On reversible maps and symmetric periodic points

TL;DR

This paper investigates symmetric periodic points in reversible dynamical systems, extending classical theorems to include symmetric cases, and reveals that non-symmetric periodic points imply infinitely many symmetric ones, with applications to systems with two degrees of freedom.

Contribution

It extends Franks' theorem to symmetric periodic points in reversible maps and establishes conditions for symmetric fixed points, providing a simpler proof and broader applicability.

Findings

Non-symmetric periodic points imply infinitely many symmetric periodic points.

A reversible map has a symmetric fixed point if and only if it is a twist map.

The approach is elementary and extends classical results to symmetric settings.

Abstract

In reversible dynamical systems, it is frequently of importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks' theorem on a dichotomy of the number of periodic points of area preserving maps on the annulus to symmetric periodic points of area preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area preserving reversible maps isotopic to the identity, which can be applied to dynamical systems with double symmetries. Our approach is simple, elementary and far from Franks' proof. We also show that a reversible map has a symmetric fixed point if and only if it is a twist map which generalizes a boundary twist condition on the closed annulus in the sense of Poincar\'e-Birkhoff. Applications to symmetric periodic orbits in reversible dynamical systems with two degrees of freedom are briefly discussed.