Geometric and measure-theoretical structures of maps with mostly contracting center

TL;DR

This paper introduces a geometric-combinatorial structure called the skeleton for diffeomorphisms with mostly contracting center, enabling analysis of physical measures, their bifurcations, and basin continuity.

Contribution

It develops the skeleton framework to understand physical measures, their bifurcations, and basin behavior in systems with mostly contracting centers, providing new tools for dynamical analysis.

Findings

Constructed examples with any number of physical measures

Analyzed basin intermingling and measure collapse

Proved basin continuity in absence of collapses

Abstract

We show that every diffeomorphism with mostly contracting center direction exhibits a geometric-combinatorial structure, which we call \emph{skeleton}, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how the physical measure bifurcate as the diffeomorphism changes. In particular, we use this to construct examples with any given number of physical measures, with basins densely intermingled, and to analyse how these measures collapse into each other - through explosions of their basins - as the dynamics varies. This theory also allows us to prove that, in the absence of collapses, the basins are continuous functions of the diffeomorphism.