On the Continuity of the Topological Entropy of Non-autonomous Dynamical Systems

TL;DR

This paper investigates the continuity properties of topological entropy in non-autonomous dynamical systems, showing it is discontinuous under the compact topology but continuous under the strong topology for $r \\geq 1$.

Contribution

It establishes the conditions under which the topological entropy of non-autonomous systems is continuous or discontinuous depending on the topology used.

Findings

Entropy discontinuous under compact topology

Entropy continuous under strong topology for $r \\geq 1$

Provides a topological characterization of entropy behavior

Abstract

Let $M$ be a compact Riemannian manifold. The set $\text{F}^{r}(M)$ consisting of sequences $(f_{i})_{i\in\mathbb{Z}}$ of $C^{r}$-diffeomorphisms on $M$ can be endowed with the compact topology or with the strong topology. A notion of topological entropy is given for these sequences. I will prove this entropy is discontinuous at each sequence if we consider the compact topology on $\text{F}^{r}(M)$. On the other hand, if $ r\geq 1$ and we consider the strong topology on $\text{F}^{r}(M)$, this entropy is a continuous map.