Dirac Physical Measures on Saddle-Type Fixed Points

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 statistical basin of attraction is dense in some open set and has positive Lebesgue measure, must be supported in the orbit of a sink. We then construct an example of a $$C^1$$ -diffeomorphism having a Dirac invariant measure, supported on a saddle-type hyperbolic fixed point, whose statistical basin of attraction is a nowhere dense set with positive Lebesgue measure. Our technique can be applied also to construct a $$C^1$$ diffeomorphism whose set of points with historic behaviour has positive measure and is nowhere dense.