Continuum-wise expansiveness and specification for set-valued functions and topological entropy

TL;DR

This paper introduces continuum-wise expansiveness and pointwise specification for set-valued functions, demonstrating that these properties imply positive topological entropy, thus linking dynamical complexity with these new concepts.

Contribution

It defines continuum-wise expansiveness and pointwise specification for set-valued functions and establishes their connection to positive topological entropy.

Findings

Set-valued cw-expansive functions imply positive entropy on compact metric spaces.

Set-valued functions with pointwise specification also have positive entropy.

Abstract

We define the concept of continuum wise expansive for set-valued functions and prove that if a compact metric space admit a set-valued $cw$-expansive function then the topological entropy of $X$ is positive.} We also introduce the notion of pointwise specification property for set-valued functions and prove that set-valued functions with this property has positive entropy.