Entropy and Emergence of Topological Dynamical Systems

TL;DR

This paper introduces the concept of entropy order for topological dynamical systems and their induced spaces, establishing its equivalence with topological entropy and a variational principle for invariant measures.

Contribution

It defines entropy order for induced systems and proves its equivalence to topological entropy, extending the variational principle to this new framework.

Findings

Entropy order coincides with topological entropy.

A variational principle for entropy order of invariant measures.

Extension of entropy concepts to induced systems.

Abstract

A topological dynamical system $(X,f)$ induces two natural systems, one is on the probability measure spaces and other one is on the hyperspace. We introduce a concept for these two spaces, which is called entropy order, and prove that it coincides with topological entropy of $(X,f)$. We also consider the entropy order of an invariant measure and a variational principle is established.