Mostly Nonuniformly Sectional Expanding Systems

Abstract

Abstract We introduce the notion of mostly nonuniform sectional expanding (MNUSE) for singular flows which encompasses the notions of sectional hyperbolicity, asymptotically sectional and multisingular hyperbolicity. We construct examples of a vector field of class $$C^r, r \ge 1$$ 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 codimension, extending the codimension 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 codimensional (i.e. with central direction of dimension greater than 2) nonuniformly sectional expanding attractors.