Topological sequence entropy for maps of the interval

Abstract

Abstract. A result by Franzová and Smítal shows that a continuous map of the interval into itself is chaotic if and only if its topological sequence entropy relative to a suitable increasing sequence of nonnegative integers is positive. In the present paper we prove that for any increasing sequence of nonnegative integers there exists a chaotic continuous map with zero topological sequence entropy relative to this sequence.

Topological sequence entropy and topological dynamics of tree maps

“…For continuous maps of the interval, positive topological sequence entropy characterizes the existence of chaos in the sense of Li and Yorke [10]. A similar result is obtained for continuous circle maps [14]. In addition, for both interval and circle cases, zero topological entropy maps have either log 2 or zero as the supremum of their topological sequence entropies [6,5].…”

“…Topological sequence entropy [12] is a conjugacy invariant that is useful for distinguishing among continuous maps with zero topological entropy. Thus, it has been used to study zero entropy systems as substitution shifts [9,19] and one-dimensional continuous maps [6,10,13,14].…”

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].…”

Graph maps with zero topological entropy and sequence entropy pairs

“…In 1990, Kuchta [16] proved that a circle map with zero topological entropy is Li-Yorke chaotic if and only if it has an infinite -limit set containing two non-separable points if and only if it has a scrambled pair. In 2000, Hric [11] proved that a circle map is Li-Yorke chaotic if and only if it has positive topological sequence entropy. Analogous problems for graph maps were considered in [25] by Ruette and Snoha, who proved that a scrambled pair for a graph map implies that the map is Li-Yorke chaotic.…”