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Pré-Publication, Document De Travail Année : 2014

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

Jingzhi Yan
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Résumé

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.
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Dates et versions

hal-01063331 , version 1 (12-09-2014)

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Jingzhi Yan. Existence of periodic points near an isolated fixed point with Lefschetz index $1$ and zero rotation for area preserving surface homeomorphisms. 2014. ⟨hal-01063331⟩
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