On the Structure of Lorenz Maps

TL;DR

This paper investigates the structure of non-wandering sets in contracting Lorenz maps, establishing conditions for the existence of a unique topological attractor and extending spectral decomposition theory to this context.

Contribution

It introduces a detailed analysis of non-wandering sets in Lorenz maps and extends classical spectral decomposition results to these maps.

Findings

Existence of a unique topological attractor when no attracting periodic orbit is present.

Residual set of points have omega-limit set equal to the attractor.

Extension of spectral decomposition theory to Lorenz maps.

Abstract

We study the non-wandering set of $C^3$ contracting Lorenz maps $f$ with negative Schwarzian derivative. We show that if $f$ doesn't have attracting periodic orbit, then there is a unique topological attractor. Precisely, there is a transitive compact set $\Lambda$ such that $\omega_f(x)=\Lambda$ for a residual set of points $x \in [0,1]$. We also develop in the context of Lorenz maps the classical theory of spectral decomposition constructed for Axiom A maps by Smale.