Unique equilibrium states for geodesic flows over manifolds without focal-points | Vidéo | Carmin.tv

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Unique equilibrium states for geodesic flows over manifolds without focal-points

De Lien-Yung Kao

Apparaît dans la collection : Jean-Morlet Chair 2019 - Research School: Thermodynamic Formalism: Modern Techniques in Smooth Ergodic Theory / Chaire Jean-Morlet 2019 - Ecole : Formalisme thermodynamique : techniques modernes en théorie ergodique

We study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted regular periodic orbits are equidistributed relative to these unique equilibrium states.

Informations sur la vidéo

Date de captation 01/07/2019
Date de publication 31/07/2019
Institut CIRM
Licence CC BY NC ND
Langue Anglais
Audience Chercheurs
Réalisateur(s) Luca Recanzone
Format MP4

Données de citation

DOI 10.24350/CIRM.V.19541703
Citer cette vidéo Kao, Lien-Yung (01/07/2019). Unique equilibrium states for geodesic flows over manifolds without focal-points. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19541703
URL https://dx.doi.org/10.24350/CIRM.V.19541703

Domaine(s)

Bibliographie

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
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. - http://dx.doi.org/10.1007/s00039-018-0465-8
BURNS, Keith et GELFERT, Katrin. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems-A, 2014, vol. 34, no 5, p. 1841-1872. - http://dx.doi.org/10.3934/dcds.2014.34.1841
BOWEN, Rufus. Some systems with unique equilibrium states. Theory of computing systems, 1974, vol. 8, no 3, p. 193-202. - http://dx.doi.org/10.1007/BF01762666
BURNS, Keith. Hyperbolic behaviour of geodesic flows on manifolds with no focal points. Ergodic Theory and Dynamical Systems, 1983, vol. 3, no 1, p. 1-12. - https://doi.org/10.1017/S0143385700001796
CLIMENHAGA, Vaughn, FISHER, Todd, et THOMPSON, Daniel J. Unique equilibrium states for Bonatti–Viana diffeomorphisms. Nonlinearity, 2018, vol. 31, no 6, p. 2532. - https://doi.org/10.1088/1361-6544/aab1cd
CLIMENHAGA, Vaughn et THOMPSON, Daniel J. Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors. Israel Journal of Mathematics, 2012, vol. 192, no 2, p. 785-817. - http://dx.doi.org/10.1007/s11856-012-0052-x
CLIMENHAGA, Vaughn et THOMPSON, Daniel J. Equilibrium states beyond specification and the Bowen property. Journal of the London Mathematical Society, 2013, vol. 87, no 2, p. 401-427. - https://doi.org/10.1112/jlms/jds054
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
P DO CARMO, Manfredo, Riemannian Geometry, Birkhäuser, 2013
EBERLEIN, Patrick, et al. When is a geodesic flow of Anosov type? I. Journal of Differential Geometry, 1973, vol. 8, no 3, p. 437-463. - http://dx.doi.org/10.4310/jdg/1214431801
ESCHENBURG, Jost-Hinrich. Horospheres and the stable part of the geodesic flow. Mathematische Zeitschrift, 1977, vol. 153, no 3, p. 237-251. - http://dx.doi.org/10.1007/BF01214477
FRANCO, Ernesto. Flows with unique equilibrium states. American Journal of Mathematics, 1977, vol. 99, no 3, p. 486-514. - http://dx.doi.org/10.2307/2373927
GERBER, Marlies. On the existence of focal points near closed geodesics on surfaces. Geometriae Dedicata, 2003, vol. 98, no 1, p. 123-160. - http://dx.doi.org/10.1023/A:1024085309422
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
GELFERT, Katrin et SCHAPIRA, Barbara. Pressures for geodesic flows of rank one manifolds. Nonlinearity, 2014, vol. 27, no 7, p. 1575. - https://doi.org/10.1088/0951-7715/27/7/1575
GULLIVER, Robert. On the variety of manifolds without conjugate points. Transactions of the American Mathematical Society, 1975, vol. 210, p. 185-201. - http://dx.doi.org/10.1090/S0002-9947-1975-0383294-0
HOPF, Eberhard. Closed surfaces without conjugate points. Proceedings of the National Academy of Sciences of the United States of America, 1948, vol. 34, no 2, p. 47. - https://dx.doi.org/10.1073%2Fpnas.34.2.47
HURLEY, Donal. Ergodicity of the geodesic flow on rank one manifolds without focal points. In : Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences. Royal Irish Academy, 1986. p. 19-30. - https://www.jstor.org/stable/20489231
KNIEPER, Gerhard. The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds. Annals of mathematics, 1998, p. 291-314. - http://dx.doi.org/10.2307/120995
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
LIMA, Yuri et SARIG, Omri. Symbolic dynamics for three dimensional flows with positive topological entropy. arXiv preprint arXiv:1408.3427, 2014. - https://arxiv.org/abs/1408.3427
LIU, Fei et WANG, Fang. Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points. Acta Mathematica Sinica, English Series, 2016, vol. 32, no 4, p. 507-520. - http://dx.doi.org/10.1007/s10114-016-5200-5
O'SULLIVAN, John J., et al. Riemannian manifolds without focal points. Journal of Differential Geometry, 1976, vol. 11, no 3, p. 321-333. - http://dx.doi.org/10.4310/jdg/1214433590
PESIN, Ja B. Geodesic flows on closed Riemannian manifolds without focal points. Mathematics of the USSR-Izvestiya, 1977, vol. 11, no 6, p. 1195. - https://doi.org/10.1070/IM1977v011n06ABEH001766
POLLICOTT, Mark. Closed geodesic distribution for manifolds of non-positive curvature. Discrete & Continuous Dynamical Systems-A, 1996, vol. 2, no 2, p. 153-161. - http://dx.doi.org/10.3934/dcds.1996.2.153
PARRY, William et POLLICOTT, Mark. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque, 1990, vol. 187, no 188, p. 1-268. - http://www.numdam.org/item?id=AST_1990__187-188__1_0
WALTERS, Peter. An introduction to ergodic theory. Springer Science & Business Media, 2000.
WALTERS, Peter. Differentiability properties of the pressure of a continuous transformation on a compact metric space. Journal of the London Mathematical Society, 1992, vol. 2, no 3, p. 471-481. - https://doi.org/10.1112/jlms/s2-46.3.471

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