Dirac Physical Measures on Saddle-Type Fixed Points
2020
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.
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