Hausdorff Dimension and Flexibility for Attractors of Non-Uniformly Hyperbolic Flows

Abstract

We study the Hausdorff dimension of attracting sets of three-dimensional flows whose one-dimensional quotient dynamics is non-uniformly expanding. Our main assumption is that the quotient map admits a full-branch uniformly expanding Gibbs--Markov induced map. We first prove that every such attractor contains a hyperbolic horseshoe of Hausdorff dimension strictly larger than two. As a consequence, after an arbitrarily small $C^2$ perturbation, the associated dynamical Lagrange and Markov spectra have robustly non-empty interior for an open and dense set of $C^1$-observables. We then establish intermediate dimension properties. Every admissible intermediate Hausdorff dimension is realized by an ergodic invariant measure supported on the attractor. Moreover, every intermediate stable and unstable dimension is realized by a strictly ergodic hyperbolic subsystem. The approach extends the dimension theory previously developed for geometric Lorenz attractors to more general non-uniformly hyperbolic attractors including singular-hyperbolic attractors, Rovella attractors, and broad classes of skew-product suspension flows modeled by intermittent and Benedicks--Carleson maps.