Topological sequence entropy and topological dynamics of tree maps

TL;DR

This paper investigates the relationship between topological sequence entropy and the dynamics of tree maps, establishing that zero topological entropy implies zero sequence entropy on certain invariant sets, but this does not extend to dendrites.

Contribution

It proves a new result linking zero topological entropy and sequence entropy for tree maps, and shows the limitation of this relationship for dendrites.

Findings

Zero topological entropy implies zero sequence entropy on non-wandering and chain recurrent sets of tree maps.

The result does not hold for dendrites, even when restricting to periodic points.

The paper clarifies the differences in dynamical complexity between tree maps and dendrites.

Abstract

We prove that a zero topological entropy continuous tree map always displays zero topological sequence entropy when it is restricted to its non-wandering and chain recurrent sets. In addition, we show that a similar result is not possible when the phase space is a dendrite even when we consider only the restriction on the set of periodic points.