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Hyperbolicity and Rigidity for Fibred Partially Hyperbolic Systems

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Abstract

Every volume-preserving accessible centre-bunched fibred partially hyperbolic system with 2-dimensional centre either (a) has two distinct centre Lyapunov exponents, or (b) exhibits an invariant continuous line field (or pair of line fields) tangent to the centre leaves, or (c) admits a continuous conformal structure on the centre leaves invariant under both the dynamics and the stable and unstable holonomies. The last two alternatives carry strong restrictions on the topology of the centre leaves: (b) can only occur on tori, and for (c) the centre leaves must be either tori or spheres. Moreover, under some additional conditions, such maps are rigid, in the sense that they are topologically conjugate to specific algebraic models. When the system is symplectic (a) implies that the centre Lyapunov exponents are non-zero, and thus the system is (non-uniformly) hyperbolic.

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Notes

  1. In this paper all measures are finite Borel measures.

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Acknowledgements

This paper corresponds to the doctoral thesis presented at IMPA by the first author. We are grateful to the committee members L. Backes, L. Lomonaco, K. Marín, K. War and J. Yang for numerous comments and suggestions. Our understanding of the measurable Riemann mapping theorem benefitted from a discussion with J.V. Pereira. Remarks and corrections by the anonymous reviewer are also gratefully acknowledged.

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  1. Sankhadip Chakraborty

    Present address: Department of Mathematics, Southern University of Science and Technology, 1088 Xueyuan Avenue, Shenzhen, 518055, People’s Republic of China

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  1. IMPA - Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

    Sankhadip Chakraborty & Marcelo Viana

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S.C. was supported by a scholarship from CNPq - National Research Council of Brazil. M.V. was partially supported by CNPq and FAPERJ - State Research Agency of Rio de Janeiro. This project was also supported by Fondation Louis D. - Institut de France (project coordinated by M. Viana).

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Chakraborty, S., Viana, M. Hyperbolicity and Rigidity for Fibred Partially Hyperbolic Systems. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-023-10343-6

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