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

2001 ◽  
Vol 7 (4) ◽  
pp. 781-786
Author(s):  
José S. Cánovas ◽  
2000 ◽  
Vol 20 (1) ◽  
pp. 15-45 ◽  
Author(s):  
ALEXANDER BLOKH ◽  
MICHAŁ MISIUREWICZ

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 $C^2$.


1999 ◽  
Vol 09 (09) ◽  
pp. 1731-1742 ◽  
Author(s):  
F. BALIBREA ◽  
V. JIMÉNEZ LÓPEZ ◽  
J. S. CÁNOVAS PEÑA

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.


2017 ◽  
Vol 27 (07) ◽  
pp. 1750107 ◽  
Author(s):  
Xinxing Wu

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.


1999 ◽  
Vol 09 (09) ◽  
pp. 1719-1729 ◽  
Author(s):  
LLUÍS ALSEDÀ ◽  
MOIRA CHAS ◽  
JAROSLAV SMÍTAL

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].


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

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