Area-Preserving Surface Dynamics and S. Saito's Fixed Point Formula

TL;DR

This paper demonstrates how Saito's fixed point formula can be used to analyze the distribution and growth of periodic points in area-preserving surface maps, revealing infinite periodic points with exponential growth.

Contribution

It introduces new stability results for local indices, finiteness of certain periodic curves, and links these to exponential growth of periodic points using Saito's formula.

Findings

Existence of infinitely many isolated periodic points

Exponential growth of periodic points with period

Finiteness of type II periodic curves

Abstract

We show that S. Saito's fixed point formula serves as a powerful tool for counting the number of isolated periodic points of an area-preserving surface map admitting periodic curves. His notion of periodic curves of types I and II plays a central role in our discussion. We establish a Shub-Sullivan type result on the stability of local indices under iterations of the map, the finiteness of the number of periodic curves of type II, and the absence of periodic curves of type I. Combined with these results, Saito's formula implies the existence of infinitely many isolated periodic points whose cardinality grows exponentially as period tends to infinity.