Shadowing, topological entropy and recurrence of induced Morse-Smale diffeomorphisms

TL;DR

This paper investigates the properties of induced maps from Morse-Smale diffeomorphisms on manifolds, revealing that the induced map on the hyperspace of subcontinua has only two possible entropy values and lacks the shadowing property on the circle.

Contribution

It provides new insights into the topological entropy and shadowing properties of induced maps on hyperspaces for Morse-Smale diffeomorphisms, especially on the circle and higher-dimensional manifolds.

Findings

Induced map on the hyperspace of subcontinua has entropy 0 or infinity.

The induced map on the circle does not have the shadowing property.

Entropy is infinite for manifolds of dimension greater than two.

Abstract

Let $f : M \rightarrow M$ be a Morse-Smale diffeomorphism defined on a compact and connected manifold without boundary. Let $C(M)$ denote the hyperspace of all subcontinua of M endowed with the Hausdorff metric and $C(f) : C(M) \rightarrow C(M)$ denote the induced homeomorphism of $f$. We show in this paper that if $M$ is the unit circle $S^1$ then the induced map $C(f)$ has not the shadowing property. Also we show that the topological entropy of $C(f)$ has only two possible values: $0$ or $\infty$. In particular, we show that the entropy of $C(f)$ is $0$ when $M$ is the unit circle $S^1$ and it is $\infty$ if the dimension of the manifold $M$ is greater than two. Furthermore, we study the recurrence of the induced maps $2^f$ and $C(f)$ and sufficient conditions to obtain infinite topological entropy in the hyperspace.