Topological entropy of diagonal maps on inverse limit spaces

TL;DR

This paper investigates the topological entropy of diagonal maps on inverse limit spaces, providing bounds and exact values in special cases, and introduces techniques for entropy computation of set-valued maps.

Contribution

It establishes an upper bound for entropy of inverse limit space maps and characterizes entropy for diagonal maps with commuting components, along with new computational techniques.

Findings

Entropy of diagonal maps equals the maximum entropy of the component maps when they strongly commute.

Provides an explicit formula for entropy in a special case of inverse limit spaces.

Develops new methods for computing topological entropy of set-valued maps.

Abstract

We give an upper bound for the topological entropy of maps on inverse limit spaces in terms of their set-valued components. In a special case of a diagonal map on the inverse limit space $\underleftarrow{\lim}(I,f)$, where every diagonal component is the same map $g\colon I\to I$ which strongly commutes with $f$ (i.e. $f^{-1}\circ g=g\circ f^{-1}$), we show that the entropy equals $\max\{\textrm{Ent}(f),\textrm{Ent}(g)\}$. As a side product, we develop some techniques for computing topological entropy of set-valued maps.