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Singular robustly chain transitive sets are singular volume partial hyperbolic

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Abstract

For diffeomorphisms or for non-singular flows, there are many results relating properties persistent under \(\mathcal{C}^1\) perturbations and a global structures for the dynamics (such as hyperbolicity, partial hyperbolicity, dominated splitting). However, a difficulty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point. In Bonatti (Star flows and multisingular hyperbolicity. arXiv:1705.05799, 2017) with Christan Bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities. Using this tool, we recover the results in Bonatti et al. (Ann Math 158(2):355–418, 2003) 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 Bonatti et al. (J Inst Math Jussieu 12(3):449–501, 2013) (a robust chain transitive singular attractor with periodic orbits of different indexes). Along with the results in [8], 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 coexistence of singularities and regular orbits with the goal of re-obtaining the results that we already know for diffeomorphisms.

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Acknowledgements

This work was done in the context of the authorś PHD thesis (under the supervention of Christian Bonatti and Martin Sambarino). The author would like to thank Christian Bonatti, Martin Sambario, Rafael Potrie and Sylvain Crovisier for their comments and suggestions. The author was supported by the ecole doctorale Carnot Pasteur, Centro de Matemáticas UdelaR, ANII FCE,and CAP UdelaR.

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    Adriana da Luz

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da Luz, A. Singular robustly chain transitive sets are singular volume partial hyperbolic. Math. Z. 294, 687–712 (2020). https://doi.org/10.1007/s00209-019-02291-z

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