  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.
