Physical measures for asymptotically sectional expanding flows in higher co-dimensions

TL;DR

This paper establishes conditions for the existence of physical measures in higher co-dimensional asymptotically sectional hyperbolic attractors, extending previous two-dimensional results to more complex, higher-dimensional systems.

Contribution

It generalizes the theory of physical measures to higher co-dimensional hyperbolic attractors, including new examples with mixed-type singularities and non-uniform expansion.

Findings

Existence of physical/SRB measures in higher co-dimensional settings.

Construction of examples with mixed hyperbolic singularities.

Extension to non-uniformly sectional expanding attractors.

Abstract

We obtain sufficient conditions for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite co-dimension, extending the co-dimension two case. We provide examples of such attractors, either with non-sectional hyperbolic equilibria, or with sectional-hyperbolic equilibria of mixed type, i.e., with a Lorenz-like singularity together with a Rovella-like singularity in a transitive set. These are higher-dimensional versions of contracting Lorenz-like attractors (also known as Rovella-like attractors) to which we apply our criteria to obtain a physical/SRB measure with full ergodic basin. We also adapt the previous examples to obtain higher co-dimensional non-uniformly sectional expanding attractors; and also asymptotical $p$-sectional hyperbolic attractors which are \emph{not} non-uniformly $(p-1)$-expanding, for any finite $p>2$.