Topological Attractors of Contracting Lorenz Maps

TL;DR

This paper investigates the structure of non-wandering sets in contracting Lorenz maps, establishing the existence of a unique topological attractor under certain conditions and classifying possible attractor types.

Contribution

It proves the existence and uniqueness of a topological attractor for contracting Lorenz maps without attracting periodic orbits and classifies the attractors.

Findings

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

Residual set of points whose omega-limit set equals the attractor.

Classification of possible attractor types in this setting.

Abstract

We study the non-wandering set of contracting Lorenz maps. We show that if such a map $f$ doesn't have any attracting periodic orbit, then there is a unique topological attractor. Precisely, there is a compact set $\Lambda$ such that $\omega_f(x)=\Lambda$ for a residual set of points $x \in [0,1]$. Furthermore, we classify the possible kinds of attractors that may occur.