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

TL;DR

The article advances the theory of physical/SRB measures for dynamical systems by establishing sufficient conditions for their existence on asymptotically sectionally hyperbolic attracting sets of arbitrary finite co-dimension. It generalizes prior codimension-two results and provides concrete high-dimensional attractors with non-sectionally hyperbolic and mixed Lorenz-like Rovella-like singularities, using NU2SE and ASE-type conditions to guarantee full-branch ergodic behavior. The authors develop a versatile construction framework based on solenoidal suspensions and attached flows to realize ASH and NU2SE attractors across codimensions, including proofs that leverage a generalized Pesin formula framework. They also demonstrate the sharpness of these notions through examples of $p$-sectional expansion that do not exhibit asymptotic $(p-1)$-sectional expansion, highlighting the nuanced landscape of higher-dimensional hyperbolic-like dynamics. The work has implications for understanding physical measures in complex flows, offering robust criteria and explicit constructions that extend the reach of sectional and p-sectional hyperbolicity theories.

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 $p>2$.

Figures (3)

• Figure 1: A sketch of the vector field $Y$ with its Poincaré map from $\Sigma_{-2}$ to $\Sigma_2$ compared with the vector field $X$ in the square $[-2,2]\times[-2,2]$.
• Figure 2: Sketch of the flow of the vector field $\widehat{Y}$ with some trajectories, showing that trajectories starting close to $(0,-2)$ (that is, close to $p$ in the original suspension flow) will fall into sinks.
• Figure 3: Sketch of the trajectories $\gamma$, $\widetilde{\gamma}$ and $\widehat{\gamma}$ of $Y_1$ if $0<x_0<x_1$.

Theorems & Definitions (30)

• Remark 1.1: domination and partial hyperbolicity for vector fields
• Lemma 1.2: Hyperbolic Lemma
• proof
• Theorem 1.3
• Remark 1.4: wNU2SE implies wASE
• Theorem 1.5: Physical/SRB measure for weak ASH attracting sets
• Theorem 1.6: Physical/SRB measures for non-uniformly sectionally expanding flows
• Remark 1.7: NU2SE implies ASE
• Remark 1.8: no atoms at equilibria
• Theorem A: Physical measure for higher co-dimensional weak ASH