Topological Sequence Entropy and Chaos

2014 ◽  
Vol 24 (07) ◽  
pp. 1450100
Author(s):  
Xin Liu ◽  
Huoyun Wang ◽  
Heman Fu
Keyword(s):  
Dynamical System ◽  
Sequence Entropy ◽  
Null System ◽  
Upper Density ◽  
Topological Sequence Entropy

A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Given 0 ≤ p ≤ q ≤ 1, a dynamical system is [Formula: see text] chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. It shows that, for any 0 ≤ p < q ≤ 1 or p = q = 0 or p = q = 1, a dynamical system which is null and [Formula: see text] chaotic can be realized.

A Remark on Topological Sequence Entropy

2017 ◽  
Vol 27 (07) ◽  
pp. 1750107 ◽  
Author(s):  
Xinxing Wu
Keyword(s):  
Dynamical System ◽  
Connected Space ◽  
Sequence Entropy ◽  
Topological Sequence Entropy ◽  
Suitable Sequence

Let [Formula: see text] be the supremum of all topological sequence entropies of a dynamical system [Formula: see text]. This paper obtains the iteration invariance and commutativity of [Formula: see text] and proves that if [Formula: see text] is a multisensitive transformation defined on a locally connected space, then [Formula: see text]. As an application, it is shown that a Cournot map is Li–Yorke chaotic if and only if its topological sequence entropy relative to a suitable sequence is positive.

Sequence Entropy and Mild Mixing

1992 ◽  
Vol 44 (1) ◽  
pp. 215-224 ◽  
Author(s):  
Qing Zhang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Infinite Subset ◽  
Mixing Properties ◽  
Sequence Entropy ◽  
Number Of Authors

Entropy characterizations of different spectral and mixing properties of dynamical systems were dealt with by a number of authors (see [5], [6] and [8]).Given an infinite subset Γ = {tn}of N and a dynamical system (X, β,μ, T) one can define sequence entropy along for any finite Petition ξ, and hΓ(T) —supξ hΓ(T,ξ). In [6] Kushnirenko used sequence entropy to give a characterization of systems with discrete spectrum.

SOME RESULTS ON ENTROPY AND SEQUENCE ENTROPY

1999 ◽  
Vol 09 (09) ◽  
pp. 1731-1742 ◽  
Author(s):  
F. BALIBREA ◽  
V. JIMÉNEZ LÓPEZ ◽  
J. S. CÁNOVAS PEÑA
Keyword(s):  
Topological Entropy ◽  
Elementary Proof ◽  
Type Inequality ◽  
Topological Invariants ◽  
Metric Entropy ◽  
Sequence Entropy ◽  
Piecewise Monotone ◽  
Monotone Maps ◽  
Topological Sequence Entropy

In this paper we study some formulas involving metric and topological entropy and sequence entropy. We summarize some classical formulas satisfied by metric and topological entropy and ask the question whether the same or similar results hold for sequence entropy. In general the answer is negative; still some questions involving these formulas remain open. We make a special emphasis on the commutativity formula for topological entropy h(f ◦ g)=h(g ◦ f) recently proved by Kolyada and Snoha. We give a new elementary proof and use similar ideas to prove commutativity formulas for metric entropy and other topological invariants. Finally we prove a Misiurewicz–Szlenk type inequality for topological sequence entropy for piecewise monotone maps on the interval I=[0, 1]. For this purpose we introduce the notion of upper entropy.

scholarly journals Topology and topological sequence entropy

2019 ◽  
Vol 63 (2) ◽  
pp. 205-296
Author(s):  
L’ubomír Snoha ◽  
Xiangdong Ye ◽  
Ruifeng Zhang
Keyword(s):  
Sequence Entropy ◽  
Topological Sequence Entropy

scholarly journals Computing explicitly topological sequence entropy: the unimodal case

2002 ◽  
Vol 52 (4) ◽  
pp. 1093-1133
Author(s):  
Victor Jiménez López ◽  
Jose Salvador Cánovas Peña
Keyword(s):  
Sequence Entropy ◽  
Topological Sequence Entropy
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