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Cours Sorbonne Université
30 videos

Cours Sorbonne Université

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Dynamique au-delà de l’hyperbolicité uniforme / Dynamics Beyond Uniform Hyperbolicity

Collection Dynamique au-delà de l’hyperbolicité uniforme / Dynamics Beyond Uniform Hyperbolicity

Organizer(s) Bonatti, Christian ; Buzzi, Jérôme ; Crovisier, Sylvain ; Gan, Shaobo ; Pacifico, Maria José
Date(s) 13/05/2019 - 24/05/2019
linked URL https://conferences.cirm-math.fr/1947.html
00:00:00 / 00:00:00
24 32

Beyond Bowen specification property - lecture 2

By Daniel J. Thompson

These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.

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Citation data

  • DOI 10.24350/CIRM.V.19525603
  • Cite this video Thompson, Daniel J. (21/05/2019). Beyond Bowen specification property - lecture 2. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19525603
  • URL https://dx.doi.org/10.24350/CIRM.V.19525603

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Bibliography

  • BABILLOT, Martine. On the mixing property for hyperbolic systems. Israel journal of mathematics, 2002, vol. 129, no 1, p. 61-76. - https://doi.org/10.1007/BF02773153
  • 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://arxiv.org/abs/1703.10878
  • CHERNOV, N. I. et HASKELL, C. Nonuniformly hyperbolic K-systems are Bernoulli. Ergodic theory and dynamical systems, 1996, vol. 16, no 1, p. 19-44. - https://doi.org/10.1017/S0143385700008695
  • 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 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
  • LEDRAPPIER, François. Mesures d’équilibre d’entropie complètement positive. Astérisque, 1977, vol. 50, p. 251-272. - http://www.numdam.org/item/AST_1977__50__251_0/
  • LEDRAPPIER, François, LIMA, Yuri, et SARIG, Omri. Ergodic properties of equilibrium measures for smooth three dimensional flows. arXiv preprint arXiv:1504.00048, 2015. - https://arxiv.org/abs/1504.00048
  • ORNSTEIN, Donald, WEISS, Benjamin. Geodesic flows are Bernoullian. Israel journal of mathematics, 1973, vol. 14, no 2, p. 184-198. - https://doi.org/10.1007/BF02762673
  • ORNSTEIN, D., & WEISS, B. (1998). On the Bernoulli nature of systems with some hyperbolic structure. Ergodic Theory and Dynamical Systems, 18(2), 441-456. - https://doi.org/10.1017/S0143385798100354
  • PESIN, Ya B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russian mathematical surveys, 1977, vol. 32, no 4. - https://doi.org/10.1070/RM1977v032n04ABEH001639ISTEX

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