Complexity function and entropy of induced maps on hyperspaces of continua

TL;DR

This paper links the complexity function of invariant subsets in shift spaces to the polynomial entropy of induced hyperspace dynamics, offering criteria for infinite topological entropy in certain systems.

Contribution

It introduces a method to compute polynomial entropy of induced hyperspace maps using the complexity function and provides a criterion for infinite entropy.

Findings

Polynomial entropy can be computed via the complexity function of invariant subsets.

A simple criterion is provided for when the hyperspace map has infinite topological entropy.

The approach applies to certain one-dimensional dynamical systems.

Abstract

We use the complexity function of an invariant, not necessary closed, subset of a two-sided shift space to compute the polynomial entropy of the induced dynamics on the hyperspace of continua for certain one-dimensional dynamical systems. We also provide a simple criterion for $f$ that implies $C(f)$ has infinite topological entropy.