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-1-notes.pdf

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Film maker :

Hennenfent, Guillaume

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https://videos.cirm-math.fr/2019-05-20_Thompson_Part1.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.19525503
Cite this video as: Thompson, Daniel J. (2019). Beyond Bowen specification property - lecture 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19525503
URI : http://dx.doi.org/10.24350/CIRM.V.19525503

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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

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://arxiv.org/abs/1505.03803

EBERLEIN, Patrick. Geometry of nonpositively curved manifolds. University of Chicago Press, 1996. -

EBERLEIN, Patrick. Geodesic flows in manifolds of nonpositive curvature, Smooth ergodictheory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 2001, vol. 69, p. 525-571. - https://doi.org/10.1090/pspum/069

GELFERT, Katrin et SCHAPIRA, Barbara. Pressures for geodesic flows of rank one manifolds. Nonlinearity, 2014, vol. 27, no 7, p. 1575. - https://hal.archives-ouvertes.fr/hal-00881421/

GERBER, Marlies, WILKINSON, Amie, et al. Hölder regularity of horocycle foliations. J. Differential Geom, 1999, vol. 52, no 1, p. 41-72. - https://doi.org/10.4310/jdg/1214425216

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

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