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Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

Part of: Dynamical systems with hyperbolic behavior Ergodic theory

Published online by Cambridge University Press:  04 June 2021

WAEL BAHSOUN  and
CARLANGELO LIVERANI
WAEL BAHSOUN*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK
CARLANGELO LIVERANI
Affiliation:
Dipartimento di Matematica, Università di Roma II, Tor Vergata, Via della Ricerca Scientifica, 00133Roma, Italy (e-mail: liverani@mat.uniroma2.it)
*
e-mail: W.Bahsoun@lboro.ac.uk
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Abstract

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Given any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.

Keywords

anisotropic Banach spacesAnosov diffeomorphsimsfoliation regularitytransfer operators

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2431 - 2467
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Avila, A., Gouëzel, S. and Tsujii, M.. Smoothness of solenoidal attractors. Discrete Contin. Dyn. Syst. 15(1) (2006), 21–35.CrossRefGoogle Scholar
Baladi, V.. Anisotropic Sobolev spaces and dynamical transfer operators: ${\boldsymbol{\mathcal{C}}}^{\infty }$ foliations. Algebraic and Topological Dynamics (Contemporary Mathematics, 385). American Mathematical Society, Providence, RI, 2005, pp. 123–135.CrossRefGoogle Scholar
Baladi, V.. The quest for the ultimate anisotropic Banach space. J. Stat. Phys. 166(3–4) (2017), 525–557.CrossRefGoogle Scholar
Baladi, V.. Characteristic functions as bounded multipliers on anisotropic spaces. Proc. Amer. Math. Soc. 146(10) (2018), 4405–4420.CrossRefGoogle Scholar
Baladi, V.. Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics, 68). Springer International, New York, 2018.CrossRefGoogle Scholar
Baladi, V.. Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps. A Functional Approach (A Series of Modern Surveys in Mathematics, 68). Springer, Cham, 2018.CrossRefGoogle Scholar
Baladi, V. and Liverani, C.. Exponential decay of correlations for piecewise cone hyperbolic contact flows. Comm. Math. Phys. 314(3) (2012), 689–773.CrossRefGoogle Scholar
Baladi, V., Demers, M. F. and Liverani, C.. Exponential decay of correlations for finite horizon Sinai billiard flows. Invent. Math. 211(1) (2018), 39–177.CrossRefGoogle Scholar
Baladi, V. and Gouëzel, S.. Good Banach spaces for piecewise hyperbolic maps via interpolation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4) (2009), 1453–1481.CrossRefGoogle Scholar
Baladi, V. and Gouëzel, S.. Banach spaces for piecewise cone-hyperbolic maps. J. Mod. Dyn. 4(1) (2010), 91–137.CrossRefGoogle Scholar
Blank, M., Keller, G. and Liverani, C.. Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15(6) (2001), 1905–1973.CrossRefGoogle Scholar
Baladi, V. and Tsujii, M.. Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier 57 (2007), 127–154.CrossRefGoogle Scholar
Baladi, V. and Tsujii, M.. Dynamical determinants and spectrum for hyperbolic diffeomorphisms. Geometric and Probabilistic Structures in Dynamics (Contemporary Mathematics, 469). Ed. Burns, K., Dolgopyat, D. and Pesin, Y.. American Mathematical Society, Providence, RI, 2008, pp. 29–68. Volume in honour of M. Brin’s 60th birthday.CrossRefGoogle Scholar
Butterley, O.. An alternative approach to generalised BV and the application to expanding interval maps. Discrete Contin. Dyn. Syst. 33(8) (2013), 3355–3363.CrossRefGoogle Scholar
Butterley, O. and Liverani, C.. Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1 (2007), 301–322.CrossRefGoogle Scholar
Butterley, O. and Liverani, C.. Robustly invariant sets in fiber contracting bundle flows. J. Mod. Dyn. 7(2) (2013), 255–267.CrossRefGoogle Scholar
Drouot, A.. Stochastic stability of Pollicott–Ruelle resonances. Comm. Math. Phys. 356(2) (2017), 357–396.CrossRefGoogle Scholar
Demers, M. F. and Liverani, C.. Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9) (2008), 4777–4814.CrossRefGoogle Scholar
Dyatlov, S. and Zworski, M.. Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci. Éc. Norm. Supér. 49(3) (2016), 543–577.CrossRefGoogle Scholar
Demers, M. F. and Zhang, H.-K.. Spectral analysis of hyperbolic systems with singularities. Nonlinearity 27(3) (2014), 379–433.CrossRefGoogle Scholar
Demers, M. F. and Zhang, H.-K.. A functional analytic approach to perturbations of the Lorentz gas. Comm. Math. Phys. 324(3) (2013), 767–830.CrossRefGoogle Scholar
Demers, M. F. and Zhang, H.-K.. Spectral analysis of the transfer operator for the Lorentz gas. J. Mod. Dyn. 5(4) (2011), 665–709.CrossRefGoogle Scholar
Faure, F.. Semiclassical origin of the spectral gap for transfer operators of a partially expanding map. Nonlinearity 24(5) (2011), 1473–1498.CrossRefGoogle Scholar
Faure, F. and Roy, N.. Ruelle–Pollicott resonances for real analytic hyperbolic maps. Nonlinearity 19(6) (2006), 1233–1252.CrossRefGoogle Scholar
Faure, F., Roy, N. and Sjöstrand, J.. Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances. Open Math. J. 1 (2008), 35–81.CrossRefGoogle Scholar
Faure, F. and Tsujii, M.. Semiclassical approach for the Ruelle–Pollicott spectrum of hyperbolic dynamics. Analytic and Probabilistic Approaches to Dynamics in Negative Curvature (Springer INdAM Series, 9). Springer, Cham, 2014, pp. 65–135.Google Scholar
Faure, F. and Tsujii, M.. The semiclassical zeta function for geodesic flows on negatively curved manifolds. Invent. Math. 208(3) (2017), 851–998.CrossRefGoogle Scholar
Gouëzel, S.. Almost sure invariance principle for dynamical systems by spectral methods. Ann. Probab. 38(4) (2010), 1639–1671.CrossRefGoogle Scholar
Gouëzel, S.. Limit theorems in dynamical systems using the spectral method. Hyperbolic Dynamics, Fluctuations and Large Deviations (Proceedings of Symposia in Pure Mathematics, 89). American Mathematical Society, Providence, RI, 2015, pp. 161–193.CrossRefGoogle Scholar
Galatolo, S.. Quantitative statistical stability, speed of convergence to equilibrium and partially hyperbolic skew products. J. Éc. Polytech. Math. 5 (2018), 377–405 CrossRefGoogle Scholar
Gouëzel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26(1) (2006), 189–217.CrossRefGoogle Scholar
Gouëzel, S. and Liverani, C.. Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differential Geom. 79(3) (2008), 433–477.CrossRefGoogle Scholar
Galatolo, S. and Lucena, R.. Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps. Discrete Contin. Dyn. Syst. 40(3) (2020), 309–1360.CrossRefGoogle Scholar
Giulietti, P. and Liverani, C.. Parabolic dynamics and anisotropic Banach spaces. J. Eur. Math. Soc. (JEMS) 21(9) (2019), 2793–2858.CrossRefGoogle Scholar
Giulietti, P., Liverani, C. and Pollicott, M.. Anosov flows and dynamical zeta functions. Ann. of Math. 178(2) (2013), 687–773.CrossRefGoogle Scholar
Hennion, H.. Sur un théorème spectral et son application aux noyaux Lipchitziens. Proc. Amer. Math. Soc. 118 (1993), 627–634.Google Scholar
Hasselblatt, B. and Wilkinson, A.. Prevalence of non-Lipschitz Anosov foliations. Ergod. Th. & Dynam. Sys. 19(3) (1999), 643–656.CrossRefGoogle Scholar
Journé, J.-L.. A regularity lemma for functions of several variables. Rev. Mat. Iberoam. 4(2) (1988), 187–193.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza.CrossRefGoogle Scholar
Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28(1) (1999), 141–152.Google Scholar
Keller, G. and Liverani, C.. Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Comm. Math. Phys. 262(1) (2006), 33–50.CrossRefGoogle Scholar
Keller, G. and Liverani, C.. Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135(3) (2009), 519–534.CrossRefGoogle Scholar
Keller, G. and Liverani, C.. Map lattices coupled by collisions. Comm. Math. Phys. 291(2) (2009), 591–597.CrossRefGoogle Scholar
Katok, A., Strelcyn, J.-M., Ledrappier, F. and Przytycki, F.. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222). Springer-Verlag, Berlin, 1986, pp. viii+283.CrossRefGoogle Scholar
Liverani, C.. On contact Anosov flows. Ann. of Math. 159(3) (2004), 1275–1312.CrossRefGoogle Scholar
Liverani, C.. A footnote on expanding maps. Discrete Contin. Dyn. Syst. 33(8) (2013), 3741–3751.CrossRefGoogle Scholar
Liverani, C.. Multidimensional expanding maps with singularities: a pedestrian approach. Ergod. Th. & Dynam. Sys. 33(1) (2013), 168–182.CrossRefGoogle Scholar
Liverani, C.. Rigorous numerical investigation of the statistical properties of piecewise expanding maps. A feasibility study. Nonlinearity 14(3) (2001), 463–490.CrossRefGoogle Scholar
Nussbaum, R. D.. The radius of the essential spectrum. Duke Math. J. 37(3) (1970), 473–478.CrossRefGoogle Scholar
Pesin, Y. B. and Sinai, Y. G.. Space-time chaos in the system of weakly interacting hyperbolic systems. J. Geom. Phys. 5(3) (1988), 483–492.CrossRefGoogle Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Hölder foliations. Duke Math. J. 86(3) (1997), 517–546.CrossRefGoogle Scholar
Saussol, B.. Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math. 116 (2000), 223–248.CrossRefGoogle Scholar
Tsujii, M.. Decay of correlations in suspension semi-flows of angle multiplying maps. Ergod. Th. & Dynam. Sys. 28(1) (2008), 291–317.CrossRefGoogle Scholar
Tsujii, M.. Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity 23(7) (2010), 1495–1545.CrossRefGoogle Scholar