Mostly nonuniformly sectional expanding systems

TL;DR

This paper introduces the concept of mostly nonuniform sectional expanding systems, providing new examples, conditions for physical measures, and extending known attractor types to higher dimensions.

Contribution

It defines MNUSE for singular flows, constructs new examples of attractors, and extends criteria for physical measures to higher co-dimension cases.

Findings

Existence of nonuniformly sectional hyperbolic sets satisfying MNUSE.

Examples of higher-dimensional Rovella-like attractors with physical measures.

Extended criteria for physical measures to higher co-dimensional attractors.

Abstract

We introduce the notion of \emph{mostly nonuniform sectional expanding} (MNUSE) for singular flows which encompasses the notions of sectional hyperbolicity, asymptotically sectional and multisingular hyperbolicity. We exhibit an example of a vector field of class $C^r, r > 1$, whose flow exhibits a nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic. 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 (i.e. with central direction of dimension greater than $2$) non-uniformly sectional expanding attractors.