Action and periodic orbits on annulus

TL;DR

This paper extends classical results on area-preserving annulus maps by showing that differences in action, even with identical rotation numbers, guarantee infinitely many periodic orbits, highlighting new dynamical invariants.

Contribution

It introduces the concept that action differences can ensure infinite periodic orbits, generalizing previous rotation number-based results for area-preserving maps.

Findings

Different actions imply infinitely many periodic orbits.

Action difference greater than one guarantees at least two fixed points.

Results apply to annulus and unit disk maps.

Abstract

We consider the classical problem of area-preserving maps on annulus $\mathbb{A} = S^1 \times [0, 1]$ . Let $\mathcal{M}_f$ be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism $f$ on $\mathbb{A}$. Given any $\mu_1$ and $\mu_2$ in $\mathcal{M}_f$, Franks \cite{Franks1988}\cite{Franks1992}, generalizing Poincar\'e's last geometric theorem (Birkhoff \cite{Birkhoff1913}), showed that if their rotation numbers are different, then $f$ has infinitely many periodic orbits. In this paper, we show that if $\mu_1$ and $\mu_2$ have different actions, even if they have the same rotation number, then $f$ has infinitely many periodic orbits. In particular, if the action difference is larger than one, then $f$ has at least two fixed points. The same result is also true for area-preserving diffeomorphisms on unit disk, where no rotation number is available.