Rates of mixing for non-Markov infinite measure semiflows
HTML articles powered by AMS MathViewer
- by Henk Bruin, Ian Melbourne and Dalia Terhesiu PDF
- Trans. Amer. Math. Soc. 371 (2019), 7343-7386 Request permission
Abstract:
We develop an abstract framework for obtaining optimal rates of mixing and higher order asymptotics for infinite measure semiflows. Previously, such results were restricted to the situation where there is a first return Poincaré map that is uniformly expanding and Markov. As illustrations of the method, we consider semiflows over non-Markov Pomeau–Manneville intermittent maps with infinite measure, and we also obtain mixing rates for semiflows over Collet–Eckmann maps with nonintegrable roof function.References
- Jon Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, 1997. MR 1450400, DOI 10.1090/surv/050
- Jon Aaronson and Manfred Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn. 1 (2001), no. 2, 193–237. MR 1840194, DOI 10.1142/S0219493701000114
- V. Araújo, O. Butterley, and P. Varandas, Open sets of axiom A flows with exponentially mixing attractors, Proc. Amer. Math. Soc. 144 (2016), no. 7, 2971–2984. MR 3487229, DOI 10.1090/proc/13055
- Vitor Araújo and Ian Melbourne, Exponential decay of correlations for nonuniformly hyperbolic flows with a $C^{1+\alpha }$ stable foliation, including the classical Lorenz attractor, Ann. Henri Poincaré 17 (2016), no. 11, 2975–3004. MR 3556513, DOI 10.1007/s00023-016-0482-9
- Artur Avila and Carlos Gustavo Moreira, Statistical properties of unimodal maps: the quadratic family, Ann. of Math. (2) 161 (2005), no. 2, 831–881. MR 2153401, DOI 10.4007/annals.2005.161.831
- Artur Avila, Sébastien Gouëzel, and Jean-Christophe Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 143–211. MR 2264836, DOI 10.1007/s10240-006-0001-5
- Viviane Baladi, Mark F. Demers, and Carlangelo Liverani, Exponential decay of correlations for finite horizon Sinai billiard flows, Invent. Math. 211 (2018), no. 1, 39–177. MR 3742756, DOI 10.1007/s00222-017-0745-1
- Viviane Baladi and Carlangelo Liverani, Exponential decay of correlations for piecewise cone hyperbolic contact flows, Comm. Math. Phys. 314 (2012), no. 3, 689–773. MR 2964773, DOI 10.1007/s00220-012-1538-4
- Viviane Baladi and Brigitte Vallée, Exponential decay of correlations for surface semi-flows without finite Markov partitions, Proc. Amer. Math. Soc. 133 (2005), no. 3, 865–874. MR 2113938, DOI 10.1090/S0002-9939-04-07671-3
- Péter Bálint and Sébastien Gouëzel, Limit theorems in the stadium billiard, Comm. Math. Phys. 263 (2006), no. 2, 461–512. MR 2207652, DOI 10.1007/s00220-005-1511-6
- Michael Benedicks and Lennart Carleson, On iterations of $1-ax^2$ on $(-1,1)$, Ann. of Math. (2) 122 (1985), no. 1, 1–25. MR 799250, DOI 10.2307/1971367
- H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys. 168 (1995), no. 3, 571–580. MR 1328254
- Henk Bruin, Stefano Luzzatto, and Sebastian Van Strien, Decay of correlations in one-dimensional dynamics, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 4, 621–646 (English, with English and French summaries). MR 2013929, DOI 10.1016/S0012-9593(03)00025-9
- Henk Bruin and Dalia Terhesiu, Upper and lower bounds for the correlation function via inducing with general return times, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 34–62. MR 3742537, DOI 10.1017/etds.2016.20
- Keith Burns, Howard Masur, Carlos Matheus, and Amie Wilkinson, Rates of mixing for the Weil-Petersson geodesic flow: exponential mixing in exceptional moduli spaces, Geom. Funct. Anal. 27 (2017), no. 2, 240–288. MR 3626613, DOI 10.1007/s00039-017-0401-3
- O. Butterley and K. War, Open sets of exponentially mixing Anosov flows, J. Eur. Math. Soc. (to appear).
- N. Chernov and H.-K. Zhang, Improved estimates for correlations in billiards, Comm. Math. Phys. 277 (2008), no. 2, 305–321. MR 2358286, DOI 10.1007/s00220-007-0360-x
- P. Collet and J.-P. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems 3 (1983), no. 1, 13–46. MR 743027, DOI 10.1017/S0143385700001802
- Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357–390. MR 1626749, DOI 10.2307/121012
- Dmitry Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1097–1114. MR 1653299, DOI 10.1017/S0143385798117431
- Michael Field, Ian Melbourne, and Andrei Török, Stability of mixing and rapid mixing for hyperbolic flows, Ann. of Math. (2) 166 (2007), no. 1, 269–291. MR 2342697, DOI 10.4007/annals.2007.166.269
- Adriano Garsia and John Lamperti, A discrete renewal theorem with infinite mean, Comment. Math. Helv. 37 (1962/63), 221–234. MR 148121, DOI 10.1007/BF02566974
- S. Gouëzel, Vitesse de décorrélation et théorèmes limites pour les applications non uniformément dilatantes, 2004. Thesis (Ph.D.)–Ecole Normale Supérieure.
- Sébastien Gouëzel, Sharp polynomial estimates for the decay of correlations, Israel J. Math. 139 (2004), 29–65. MR 2041223, DOI 10.1007/BF02787541
- Sébastien Gouëzel, Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps, Israel J. Math. 180 (2010), 1–41. MR 2735054, DOI 10.1007/s11856-010-0092-z
- Sébastien Gouëzel, Correlation asymptotics from large deviations in dynamical systems with infinite measure, Colloq. Math. 125 (2011), no. 2, 193–212. MR 2871313, DOI 10.4064/cm125-2-5
- M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys. 81 (1981), no. 1, 39–88. MR 630331
- Gerhard Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 (1989), no. 2-3, 183–200. MR 1026617, DOI 10.1007/BF01308670
- Gerhard Keller and Carlangelo Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 1, 141–152. MR 1679080
- Carlangelo Liverani, On contact Anosov flows, Ann. of Math. (2) 159 (2004), no. 3, 1275–1312. MR 2113022, DOI 10.4007/annals.2004.159.1275
- Carlangelo Liverani and Dalia Terhesiu, Mixing for some non-uniformly hyperbolic systems, Ann. Henri Poincaré 17 (2016), no. 1, 179–226. MR 3437828, DOI 10.1007/s00023-015-0399-8
- Marco Martens, Distortion results and invariant Cantor sets of unimodal maps, Ergodic Theory Dynam. Systems 14 (1994), no. 2, 331–349. MR 1279474, DOI 10.1017/S0143385700007902
- Ian Melbourne, Rapid decay of correlations for nonuniformly hyperbolic flows, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2421–2441. MR 2276628, DOI 10.1090/S0002-9947-06-04267-X
- Ian Melbourne, Decay of correlations for slowly mixing flows, Proc. Lond. Math. Soc. (3) 98 (2009), no. 1, 163–190. MR 2472164, DOI 10.1112/plms/pdn028
- I. Melbourne, Superpolynomial and polynomial mixing for semiflows and flows, Nonlinearity 31 (2018), 268–316.
- Ian Melbourne and Dalia Terhesiu, Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math. 189 (2012), no. 1, 61–110. MR 2929083, DOI 10.1007/s00222-011-0361-4
- Ian Melbourne and Dalia Terhesiu, First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure, Israel J. Math. 194 (2013), no. 2, 793–830. MR 3047092, DOI 10.1007/s11856-012-0154-5
- Ian Melbourne and Dalia Terhesiu, Operator renewal theory for continuous time dynamical systems with finite and infinite measure, Monatsh. Math. 182 (2017), no. 2, 377–431. MR 3600409, DOI 10.1007/s00605-016-0922-0
- I. Melbourne and D. Terhesiu, Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure, J. Mod. Dyn. (to appear).
- I. Melbourne and D. Terhesiu, Renewal theorems and mixing for non Markov flows with infinite measure, arXiv:1701.08440 (2017).
- W. de Melo and S. van Strien, One-dimensional dynamics, Springer, Berlin–Heidelberg–New York, 1993.
- Tomasz Nowicki, Some dynamical properties of $S$-unimodal maps, Fund. Math. 142 (1993), no. 1, 45–57. MR 1207470, DOI 10.4064/fm-142-1-45-57
- Mark Pollicott, On the rate of mixing of Axiom A flows, Invent. Math. 81 (1985), no. 3, 413–426. MR 807065, DOI 10.1007/BF01388579
- Yves Pomeau and Paul Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys. 74 (1980), no. 2, 189–197. MR 576270
- Omri Sarig, Subexponential decay of correlations, Invent. Math. 150 (2002), no. 3, 629–653. MR 1946554, DOI 10.1007/s00222-002-0248-5
- Domokos Szász and Tamás Varjú, Limit laws and recurrence for the planar Lorentz process with infinite horizon, J. Stat. Phys. 129 (2007), no. 1, 59–80. MR 2349520, DOI 10.1007/s10955-007-9367-0
- Dalia Terhesiu, Improved mixing rates for infinite measure-preserving systems, Ergodic Theory Dynam. Systems 35 (2015), no. 2, 585–614. MR 3316926, DOI 10.1017/etds.2013.59
- Masato Tsujii, Exponential mixing for generic volume-preserving Anosov flows in dimension three, J. Math. Soc. Japan 70 (2018), no. 2, 757–821. MR 3787739, DOI 10.2969/jmsj/07027595
- Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585–650. MR 1637655, DOI 10.2307/120960
- Lai-Sang Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153–188. MR 1750438, DOI 10.1007/BF02808180
- Roland Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity 11 (1998), no. 5, 1263–1276. MR 1644385, DOI 10.1088/0951-7715/11/5/005
Additional Information
- Henk Bruin
- Affiliation: Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria
- MR Author ID: 329851
- Ian Melbourne
- Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
- MR Author ID: 123300
- Dalia Terhesiu
- Affiliation: Mathematics Department, University of Exeter, Exeter EX4 4QF, United Kingdom
- MR Author ID: 826851
- Received by editor(s): May 22, 2017
- Received by editor(s) in revised form: September 22, 2017, January 31, 2018, and March 20, 2018
- Published electronically: October 24, 2018
- Additional Notes: The research of the second author was supported in part by a European Advanced Grant StochExtHomog (ERC AdG 320977).
The authors are grateful for the support of the Erwin Schrödinger International Institute for Mathematical Physics at the University of Vienna, where part of this research was carried out. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7343-7386
- MSC (2010): Primary 37A25; Secondary 37A40, 37A50, 37D25
- DOI: https://doi.org/10.1090/tran/7582
- MathSciNet review: 3939580