Dirac physical measures on saddle-type fixed points
Résumé
In this article we study some statistical aspects of surface diffeomorphisms. We first show that for a $C^1$ generic diffeomorphism, a Dirac invariant measure whose \emph{statistical basin of attraction} is dense in some open set must be supported in the orbit of a sink. We then construct an example of a $C^1$-diffeomorphism having an invariant Dirac measure, supported on a hyperbolic fixed point of saddle type, whose statistical basin of attraction is a nowhere dense set with positive Lebesgue measure.