Topological sequence entropy of interval maps

Cánovas, José S.

doi:10.1088/0951-7715/17/1/003

Cited by 18 publications

(18 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We end this section with the following observation (we thank W. Huang for suggesting this question). In [28] the authors have shown that if T : [0, 1] −→ [0, 1] is a continuous map then T is null if and only if T is not chaotic in the sense of Li and Yorke; and in [18] the author has shown that h * top (T ) ∈ {0, log 2, ∞}. In fact we can prove the following:…”

Section: An Application To Interval Mapsmentioning

confidence: 93%

Local entropy theory

Glasner

2009

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

Section: An Application To Interval Mapsmentioning

confidence: 93%

Local entropy theory

Glasner

2009

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…A similar topological concept, called topological sequence entropy, was introduced and studied several years later by Goodman [25], who also investigated the relationship between his concept and the measure-theoretic analogue defined by Kushnirenko. After that there has been a significant activity in studying both kinds of sequence entropies -see for example [9,11,22,33].…”

Section: Introductionmentioning

confidence: 99%

Metric Versus Topological Receptive Entropy of Semigroup Actions

Biś

Dikranjan

Bruno

et al. 2021

Qual. Theory Dyn. Syst.

View full text Add to dashboard Cite

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov (Adv Math 115:54–98, 1995). We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin–Katok Formula and the local Variational Principle.

show abstract

“…By [14], T ∈ C(I) is Li-Yorke chaotic if and only if h * (T ) > 0. Due to [9], there are only three possibilities for T ∈ C(I), namely, h * (T ) is either 0 or log 2 or ∞. These facts are shown in Table 1.…”

mentioning

confidence: 99%

“…. } ∪ {∞} X finite sets, zero-dimensional spaces with finite derived sets [48] interval [9], circle [8], finite trees [42], finite graphs [43] zero-dimensional spaces with infinite derived sets [42], some dendrites [42], manifolds of dimension ≥ 2 [42] Table 2. Known values of S(X) Since every dendrite is an absolute retract for the class of all compact metric spaces [7], we in fact have S(X) = log N * not only for the manifolds with dimension ≥ 2, but for any compact metric space X containing the dendrite from [42].…”

mentioning

confidence: 99%

Topology and topological sequence entropy

Snoha

Zhang

2019

Sci. China Math.

View full text Add to dashboard Cite

Let X be a compact metric space and T : X −→ X be continuous. Let h * (T ) be the supremum of topological sequence entropies of T over all subsequences of Z + and S(X) be the set of the values h * (T ) for all continuous maps T on X. It is known that {0} ⊆ S(X) ⊆ {0, log 2, log 3, . . .} ∪ {∞}. Only three possibilities for S(X) have been observed so far, namely S(X) = {0}, S(X) = {0, log 2, ∞} and S(X) = {0, log 2, log 3, . . .} ∪ {∞}.In this paper we completely solve the problem of finding all possibilities for S(X) by showing that in fact for every set {0} ⊆ A ⊆ {0, log 2, log 3, . . .} ∪ {∞} there exists a one-dimensional continuum X A with S(X A ) = A. In the construction of X A we use Cook continua. This is apparently the first application of these very rigid continua in dynamics.We further show that the same result is true if one considers only homeomorphisms rather than continuous maps. The problem for group actions is also addressed. For some class of group actions (by homeomorphisms) we provide an analogous result, but in full generality this problem remains open. The result works also for an analogous class of semigroup actions (by continuous maps).

show abstract

Topological sequence entropy of interval maps

Abstract: We establish a full classification of chaotic and non-chaotic interval maps from the point of view of topological sequence entropy. This completes the papers of Franzová and Smítal (1991 Positive sequence topological entropy characterizes chaotic maps

Cited by 18 publications

References 15 publications

Local entropy theory

Local entropy theory

Metric Versus Topological Receptive Entropy of Semigroup Actions

Topology and topological sequence entropy

Contact Info

Product

Resources

About