Dirac physical measures on saddle-type fixed points

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.

arXiv comment63 pages, 26 figures. Final version, accepted in Journal of Dynamics and Differential Equations

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• Takens' last problem and existence of non-trivial wandering domains
• Pesin Entropy Formula for C1 Diffeomorphisms with Dominated Splitting
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Cited by in corpus (5)

• Emergence via non-existence of averages
• Abundance of observable Lyapunov irregular sets
• Historic and physical wandering domains for wild blender-horseshoes
• Lyapunov exponents for random maps
• SRB measures for C $\infty$ surface diffeomorphisms