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Transfer operators for Sinai billiards - lecture 2

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Authors : Baladi, Viviane (Author of the conference)
CIRM (Publisher )

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Abstract : We will discuss an approach to the statistical properties of two-dimensional dispersive billiards (mostly discrete-time) using transfer operators acting on anisotropic Banach spaces of distributions. The focus of this part will be our recent work with Mark Demers on the measure of maximal entropy but we will also survey previous results by Demers, Zhang, Liverani, etc on the SRB measure.

Keywords : dynamical systems; billiards; transfer operators

MSC Codes :
37B40 - Topological entropy
37C30 - Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems
37D50 - Hyperbolic systems with singularities (billiards, etc.)

Additional resources :
https://www.cirm-math.fr/RepOrga/1947/Notes/Baladi-notes.pdf

    Information on the Video

    Film maker : Hennenfent, Guillaume
    Language : English
    Available date : 11/06/2019
    Conference Date : 16/05/2019
    Subseries : Research School
    arXiv category : Dynamical Systems ; Spectral Theory ; Mathematical Physics
    Mathematical Area(s) : Dynamical Systems & ODE ; Analysis and its Applications
    Format : MP4 (.mp4) - HD
    Video Time : 00:52:33
    Targeted Audience : Researchers
    Download : https://videos.cirm-math.fr/2019-05-16_Baladi_part2.mp4

Information on the Event

Event Title : Dynamique au-delà de l'hyperbolicité uniforme / Dynamics Beyond Uniform Hyperbolicity
Event Organizers : Bonatti, Christian ; Buzzi, Jérôme ; Crovisier, Sylvain ; Gan, Shaobo ; Pacifico, Maria José
Dates : 13/05/2019 - 24/05/2019
Event Year : 2019
Event URL : https://conferences.cirm-math.fr/1947.html

Citation Data

DOI : 10.24350/CIRM.V.19523803
Cite this video as: Baladi, Viviane (2019). Transfer operators for Sinai billiards - lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19523803
URI : http://dx.doi.org/10.24350/CIRM.V.19523803

See Also

Bibliography

  • BALADI, Viviane et DEMERS, Mark. On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps. arXiv preprint arXiv:1807.02330, 2018. - https://arxiv.org/abs/1807.02330

  • BALADI, Viviane, DEMERS, Mark F., et LIVERANI, Carlangelo. Exponential decay of correlations for finite horizon Sinai billiard flows. Inventiones mathematicae, 2018, vol. 211, no 1, p. 39-177. - https://doi.org/10.1007/s00222-017-0745-1

  • BOWEN, Rufus. Topological entropy for noncompact sets. Transactions of the American Mathematical Society, 1973, vol. 184, p. 125-136. - https://doi.org/10.2307/1996403

  • BOWEN, Rufus. Maximizing entropy for a hyperbolic flow. Theory of Computing Systems, 1973, vol. 7, no 3, p. 300-303. - https://doi.org/10.1007/BF01795948

  • BRIN, Michael et KATOK, Anatole. On local entropy. In : Geometric dynamics. Springer, Berlin, Heidelberg, 1983. p. 30-38. - https://doi.org/10.1007/BFb0061408

  • BUNIMOVICH, Leonid Abramovich, SINAI, Yakov G., et CHERNOV, Nikolai Ivanovich. Markov partitions for two-dimensional hyperbolic billiards. Russian Mathematical Surveys, 1990, vol. 45, no 3, p. 105. - https://doi.org/10.1070/RM1990v045n03ABEH002355

  • CHERNOV, Nikolai Ivanovich. Topological entropy and periodic points of two-dimensional hyperbolic billiards. Functional Analysis and Its Applications, 1991, vol. 25, no 1, p. 39-45. - https://doi.org/10.1007/BF01090675

  • CHERNOV, N. I. et MARKARIAN, R. Mathematical Surveys and Monographs. Chaotic Billiards, 2006, vol. 127. -

  • DEMERS, Mark F., WRIGHT, Paul, et YOUNG, Lai-Sang. Entropy, Lyapunov exponents and escape rates in open systems. Ergodic Theory and Dynamical Systems, 2012, vol. 32, no 4, p. 1270-1301. - https://doi.org/10.1017/S0143385711000344

  • DEMERS, Mark et ZHANG, Hong-Kun. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, vol. 5, no 4. - https://doi.org/10.3934/jmd.2011.5.665

  • GOUËZEL, Sébastien, LIVERANI, Carlangelo, et al. Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. Journal of Differential Geometry, 2008, vol. 79, no 3, p. 433-477. - https://arxiv.org/abs/math/0606722

  • LIMA, Yuri et MATHEUS, Carlos. Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities. arXiv preprint arXiv:1606.05863, 2016. - https://arxiv.org/abs/1606.05863

  • PESIN, Yakov B. Dimension theory in dynamical systems: contemporary views and applications. University of Chicago Press, 2008. -

  • PESIN, Ya B. et PITSKEL', B. S. Topological pressure and the variational principle for noncompact sets. Functional Analysis and its Applications, 1984, vol. 18, no 4, p. 307-318. - https://doi.org/10.1007/BF01083692

  • SINAI, Yakov G. Dynamical systems with elastic reflections. Russian Mathematical Surveys, 1970, vol. 25, no 2, p. 137. - https://doi.org/10.1070/RM1970v025n02ABEH003794

  • YOUNG, Lai-Sang. Statistical properties of dynamical systems with some hyperbolicity. Annals of Mathematics, 1998, vol. 147, p. 585-650. - https://doi.org/102307/120960



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