Existence of periodic points near an isolated fixed point with Lefschetz index $1$ and zero rotation for area preserving surface homeomorphisms

TL;DR

This paper proves that for area-preserving surface homeomorphisms with an isolated fixed point of Lefschetz index one, the fixed point is accumulated by periodic orbits, supporting Le Roux's conjecture.

Contribution

It establishes the accumulation of periodic orbits near a degenerate fixed point with index one under topological conditions, extending previous conjectures.

Findings

Fixed point is accumulated by periodic orbits in measure sense.

Dirac measure at fixed point is a limit of measures supported on periodic orbits.

Proof applies to homeomorphisms and uses local rotation set concepts.

Abstract

Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_0$ with Lefschetz index one. Le Roux conjectured that $z_0$ is accumulated by periodic orbits. In this article, we will approach Le Roux's conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_0$ and if the area of $M$ is finite, then $z_0$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_0$ is the limit in weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological and will works for homeomorphisms and is related to the notion of local rotation set.