Inverse limits as attractors in parameterized families

TL;DR

This paper demonstrates how parameterized map families can generate homeomorphisms with inverse limits as attractors, with applications to interval maps, rotation sets, and circle maps.

Contribution

It introduces a method to construct homeomorphisms with inverse limit attractors from parameterized spine maps, extending previous dynamical systems techniques.

Findings

Inverse limits serve as global attractors in constructed homeomorphisms

Applications include unimodal interval maps and rotation sets

Method connects spine maps to ambient manifold dynamics

Abstract

We show how a parameterized family of maps of the spine of a manifold can be used to construct a family of homeomorphisms of the ambient manifold which have the inverse limits of the spine maps as global attractors. We describe applications to unimodal families of interval maps, to rotation sets, and to the standard family of circle maps.