Global Dynamics for Symmetric Planar Maps

TL;DR

This paper establishes conditions under which the global dynamics of symmetric planar maps with a hyperbolic fixed point can be determined, especially when symmetry includes reflections, using theories of free homeomorphisms.

Contribution

It provides new criteria for analyzing global dynamics of equivariant planar maps with symmetry, emphasizing the role of reflections in simplifying the analysis.

Findings

Presence of reflection simplifies global dynamics analysis.

Conditions identified for global behavior based on local hyperbolic fixed point.

Discussion on complexities when reflections are absent.

Abstract

We consider sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic. When the map is equivariant under the action of a compact Lie group, it is possible to describe the local dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. This allows us to use results based on the theory of free homeomorphisms to describe the global dynamical behaviour. We briefly discuss the case when reflections are absent, for which global dynamics may not follow from local dynamics near the unique fixed point.