Dirac physical measures on saddle-type fixed points

TL;DR

This paper investigates the statistical properties of surface diffeomorphisms, revealing conditions under which Dirac measures are supported on sinks or saddle points, and constructing examples with complex basins of attraction.

Contribution

It characterizes when Dirac invariant measures are supported on sinks versus saddle points and constructs examples with intricate statistical basins in surface diffeomorphisms.

Findings

Dirac measures with dense basins are supported on sinks.

Existence of saddle fixed points with nowhere dense basins of positive measure.

Construction of diffeomorphisms with positive measure historic behavior sets.

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.