Physical measures for mostly sectional expanding flows

TL;DR

This paper establishes conditions under which certain partially hyperbolic attracting sets support ergodic physical measures, extending known results from diffeomorphisms to continuous-time systems with equilibria.

Contribution

It extends the theory of physical measures to continuous-time systems with equilibria, providing new criteria based on sectional expansion and applying to various attractors.

Findings

Supports existence of physical measures under sectional expansion conditions

Finite ergodic physical measures supported on attracting sets

Unified approach for Lorenz-like and sectional-hyperbolic attractors

Abstract

We prove that a partially hyperbolic attracting set for a C2 vector field, having slow recurrence to equilibria, supports an ergodic physical/SRB measure if, and only if, the trapping region admits non-uniform sectional expansion on a positive Lebesgue measure subset. Moreover, in this case, the attracting set supports at most finitely many ergodic physical/SRB measures, which are also Gibbs states along the central-unstable direction. This extends to continuous time systems a similar well-known result obtained for diffeomorphisms, encompassing the presence of equilibria accumulated by regular orbits within the attracting set. In codimension two the same result holds, assuming only the trajetories on the trapping region admit a sequence of times with asymptotical sectional expansion, on a positive volume subset. We present several examples of application, including the existence of physical measures for asymptotically sectional hyperbolic attracting sets, and obtain physical measures in an alternative unified way for many known examples: Lorenz-like and Rovella attractors, and sectional-hyperbolic attracting sets (including the multidimensional Lorenz attractor).