CIRM - Videos & books Library - Beyond Bowen specification property

Multi angle

Authors : Thompson, Daniel J. (Author of the conference)
CIRM (Publisher )

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Abstract : 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.

Keywords : equilibrium states; geodesic flows; topological pressure

MSC Codes :
37C40 - Smooth ergodic theory, invariant measures
37D25 - Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D35 - Thermodynamic formalism, variational principles, equilibrium states
37D40 - Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Additional resources :
https://www.cirm-math.fr/RepOrga/2264/Notes/Thompson-2-notes.pdf

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Film maker :

Hennenfent, Guillaume

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https://videos.cirm-math.fr/2019-05-21_Thompson_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.19525603
Cite this video as: Thompson, Daniel J. (2019). Beyond Bowen specification property - lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19525603
URI : http://dx.doi.org/10.24350/CIRM.V.19525603

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Bibliography

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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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