, and let Ube a non-empty family of open subsets of Qn

i=1Xsuch that K⊆SU.

We use N(K,U)to denote

N(K,U)=minn|V| | V is a non-empty subfamily of Usuch that K⊆[Vo.

4 IZTOK BANI ˇ

C, RENE GRIL ROGINA, JUDY KENNEDY, VAN NALL

Theorem 3.5. [4, Theorem 3.7] Let X be a compact metric space, let G be a closed relation

on X, and let αbe a non-empty open cover for X. Then the limit

lim

m→∞

log N(?m

i=1G−1, αm+1)

exists.

Deﬁnition 3.6. Let Xbe a compact metric space, let Gbe a closed relation on X, and let

αbe a non-empty open cover for X. We deﬁne the entropy of G with respect to the open

cover αby

ent(G, α)=lim

m→∞

log N(?m

i=1G−1, αm+1)

Deﬁnition 3.7. Let Xbe a metric space and let Sand Tbe families of subsets of X. We

say that the family Sreﬁnes the family T, if for each S∈ S there is T∈ T such that S⊆T.

The notation

T ≤ S

means that the family Sreﬁnes the family T.

Proposition 1. [4, Proposition 1] Let X be a compact metric space and let G be a closed

relation on X. For all non-empty open covers αand β,

α≤β=⇒ent(G, α)≤ent(G, β).

Proposition 2. [4, Proposition 2] Let X be a compact metric space and let αbe a non-empty

open cover for X. For all closed relations H and G on X,

H⊆G=⇒ent(H, α)≤ent(G, α).

Deﬁnition 3.8. Let Xbe a compact metric space, let Gbe a closed relation on X, and let

E={ent(G, α)|αis a non-empty open cover for X}.

We deﬁne the entropy of G by

ent(G)=









sup(E); G,∅and Eis bounded in R

∞;G,∅and Eis not bounded in R.

The following three theorems from [4] summarize what we need to know about ent(G).

Theorem 3.9. [4, Theorem 3.11] Let X be a compact metric space. For all closed relations

H and G on X,

H⊆G=⇒ent(H)≤ent(G).

Theorem 3.10. [4, Theorem 3.12] Let X be a compact metric space and let G be a closed

relation on X. Then

ent(G−1)=ent(G).

In [4] it is shown that the entropy of closed relations on Xis a generalization of the well-

known topological entropy of continuous functions f:X→X. For a continuous function

f:X→Xon the compact metric space Xthe entropy of fis usually denoted h(f). To suit

the purposes of this paper it is enough to note that it is shown in [4, Theorem 3.19] that

h(f)=ent(Γ(f)). For more information on h(f) see [4] and [27]. Finally, we need Theorem

3.12, also from [4], where the following notation is used.

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SUFFICIENTCONDITIONSFORNON-ZEROENTROPYANDFINITERELATIONSIZTOKBANIC,RENEGRILROGINA,JUDYKENNEDY,VANNALLAbstract.Weintroducethenotionsofreturns,dispersionsandwell-alignedsetsforclosedrelationsoncompactmetricspacesandthenweusethemtoobtainnon-trivialsucientconditionsforsucharelationtohavenon-zeroentrop...

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SUFFICIENT CONDITIONS FOR NON-ZERO ENTROPY AND FINITE RELATIONS IZTOK BANI ˇC RENE GRIL ROGINA JUDY KENNEDY V AN NALL

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