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 C² perturbation, the associated dynamical Lagrange and Markov spectra have robustly non-empty interior for an open and dense set of C¹-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.

Uniform L^∞-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains
Antonio L. Pereira
arXiv29 Sep 2026PreprintCC BY 4.0Full textAnalysis & PDEs Dynamical systemsarXiv
Flexibility of topological entropy and factor persistence in the C⁰-Gromov Hausdorff topology
Richard Cubas, Jorge Crisostomo
arXiv22 Oct 2026PreprintarXiv licenceFull textDynamical systemsarXiv
Behavior of the SRB Entropy Functional in Families of Hyperbolic Attractors and Expanding Maps
Miaohua Jiang
arXiv1 Sep 2026PreprintCC BY-NC-ND 4.0Full textDynamical systemsarXiv
Uniformly Lipschitz Hyperbolic Group Actions on ℓ^p
Chris Gartland, Tianyi Zheng
arXiv24 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Analysis & PDEsarXiv
Uniform Exponential Attractors for Non-Autonomous Dynamical Systems and an Application to the damped hyperbolic equation with sub-quintic growth
Chang Qingquan, Li Dandan
arXiv9 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEs Dynamical systemsarXiv

Cite

BibTeX RIS

BibTeX

@misc{rego2026hausdorff,
  title = {{Hausdorff Dimension and Flexibility for Attractors of Non-Uniformly Hyperbolic Flows}},
  author = {Elias Rego and Paulo Varandas},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13160},
  eprint = {2610.13160},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13160},
}

RIS

TY  - UNPB
TI  - Hausdorff Dimension and Flexibility for Attractors of Non-Uniformly Hyperbolic Flows
AU  - Elias Rego
AU  - Paulo Varandas
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13160
UR  - https://arxiv.org/abs/2610.13160
AB  - 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.
ER  -

Abstract

Related

Cite