Equilibrium states in dynamical systems via geometric measure theory

Climenhaga, Vaughn; Pesin, Yakov; Zelerowicz, Agnieszka

doi:10.1090/bull/1659

Equilibrium states in dynamical systems via geometric measure theory
HTML articles powered by AMS MathViewer

by Vaughn Climenhaga, Yakov Pesin and Agnieszka Zelerowicz PDF

Bull. Amer. Math. Soc. 56 (2019), 569-610 Request permission

Abstract:

Given a dynamical system with a uniformly hyperbolic (chaotic) attractor, the physically relevant Sinaĭ–Ruelle–Bowen (SRB) measure can be obtained as the limit of the dynamical evolution of the leaf volume along local unstable manifolds. We extend this geometric construction to the substantially broader class of equilibrium states corresponding to Hölder continuous potentials; these states arise naturally in statistical physics and play a crucial role in studying stochastic behavior of dynamical systems. The key step in our construction is to replace leaf volume with a reference measure that is obtained from a Carathéodory dimension structure via an analogue of the construction of Hausdorff measure. In particular, we give a new proof of existence and uniqueness of equilibrium states that does not use standard techniques based on Markov partitions or the specification property; our approach can be applied to systems that do not have Markov partitions and do not satisfy the specification property.

References

Rufus Bowen and Brian Marcus, Unique ergodicity for horocycle foliations, Israel J. Math. 26 (1977), no. 1, 43–67. MR 451307, DOI 10.1007/BF03007655
Rufus Bowen, Markov partitions for Axiom $\textrm {A}$ diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. MR 277003, DOI 10.2307/2373370
Rufus Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 184 (1973), 125–136. MR 338317, DOI 10.1090/S0002-9947-1973-0338317-X
Rufus Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no. 3, 193–202. MR 399413, DOI 10.1007/BF01762666
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989, DOI 10.1007/BFb0081279
Rufus Bowen, A horseshoe with positive measure, Invent. Math. 29 (1975), no. 3, 203–204. MR 380890, DOI 10.1007/BF01389849
Luis Barreira and Yakov Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge, 2007. Dynamics of systems with nonzero Lyapunov exponents. MR 2348606, DOI 10.1017/CBO9781107326026
Luis Barreira and Yakov Pesin, Introduction to smooth ergodic theory, Graduate Studies in Mathematics, vol. 148, American Mathematical Society, Providence, RI, 2013. MR 3076414, DOI 10.1090/gsm/148
N. Chernov and D. Dolgopyat, Brownian Brownian motion. I, Mem. Amer. Math. Soc. 198 (2009), no. 927, viii+193. MR 2499824, DOI 10.1090/memo/0927
Vaughn Climenhaga, Dmitry Dolgopyat, and Yakov Pesin, Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps, Comm. Math. Phys. 346 (2016), no. 2, 553–602. MR 3535895, DOI 10.1007/s00220-016-2710-z
Yves Coudène, Boris Hasselblatt, and Serge Troubetzkoy, Multiple mixing from weak hyperbolicity by the Hopf argument, Stoch. Dyn. 16 (2016), no. 2, 1660003, 15. MR 3470552, DOI 10.1142/S0219493716600030
Vaughn Climenhaga, Bowen’s equation in the non-uniform setting, Ergodic Theory Dynam. Systems 31 (2011), no. 4, 1163–1182. MR 2818690, DOI 10.1017/S0143385710000362
Vaughn Climenhaga and Yakov Pesin, Building thermodynamics for non-uniformly hyperbolic maps, Arnold Math. J. 3 (2017), no. 1, 37–82. MR 3646530, DOI 10.1007/s40598-016-0052-8
V. Climenhaga, Ya. Pesin, and A. Zelerowicz, Equilibrium measures for some partially hyperbolic systems, arXiv:1810.08663v1 (2018).
Tushar Das and David Simmons, The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result, Invent. Math. 210 (2017), no. 1, 85–134. MR 3698340, DOI 10.1007/s00222-017-0725-5
J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys. 57 (1985), no. 3, 617–656. MR 800052, DOI 10.1103/RevModPhys.57.617
Manfred Einsiedler and Thomas Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. MR 2723325, DOI 10.1007/978-0-85729-021-2
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Ursula Hamenstädt, A new description of the Bowen-Margulis measure, Ergodic Theory Dynam. Systems 9 (1989), no. 3, 455–464. MR 1016663, DOI 10.1017/S0143385700005095
Boris Hasselblatt, A new construction of the Margulis measure for Anosov flows, Ergodic Theory Dynam. Systems 9 (1989), no. 3, 465–468. MR 1016664, DOI 10.1017/S0143385700005101
Nicolai T. A. Haydn, Canonical product structure of equilibrium states, Random Comput. Dynam. 2 (1994), no. 1, 79–96. MR 1265227
Eberhard Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 91 (1939), 261–304 (German). MR 1464
Vadim A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), no. 4, 361–393 (English, with French summary). Hyperbolic behaviour of dynamical systems (Paris, 1990). MR 1096098
Vadim A. Kaimanovich, Bowen-Margulis and Patterson measures on negatively curved compact manifolds, Dynamical systems and related topics (Nagoya, 1990) Adv. Ser. Dynam. Systems, vol. 9, World Sci. Publ., River Edge, NJ, 1991, pp. 223–232. MR 1164891, DOI 10.1080/02681119408806180
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822, DOI 10.1007/BF02684777
Anatole Katok, Fifty years of entropy in dynamics: 1958–2007, J. Mod. Dyn. 1 (2007), no. 4, 545–596. MR 2342699, DOI 10.3934/jmd.2007.1.545
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 163–188 (French). MR 743818, DOI 10.1007/BF02698772
Renaud Leplaideur, Local product structure for equilibrium states, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1889–1912. MR 1661262, DOI 10.1090/S0002-9947-99-02479-4
François Ledrappier and Jean-Marie Strelcyn, A proof of the estimation from below in Pesin’s entropy formula, Ergodic Theory Dynam. Systems 2 (1982), no. 2, 203–219 (1983). MR 693976, DOI 10.1017/S0143385700001528
Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
G. A. Margulis, Certain measures that are connected with U-flows on compact manifolds, Funkcional. Anal. i Priložen. 4 (1970), no. 1, 62–76 (Russian). MR 0272984
S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241–273. MR 450547, DOI 10.1007/BF02392046
Yakov Pesin and Vaughn Climenhaga, Lectures on fractal geometry and dynamical systems, Student Mathematical Library, vol. 52, American Mathematical Society, Providence, RI, 2009. MR 2560337, DOI 10.1090/stml/052
Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
Ya. B. Pesin, Dimension-like characteristics for invariant sets of dynamical systems, Uspekhi Mat. Nauk 43 (1988), no. 4(262), 95–128, 255 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 4, 111–151. MR 969568, DOI 10.1070/RM1988v043n04ABEH001892
Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. MR 1489237, DOI 10.7208/chicago/9780226662237.001.0001
Ya. B. Pesin and B. S. Pitskel′, Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen. 18 (1984), no. 4, 50–63, 96 (Russian, with English summary). MR 775933
Frédéric Paulin, Mark Pollicott, and Barbara Schapira, Equilibrium states in negative curvature, Astérisque 373 (2015), viii+281 (English, with English and French summaries). MR 3444431
Ya. B. Pesin and Ya. G. Sinaĭ, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 417–438 (1983). MR 721733, DOI 10.1017/S014338570000170X
V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation 1952 (1952), no. 71, 55. MR 0047744
David Ruelle and Dennis Sullivan, Currents, flows and diffeomorphisms, Topology 14 (1975), no. 4, 319–327. MR 415679, DOI 10.1016/0040-9383(75)90016-6
David Ruelle, A measure associated with axiom-A attractors, Amer. J. Math. 98 (1976), no. 3, 619–654. MR 415683, DOI 10.2307/2373810
David Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), no. 1, 83–87. MR 516310, DOI 10.1007/BF02584795
David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
Omri M. Sarig, Thermodynamic formalism for countable Markov shifts, Hyperbolic dynamics, fluctuations and large deviations, Proc. Sympos. Pure Math., vol. 89, Amer. Math. Soc., Providence, RI, 2015, pp. 81–117. MR 3309096, DOI 10.1090/pspum/089/01485
Ja. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen 2 (1968), no. 1, 64–89 (Russian). MR 0233038
Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
Steve Smale, Finding a horseshoe on the beaches of Rio, Math. Intelligencer 20 (1998), no. 1, 39–44. MR 1601831, DOI 10.1007/BF03024399
Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. MR 556586, DOI 10.1007/BF02684773
Marcelo Viana, Stochastic dynamics of deterministic systems, Lecture notes XXI Braz. Math. Colloq., IMPA, Rio de Janeiro, 1997.
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473–487. MR 217808, DOI 10.1016/0040-9383(67)90005-5