Singular robustly chain transitive sets are singular volume partial hyperbolic
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 flo...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.1810.09904 |