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

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.