Singular robustly chain transitive sets are singular volume partial hyperbolic

TL;DR

This paper extends the understanding of hyperbolic structures in dynamical systems with singularities, showing that robust chain transitive sets exhibit a form of partial hyperbolicity, using a new approach to handle singularities.

Contribution

It introduces a method to adapt hyperbolic structures to singularities, recovering known results for flows and demonstrating a form of weak hyperbolicity in robust chain transitive sets.

Findings

Robust chain transitive sets have a weak form of hyperbolicity.

The proposed method adapts hyperbolic structures to singularities.

Results apply to singular attractors with periodic orbits of different indices.

Abstract

For diffeomorphisms or for non-singular flows, there are many results relating properties persistent under C1 perturbations and global structures for the dynamics ( such as hyperbolicity, partial hyperbolicity, dominated splitting). However, a dif\'iculty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point. In [BdL] with Christan Bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities. In this paper, using this tool, we recover the results in [BDP] for flows, showing that robustly chain transitive sets have a weak form of hyperbolicity. allowing us to conclude as well the kind of hyperbolicity carried by the examples in \cite{BLY} (a robust chain transitive singular attractor with periodic orbits of different indexes). Along with the results in [BdL], this shows that the way we propose to interpret the effect of singularities has the potential to adapt to other settings in which there is a coexistence of singularities and regular orbits with the goal of re-obtaining the results that we already know for diffeomorphisms.