In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic systems—center isometries—that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying on symbolic dynamics) and even provide new information in the classical case.

For such systems, we give in particular a constructive proof of the existence of the SRB measure and of the entropy maximizing measure. We also establish very fine statistical properties (Bernoulliness), and we give a characterization of equilibrium states in terms of their conditional measures in the stable/unstable lamination, similar to the SRB case. The construction is applied to obtain the uniqueness of quasi-invariant measures associated to Hölder Jacobians for the horocyclic flow.

Received:

Revised:

Published online:

DOI:
10.5802/mrr.9

Classification: 37D35, 37D30

Keywords: equilibrium states, conditional measures, unique ergodicity

Author's affiliations:

Revised:

Published online:

Classification: 37D35, 37D30

Keywords: equilibrium states, conditional measures, unique ergodicity

Author's affiliations:

Pablo D. Carrasco ^{1};
Federico Rodriguez-Hertz ^{2}

@article{MRR_2021__2__45_0, author = {Pablo D. Carrasco and Federico Rodriguez-Hertz}, title = {Geometrical constructions of equilibrium states}, journal = {Mathematics Research Reports}, pages = {45--54}, publisher = {MathOA foundation}, volume = {2}, year = {2021}, doi = {10.5802/mrr.9}, language = {en}, url = {https://mrr.centre-mersenne.org/articles/10.5802/mrr.9/} }

TY - JOUR TI - Geometrical constructions of equilibrium states JO - Mathematics Research Reports PY - 2021 DA - 2021/// SP - 45 EP - 54 VL - 2 PB - MathOA foundation UR - https://mrr.centre-mersenne.org/articles/10.5802/mrr.9/ UR - https://doi.org/10.5802/mrr.9 DO - 10.5802/mrr.9 LA - en ID - MRR_2021__2__45_0 ER -

Pablo D. Carrasco; Federico Rodriguez-Hertz. Geometrical constructions of equilibrium states. Mathematics Research Reports, Volume 2 (2021), pp. 45-54. doi : 10.5802/mrr.9. https://mrr.centre-mersenne.org/articles/10.5802/mrr.9/

[1] Geodesic paths and horocycle flow on abelian covers, Lie groups and ergodic theory, Tata Inst. Fund. Res. Stud. Math., Bombay, 1998, pp. 1-32 | Zbl: 0967.37020

[2] Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Physics, 102, Springer-Verlag, 2005 | MR: 2105774 | Zbl: 1060.37020

[3] SRB measures for Anosov actions, 2021 (Preprint. https://arxiv.org/abs/2103.12127)

[4] Markov Partitions for Axiom A Diffeomorphisms, American Journal of Mathematics, Volume 92 (1970) no. 3, pp. 725-747 | Article | MR: 277003 | Zbl: 0208.25901

[5] Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect. Notes in Math., Volume 407, Springer-Verlag, Berlin, 1975, pp. 1-73 | Zbl: 0308.28010

[6] The ergodic theory of Axiom A flows, Invent. Math., Volume 29 (1975), pp. 181-202 | Article | MR: 380889

[7] On local entropy, Lect. Notes in Math., Volume 1007, Springer-Verlag, Berlin, 1983 | Article | MR: 730261 | Zbl: 0533.58020

[8] On the Ergodicity of Partially Hyperbolic Systems, Ann. of Math, Volume 171 (2010), pp. 451-489 | Article | MR: 2630044 | Zbl: 1196.37057

[9] Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map, 2021 (Preprint. https://arxiv.org/abs/2103.07333)

[10] Equilibrium states for center isometries, 2021 (Preprint. https://arxiv.org/abs/2103.07323)

[11] Equilibrium measures for some partially hyperbolic systems, Journal of Modern Dynamics, Volume 16 (2020) no. 0, pp. 155-205 | Article | MR: 4128833 | Zbl: 07342487

[12] Nombre de Rotation des Diffeomorphismes du Cercle et Mesures Automorphes, Regular and Chaotic Dynamics, Volume 4 (1999) no. 4, p. 19 | Article | MR: 1780302 | Zbl: 1012.37024

[13] Unipotent Flows and Applications, Homogeneous Flows, Moduli Spaces and Arithmetic (Clay Mathematics Proceedings), Volume 10 (2007), pp. 71-130 | Zbl: 1243.37004

[14] Conformal Fractals - Ergodic Theory Methods, Cambridge University Press, 2011, 366 pages | Zbl: 1202.37001

[15] The unique ergodigity of the horocycle flow, Lect. Notes in Math., Volume 318, Springer-Verlag, Berlin, 1973, pp. 95-115 | Article

[16] Stably Ergodic Diffeomorphisms, Annals of Mathematics, Volume 140 (1994) no. 2, pp. 295-329 | Article | MR: 1298715 | Zbl: 0824.58032

[17] Differentiation of Integrals in R${}^{\text{n}}$, Lect. Notes in Math., Volume 481, Springer-Verlag, Berlin, 1975 | Article | MR: 457661

[18] Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., Volume 172 (2008), pp. 353-381 | Article | MR: 2390288 | Zbl: 1136.37020

[19] A Survey of Partially Hyperbolic Dynamics, Partially Hyperbolic Dynamics, Laminations and Teichmuller Flow (Fields Institute Communications), Volume 51 (2007), pp. 35-88 | MR: 2388690

[20] Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory and Dynamical Systems, Volume 16 (1996) no. 4, pp. 751-778 | Article | MR: 1406432 | Zbl: 0859.58021

[21] The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Mathematics, Volume 122 (1985), pp. 509-539 | Article | MR: 819556 | Zbl: 0605.58028

[22] The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Mathematics, Volume 122 (1985), pp. 540-574 | Article | MR: 819557 | Zbl: 1371.37012

[23] Unique ergodicity of the horocycle flow: Variable negative curvature case, Israel Journal of Mathematics, Volume 21 (1975) no. 2-3, pp. 133-144 | Article | MR: 407902 | Zbl: 0314.58013

[24] On some aspects of the theory of Anosov systems, Springer, 2004 | Zbl: 1140.37010

[25] Geodesic flows are Bernoullian, Israel Journal of Mathematics, Volume 14 (1973) no. 2, pp. 184-198 | Article | MR: 325926 | Zbl: 0256.58006

[26] The limit set of a Fuchsian group, Acta Mathematica, Volume 136 (1976) no. 0, pp. 241-273 | Article | MR: 450547 | Zbl: 0336.30005

[27] Stable Ergodicity and Julienne Quasi-Conformality, JEMS, Volume 2 (2000) no. 1, pp. 1-52 | Article | MR: 1750453 | Zbl: 0964.37017

[28] An overview of Patterson-Sullivan theory, The barycenter method, FIM, Zurich (2006)

[29] Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Mathematical Journal, Volume 63 (1991) no. 1, pp. 235-280 | MR: 1106945 | Zbl: 0733.22007

[30] On the Fundamental Ideas of Measure Theory, Transl. Amer. Math. Soc., Volume 10 (1962), pp. 1-52

[31] A measure associated with Axiom-A attractors, American Journal of Mathematics, Volume 98 (1976) no. 3, p. 619 | Article | MR: 415683 | Zbl: 0355.58010

[32] Thermodynamic formalism. The mathematical structures of equilibrium statistical mechanics, Cambridge University Press, Cambridge, 2004 | Zbl: 1062.82001

[33] Symbolic dynamics for surface diffeomorphisms with positive entropy, Journal of the AMS, Volume 26 (2013), pp. 341-426 | MR: 3011417 | Zbl: 1280.37031

[34] On quasi-invariant transverse measures for the horospherical foliation of a negatively curved manifold, Ergodic Theory and Dynamical Systems, Volume 24 (2004), pp. 227-255 | Article | MR: 2041270 | Zbl: 1115.37028

[35] Markov partitions and C-diffeomorphisms, Functional Analysis and Its Applications, Volume 2 (1968) no. 1, pp. 61-82 | Article | MR: 233038 | Zbl: 0182.55003

[36] Equilibrium measures for certain isometric extensions of Anosov systems, Ergodic Theory and Dynamical Systems, Volume 38 (2016) no. 3, pp. 1154-1167 | Article | MR: 3784258

[37] An introduction to Ergodic Theory, Springer, 1982, 250 pages | Article | Zbl: 0475.28009

[38] What Are SRB Measures, and Which Dynamical Systems Have Them?, Journal of Statistical Physics, Volume 108 (2002), pp. 733-754 | Article | MR: 1933431 | Zbl: 1124.37307

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