Polynomial entropy of Morse-Smale diffeomorphisms on surfaces

TL;DR

This paper computes the generalized entropy, an extension of classical entropy, for Morse-Smale diffeomorphisms on surfaces to measure their dynamical complexity.

Contribution

It provides the first explicit calculation of generalized entropy for Morse-Smale diffeomorphisms on surfaces, expanding understanding of their complexity.

Findings

Generalized entropy of Morse-Smale diffeomorphisms is computed.

Classical entropy is zero for these systems, but generalized entropy captures their complexity.

Results extend the concept of entropy to systems previously considered simple.

Abstract

A classical problem in dynamical systems is to measure the complexity of a map in terms of their orbits. One of the main tools we have to achieve this goal is entropy. However, many interesting families of dynamical systems have every element with zero-entropy. One of said families are the Morse-Smale diffeomorphisms. The first author and E. Pujals introduced the concept of generalized entropy that extends the classical notion of entropy. In this work, we compute the generalized entropy of Morse-Smale diffeomorphisms on surfaces.