On the dynamics of composition of commuting interval maps

Abstract

Let f, g : [0, 1] → [0, 1] be two commuting continuous maps. We establish some results on the topological dynamic shared by both maps and state some conditions to get that the topological entropy of the composition f • g will be positive. Published by Elsevier Inc.

Advances in Periodic Difference Equations with Open Problems

“…Following the Sharkovsky's order of natural numbers, let T 1 = {f ∈ C(I) : P(f ) is closed}, T 2 = {f ∈ C(I) : f has periodic points of period 2 n , n ≥ 0} and T 3 = {f ∈ C(I) : f has a periodic point which is not a power of two}. The next result was proved in [23].…”

“…Note that maps of type T 3 have positive topological entropy, and therefore, they are chaotic in the sense of Li and Yorke. Chaotic maps in the sense of Li and Yorke may also exist in the family T 2 , but they cannot be found in T 1 , which contains the set of continuous interval maps with finite set of periods (see [18] or [23]). So, if one has to look for counterexamples for Open Problem 5.1, he/she should construct maps having both infinitely many periodic points.…”

A Note on Topological Entropy of Continuous Self-Maps

“…Other papers dealing with extensions of topological entropy to non-compact spaces are Hofer [8], Bowen [3], Canovas [4], Canovas& Rodriguez [7], and Canovas&Linero [5,6].…”

Fractal entropy of nonautonomous systems

“…This kind of problem for nonautonomous dynamical systems has been studied for many years by several authors. A good discussion of these properties and applications appears in [Balibrea et al 1999;Cánovas and Linero 2002;2005;Hric 1999;2000;Kolyada and Snoha 1996;Zhu et al 2006]. Although systems with positive entropy are much more complicated than those with zero entropy, zero entropy systems have various complexities; see [de Carvalho 1997;Dou et al 2011;Ferenczi and Park 2007;Huang et al 2007;Misiurewicz 1981;Misiurewicz and Smítal 1988;Misiurewicz and Szlenk 1980].…”