scholarly journals Topological sequence entropy of $\omega$–limit sets of interval maps

2001 ◽  
Vol 7 (4) ◽  
pp. 781-786
Author(s):  
José S. Cánovas ◽  
Keyword(s):  
Limit Sets ◽  
Sequence Entropy ◽  
Interval Maps ◽  
Topological Sequence Entropy

Typical limit sets of critical points for smooth interval maps

2000 ◽  
Vol 20 (1) ◽  
pp. 15-45 ◽  
Author(s):  
ALEXANDER BLOKH ◽  
MICHAŁ MISIUREWICZ
Keyword(s):  
Critical Points ◽  
Limit Sets ◽  
Interval Maps ◽  
Class C

We prove that interval maps for which $\omega$-limit sets of all critical points are minimal are dense in the space of all interval maps of class $C^2$.

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.

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.

ON THE STRUCTURE OF THE ω-LIMIT SETS FOR CONTINUOUS MAPS OF THE INTERVAL

1999 ◽  
Vol 09 (09) ◽  
pp. 1719-1729 ◽  
Author(s):  
LLUÍS ALSEDÀ ◽  
MOIRA CHAS ◽  
JAROSLAV SMÍTAL
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Discrete Dynamical Systems ◽  
Limit Sets ◽  
Interval Maps ◽  
Positive Topological Entropy ◽  
Countable Ordinal ◽  
Continuous Maps

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα∈I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

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
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