Exponential mixing for skew products with discontinuities
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- by Oliver Butterley and Peyman Eslami PDF
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Abstract:
We consider the 2D skew product $F: (x,u) \mapsto (f(x), u+\tau (x))$, where the base map $f$ is a piecewise $\mathscr {C}^{2}$, covering and uniformly expanding the map of the circle, and the fibre map $\tau$ is piecewise $\mathscr {C}^{2}$. We show that this system mixes exponentially when $\tau$ is not cohomologous (via a Lipschitz function) to a piecewise constant.References
- José F. Alves, Maria Carvalho, and Jorge Milhazes Freitas, Statistical stability for Hénon maps of the Benedicks-Carleson type, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 2, 595–637. MR 2595193, DOI 10.1016/j.anihpc.2009.09.009
- 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, Michael Benedicks, and Véronique Maume-Deschamps, Almost sure rates of mixing for i.i.d. unimodal maps, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 1, 77–126 (English, with English and French summaries). MR 1886006, DOI 10.1016/S0012-9593(01)01083-7
- 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
- Oliver Butterley, Area expanding $\scr {C}^{1+\alpha }$ suspension semiflows, Comm. Math. Phys. 325 (2014), no. 2, 803–820. MR 3148102, DOI 10.1007/s00220-013-1835-6
- Nikolai Chernov and Roberto Markarian, Chaotic billiards, Mathematical Surveys and Monographs, vol. 127, American Mathematical Society, Providence, RI, 2006. MR 2229799, DOI 10.1090/surv/127
- N. I. Chernov, Markov approximations and decay of correlations for Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 269–324. MR 1626741, DOI 10.2307/121010
- Mark F. Demers and Carlangelo Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc. 360 (2008), no. 9, 4777–4814. MR 2403704, DOI 10.1090/S0002-9947-08-04464-4
- 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 and Carlangelo Liverani, Energy transfer in a fast-slow Hamiltonian system, Comm. Math. Phys. 308 (2011), no. 1, 201–225. MR 2842975, DOI 10.1007/s00220-011-1317-7
- P. Eslami, Stretched-exponential mixing for $\mathscr {C}^{1+\alpha }$ skew products with discontinuities, preprint, arXiv:1405.6981, 2014.
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- Jorge Milhazes Freitas, Continuity of SRB measure and entropy for Benedicks-Carleson quadratic maps, Nonlinearity 18 (2005), no. 2, 831–854. MR 2122687, DOI 10.1088/0951-7715/18/2/019
- S. Galatolo and I. Nisoli, Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps, To appear in Ergod. Th. & Dynam. Sys., 2014.
- Stefano Galatolo, Jérôme Rousseau, and Benoit Saussol, Skew products, quantitative recurrence, shrinking targets and decay of correlations, Ergodic Theory Dynam. Systems 35 (2015), no. 6, 1814–1845. MR 3377286, DOI 10.1017/etds.2014.10
- Sébastien Gouëzel, Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences, Duke Math. J. 147 (2009), no. 2, 193–284. MR 2495076, DOI 10.1215/00127094-2009-011
- Gerhard Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math. 94 (1982), no. 4, 313–333. MR 685377, DOI 10.1007/BF01667385
- G. Keller, Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems, Trans. Amer. Math. Soc. 314 (1989), no. 2, 433–497. MR 1005524, DOI 10.1090/S0002-9947-1989-1005524-4
- 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
- G. Keller and C. Liverani, A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps, Dynamics of coupled map lattices and of related spatially extended systems, Lecture Notes in Phys., vol. 671, Springer, Berlin, 2005, pp. 115–151. MR 2395729, DOI 10.1007/11360810_{6}
- Gerhard Keller and Carlangelo Liverani, Map lattices coupled by collisions, Comm. Math. Phys. 291 (2009), no. 2, 591–597. MR 2530174, DOI 10.1007/s00220-009-0835-z
- Carlangelo Liverani, Decay of correlations, Ann. of Math. (2) 142 (1995), no. 2, 239–301. MR 1343323, DOI 10.2307/2118636
- Carlangelo Liverani, Decay of correlations for piecewise expanding maps, J. Statist. Phys. 78 (1995), no. 3-4, 1111–1129. MR 1315241, DOI 10.1007/BF02183704
- Carlangelo Liverani, Rigorous numerical investigation of the statistical properties of piecewise expanding maps. A feasibility study, Nonlinearity 14 (2001), no. 3, 463–490. MR 1830903, DOI 10.1088/0951-7715/14/3/303
- 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
- Ippei Obayashi, Exponential decay of correlations for surface semiflows with an expanding direction, J. Math. Kyoto Univ. 49 (2009), no. 2, 427–440. MR 2571851, DOI 10.1215/kjm/1256219166
- 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
- David Ruelle, Flots qui ne mélangent pas exponentiellement, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 4, 191–193 (French, with English summary). MR 692974, DOI 10.1142/9789812833709_{0}024
- Omri Sarig, Subexponential decay of correlations, Invent. Math. 150 (2002), no. 3, 629–653. MR 1946554, DOI 10.1007/s00222-002-0248-5
- Masato Tsujii, Decay of correlations in suspension semi-flows of angle-multiplying maps, Ergodic Theory Dynam. Systems 28 (2008), no. 1, 291–317. MR 2380311, DOI 10.1017/S0143385707000430
- 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
Additional Information
- Oliver Butterley
- Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
- MR Author ID: 805760
- Email: oliver.butterley@univie.ac.at
- Peyman Eslami
- Affiliation: Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy
- MR Author ID: 819612
- Email: eslami@mat.uniroma2.it
- Received by editor(s): June 26, 2014
- Received by editor(s) in revised form: January 22, 2015
- Published electronically: May 6, 2016
- Additional Notes: The first author was supported by the Austrian Science Fund, Lise Meitner position M1583.
The second author was supported by an INdAM-COFUND Marie Curie fellowship
This research was partially supported by the Stiftung Aktion Österreich Ungarn (AÖU), Projekt Nr. 87öu6. - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 783-803
- MSC (2010): Primary 37A25; Secondary 37C30, 37D50
- DOI: https://doi.org/10.1090/tran/6761
- MathSciNet review: 3572254