Quasi-energy function for diffeomorphisms with wild separatrices

TL;DR

This paper introduces the concept of quasi-energy functions for Morse-Smale diffeomorphisms on the 3-sphere, providing a minimal critical point Lyapunov function for a class of such diffeomorphisms.

Contribution

It defines quasi-energy functions and constructs them for a class of Morse-Smale diffeomorphisms on the 3-sphere, extending the understanding of Lyapunov functions in complex dynamical systems.

Findings

Existence of quasi-energy functions for certain Morse-Smale diffeomorphisms

Construction method for quasi-energy functions on the 3-sphere

Extension of Lyapunov function concepts to diffeomorphisms with wild separatrices

Abstract

According to Pixton, there are Morse-Smale diffeomorphisms of the 3-sphere which have no energy function, that is a Lyapunov function whose critical points are all periodic points of the diffeomorphism. We introduce the concept of quasi-energy function for a Morse-Smale diffeomorphism as a Lyapunov function with the least number of critical points and construct a quasi-energy function for any diffeomorphism from some class of Morse-Smale diffeomorphisms on the 3-sphere.