A characterization of physical measures for systems with mixed central behavior

TL;DR

This paper characterizes when physical measures exist in certain smooth dynamical systems with mixed hyperbolic behavior, linking their existence to the presence of regular points on positive volume sets.

Contribution

It establishes necessary and sufficient conditions for physical measures in systems with mixed central behavior, extending results to robust classes of attractors.

Findings

Physical measures exist if and only if certain regular points are present.

Results apply to robust classes of non-hyperbolic attractors.

Provides criteria for ergodic hyperbolic physical measures.

Abstract

We show that the existence of physical measures for $C^\infty$ smooth instances of certain partially hyperbolic dynamics, both continuous and discrete, exhibiting mixed behavior (positive and negative Lyapunov exponents) along the central non-uniformly hyperbolic multidimensional invariant direction, is equivalent to the existence of certain types of ``regular points'' on positive volume subsets, including Lyapunov regular points. This encompasses the $C^3$ robust class of multidimensional non-hyperbolic attractors obtained by Viana, and the $C^1$ robust classes of $3$-sectionally hyperbolic wild strange attractors presented by Shilnikov and Turaev, providing necessary and sufficient conditions for the existence of ergodic hyperbolic physical measures on these and other dynamical systems.