Abundance of fast growth of the number of periodic points in 2-dimensional area-preserving dynamics

TL;DR

This paper demonstrates that in certain open subsets of real-analytic Hamiltonian and smooth area-preserving diffeomorphisms on closed surfaces, the number of periodic points grows rapidly, indicating complex dynamical behavior.

Contribution

It establishes the density and typicality of fast growth of periodic points in specific classes of area-preserving surface diffeomorphisms.

Findings

Fast growth of periodic points is dense in some real-analytic Hamiltonian diffeomorphisms.

Fast growth of periodic points is typical in smooth area-preserving diffeomorphisms.

The results highlight complex dynamics in area-preserving surface maps.

Abstract

We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there exists an open subset of the set of smooth area-preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points.