Turbulent homeomorphisms and the topological snail

TL;DR

This paper introduces the topological snail as a universal geometric object for turbulent homeomorphisms, providing insights into their fixed points, periodic orbits, and topological entropy through matrix representations.

Contribution

It defines the topological snail and demonstrates its use in analyzing turbulent homeomorphisms, linking topological, dynamical, and algebraic properties.

Findings

Characterization of fixed points and periodic orbits via the topological snail

Relation between topological entropy and spectral radius of turbulence matrices

Dynamical interpretation of Fermat's theorem extension to matrices

Abstract

The topological snail is a geometric universal plane object, described by matrices in the projective special linear group with integer coefficients. It has many nice properties : in the case of three points, it natually defines a representation of the mapping-class group. From a dynamical point of view, it gives a description of the set of fixed points and periodic obits of turbulent homeomorphims in the plane. Those homeomorphisms are the truly complicated one. They have many fixed points and periodic orbits, as much as the trace of the turbulence matrix that caracterizes their action on a finite invariant set. The topological snail generally gives a map of the fixed points and periodic orbits of a such homeomorphims. One can compute their indexes and turbulent topological types, and different Nielsen's classes of fixed points naturally give a minoration of the topological entropy of the considered homeomorphisms by the logarithm of the spectral radius of the turbulence matrix. One even get a dynamical interpretation and proof of the extension of the Fermat' theorem to matrices. I first described the topological snail and its properties at my conference the 31st of may 2024 in Paris, in the case of a purely turbulent homeomorphism, with an invariant set of three points.