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Beyond Bowen specification property - lecture 3

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Multi angle
Auteurs : Climenhaga, Vaughn (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to specification" using a decomposition of the space of finite-length orbit segments, and then survey various applications, including factors of beta-shifts, derived-from-Anosov diffeomorphisms, and geodesic flows in non-positive curvature and beyond.

Keywords : measures of maximal entropy; expansivity; specification

Codes MSC :
37B10 - Symbolic dynamics
 37B40 - Topological entropy
 37D35 - Thermodynamic formalism, variational principles, equilibrium states
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/1947/Notes/Climenhaga-3-notes.pdf
    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 11/06/2019
    Date de captation : 17/05/2019
    Sous collection : Research School
    arXiv category : Dynamical Systems
    Domaine : Dynamical Systems & ODE
    Format : MP4 (.mp4) - HD
    Durée : 00:53:16
    Audience : Researchers
    Download : https://videos.cirm-math.fr/2019-05-17_Climenhaga_Part3.mp4
Informations sur la Rencontre

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

Données de citation

DOI : 10.24350/CIRM.V.19525303
Citer cette vidéo: Climenhaga, Vaughn (2019). Beyond Bowen specification property - lecture 3. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19525303
URI : http://dx.doi.org/10.24350/CIRM.V.19525303

Bibliographie

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  • ALVES, José Ferreira. SRB measures for non-hyperbolic systems with multidimensional expansion. In : Annales scientifiques de l'Ecole normale supérieure. No longer published by Elsevier, 2000. p. 1-32. - https://doi.org/10.1016/S0012-9593(00)00101-4

  • BALLMANN, Werner, BRIN, Michael, BURNS, Keith, et al. On surfaces with no conjugate points. Journal of Differential Geometry, 1987, vol. 25, no 2, p. 249-273. - https://doi.org/10.4310/jdg/1214440852

  • BURNS, Keith, CLIMENHAGA, Vaughn, FISHER, Todd, et al. Unique equilibrium states for geodesic flows in nonpositive curvature. Geometric and Functional Analysis, 2018, vol. 28, no 5, p. 1209-1259. - https://doi.org/10.1007/s00039-018-0465-8

  • BUZZI, Jérôme et FISHER, Todd. Entropic stability beyond partial hyperbolicity. arXiv preprint arXiv:1103.2707, 2011. - https://doi.org/10.3934/jmd.2013.7.527

  • BOSCHÉ, Aurélien. Expansive geodesic flows on compact manifolds without conjugate points. 2015. Thèse de doctorat. Université Grenoble Alpes. - https://tel.archives-ouvertes.fr/tel-01691107

  • CHEN, Dong, KAO, Lien-Yung, et PARK, Kiho. Unique equilibrium states for geodesic flows over surfaces without focal points. arXiv preprint arXiv:1808.00663, 2018. - https://arxiv.org/abs/1808.00663

  • CLIMENHAGA, Vaughn, KNIEPER, Gerhard, et WAR, Khadim. Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points. arXiv preprint arXiv:1903.09831, 2019. - https://arxiv.org/abs/1903.09831

  • CLIMENHAGA, Vaughn et THOMPSON, Daniel J. Intrinsic ergodicity via obstruction entropies. Ergodic Theory and Dynamical Systems, 2014, vol. 34, no 6, p. 1816-1831. - https://doi.org/10.1017/etds.2013.16

  • CLIMENHAGA, Vaughn et THOMPSON, Daniel J. Unique equilibrium states for flows and homeomorphisms with non-uniform structure. Advances in Mathematics, 2016, vol. 303, p. 745-799. - https://doi.org/10.1016/j.aim.2016.07.029

  • GELFERT, Katrin et RUGGIERO, Rafael O. Geodesic Flows Modelled by Expansive Flows. Proceedings of the Edinburgh Mathematical Society, 2019, vol. 62, no 1, p. 61-95. - https://doi.org/10.1017/S0013091518000160

  • KNIEPER, Gerhard. The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds. 1997. - https://doi.org/10.2307/120995

  • LIU, Fei, WANG, Fang, et WU, Weisheng. On the Patterson-Sullivan measure for geodesic flows on rank $1 $ manifolds without focal points. arXiv preprint arXiv:1812.04398, 2018. - https://arxiv.org/abs/1812.04398

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