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

Published online by Cambridge University Press:  21 July 2015

JINGZHI YAN
JINGZHI YAN*
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche, 4 place Jussieu, 75252 Paris, France email jingzhi.yan@imj-prg.fr
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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 paper, 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 the weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological. It works for homeomorphisms and is related to the notion of local rotation set.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2293 - 2333
Copyright
© Cambridge University Press, 2015 

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