Topological properties of Lorenz maps derived from unimodal maps

TL;DR

This paper explores the topological properties of Lorenz maps derived from unimodal maps, establishing a Sharkovsky-like theorem and identifying conjugacy to Sturmian shifts, with implications for inverse limit spaces and planar attractors.

Contribution

It introduces a Sharkovsky-like theorem for symmetric Lorenz maps and characterizes cases where unimodal maps are conjugate to Sturmian shifts.

Findings

Established a Sharkovsky-like ordering for Lorenz maps

Identified conditions for conjugacy to Sturmian shifts

Connected unimodal inverse limit spaces to planar attractors

Abstract

A symmetric Lorenz map is obtain by ``flipping'' one of the two branches of a symmetric unimodal map. We use this to derive a Sharkovsky-like theorem for symmetric Lorenz maps, and also to find cases where the unimodal map restricted to the critical omega-limit set is conjugate to a Sturmian shift. This has connections with properties of unimodal inverse limit spaces embedded as attractors of some planar homeomorphisms.