Robust transitivity versus trapping regions for partially hyperbolic diffeomorphisms

Sylvain Crovisier; Rafael Potrie

Robust transitivity versus trapping regions for partially hyperbolic diffeomorphisms

Dynamical Systems 2026-07-05 v1

Authors: Sylvain Crovisier , Rafael Potrie

Abstract

We show that among partially hyperbolic diffeomorphisms with one dimensional center there is a $C^1$ -open and dense subset for which either there is a proper quasi-attractor or both strong foliations are minimal. The same result holds among volume-preserving partially hyperbolic diffeomorphisms and has consequences about robust transitivity beyond the conservative setting. The proofs involve a careful study of minimal $\mathcal{W}^u$ -saturated sets and interact with recent results in the subject.

Cite

@article{arxiv.2607.04496,
  title  = {Robust transitivity versus trapping regions for partially hyperbolic diffeomorphisms},
  author = {Sylvain Crovisier and Rafael Potrie},
  journal= {arXiv preprint arXiv:2607.04496},
  year   = {2026}
}