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arXiv · 1909.02172

Dirac physical measures on saddle-type fixed points

Pablo Guarino·Pierre-Antoine Guihéneuf·Bruno Santiago
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Abstract

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 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 hyperbolic fixed point of saddle type, 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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Pablo Guarino, Pierre-Antoine Guihéneuf, Bruno Santiago. 2019-09-05. Dirac physical measures on saddle-type fixed points. https://doi.org/10.1007/s10884-020-09911-x

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