Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T17:48:34.841Z Has data issue: false hasContentIssue false
  • English
  • Français

Functional norms for Young towers

Published online by Cambridge University Press:  24 August 2009

MARK F. DEMERS
MARK F. DEMERS*
Affiliation:
Department of Mathematics, Fairfield University, Fairfield CT 06824, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce functional norms for hyperbolic Young towers which allow us to directly study the transfer operator on the full tower. By eliminating the need for secondary expanding towers commonly employed in this context, this approach simplifies and expands the analysis of this class of Markov extensions and the underlying systems for which they are constructed. As an example, we prove large-deviation estimates with a uniform rate function for a large class of non-invariant measures and show how to translate these to the underlying system.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1371 - 1398
Copyright
Copyright © Cambridge University Press 2009

References

[1]Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, Singapore, 2000.CrossRefGoogle Scholar
[2]Baladi, V. and Tsujii, M.. Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier (Grenoble) 57 (2007), 127–154.CrossRefGoogle Scholar
[3]Benedicks, M. and Young, L.-S.. Markov extensions and decay of correlations for certain Hénon maps. Astérique 261 (2000), 13–56.Google Scholar
[4]Blank, M., Keller, G. and Liverani, C.. Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15(6) (2001), 1905–1973.CrossRefGoogle Scholar
[5]Bruin, H., Demers, M. and Melbourne, I.. Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[6]Chernov, N.. Statistical properties of piecewise smooth hyperbolic systems in high dimensions. Discrete Contin. Dyn. Syst. 5 (1999), 425–448.CrossRefGoogle Scholar
[7]Chernov, N.. Sinai billiards under small external forces. Ann. Henri Poincaré 2(2) (2001), 197–236.CrossRefGoogle Scholar
[8]Chernov, N.. Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122 (2006), 1061–1094.CrossRefGoogle Scholar
[9]Chernov, N. and Young, L.-S.. Decay of correlations for Lorenz gases and hard balls. Hard Ball Systems and the Lorenz Gas (Encyclopaedia of Mathematical Sciences, 101). Ed. Szasz, D.. Springer, Berlin, 2000, pp. 89–120.CrossRefGoogle Scholar
[10]Dembo, A. and Zeitouni, O.. Large Deviations Techniques and Applications, 2nd edn(Applications of Mathematics, 38). Springer, New York, 1998.CrossRefGoogle Scholar
[11]Demers, M. and Liverani, C.. Stability of statistical properties for two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9) (2008), 4777–4814.CrossRefGoogle Scholar
[12]Demers, M., Wright, P. and Young, L.-S.. Escape rates and physically relevant measures for billiards with small holes. Comm. Math. Phys. to appear.Google Scholar
[13]Doeblin, W. and Fortet, R.. Sur des chaînes à liaisons complète. Bull. Soc. Math. France 65 (1937), 132–148.CrossRefGoogle Scholar
[14]Gouëzel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26(1) (2006), 189–217.CrossRefGoogle Scholar
[15]Hennion, H. and Hervé, L.. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems via Quasi-Compactness (Lecture Notes in Mathematics, 1766). Springer, Berlin, 2001.CrossRefGoogle Scholar
[16]Holland, M. and Melbourne, I.. Central limit theorems and invariance principles for Lorenz attractors. J. Lond. Math. Soc. (2) 76 (2007), 345–364.CrossRefGoogle Scholar
[17]Ionescu-Tulcea, C. T. and Marinescu, G.. Théorème ergodique pour des classes d’opérations non complètement continues. Ann. of Math. (2) 52 (1950), 140–147.CrossRefGoogle Scholar
[18]Kato, T.. Perturbation Theory for Linear Operators, 2nd edn(Grundlehren der mathematischen Wissenchaften, 132). Springer, Berlin, 1984.Google Scholar
[19]Keller, G.. On the rate of convergence to equilibrium in one-dimensional systems. Comm. Math. Phys. 96(2) (1984), 181–193.CrossRefGoogle Scholar
[20]Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 (1999), 141–152.Google Scholar
[21]Keller, G. and Nowicki, T.. Spectral theory, zeta functions and the distribution of periodic points for Collet–Eckmann maps. Comm. Math. Phys. 149 (1992), 31–69.CrossRefGoogle Scholar
[22]Lasota, A. and Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1963), 481–488.CrossRefGoogle Scholar
[23]Melbourne, I. and Nicol, M.. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360 (2008), 6661–6676.CrossRefGoogle Scholar
[24]Orey, S. and Pelikan, S.. Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms. Trans. Amer. Math. Soc. 315(2) (1989), 741–753.Google Scholar
[25]Nagaev, S. V.. Some limit theorems for stationary Markov chains. Theory Probab. Appl. 11(4) (1957), 378–406.CrossRefGoogle Scholar
[26]Rey-Bellet, L. and Young, L.-S.. Large deviations in nonuniformly hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 28 (2008), 587–612.CrossRefGoogle Scholar
[27]Rugh, H. H.. The correlation spectrum for hyperbolic analytic maps. Nonlinearity 5(6) (1992), 1237–1263.CrossRefGoogle Scholar
[28]Walters, P.. Ergodic Theory: Introductory Lectures (Lecture Notes in Mathematics, 458). Springer, Berlin, 1975.CrossRefGoogle Scholar
[29]Wang, Q. and Young, L.-S.. Nonuniformly expanding 1D maps. Comm. Math. Phys. 264 (2006), 255–282.CrossRefGoogle Scholar
[30]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1998), 585–650.CrossRefGoogle Scholar
[31]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar