Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-27T22:32:10.789Z Has data issue: false hasContentIssue false
  • English
  • Français

Multidimensional expanding maps with singularities: a pedestrian approach

Published online by Cambridge University Press:  02 February 2012

CARLANGELO LIVERANI
CARLANGELO LIVERANI*
Affiliation:
Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

I provide a proof of the existence of absolutely continuous invariant measures (and study their statistical properties) for multidimensional piecewise expanding systems with not necessarily bounded derivative or distortion. The proof uses basic properties of multidimensional BV functions (the space of functions of bounded variations).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 168 - 182
Copyright
Copyright © Cambridge University Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, River Edge, NJ, 2000.CrossRefGoogle Scholar
[2]Baladi, V.. Anisotropic Sobolev spaces and dynamical transfer operators: 𝒞∞ foliations. Algebraic and Topological Dynamics (Contemporary Mathematics, 385). Eds. Kolyada, S., Manin, Y. and Ward, T.. American Mathematical Society, Providence, RI, 2005, pp. 123–135.CrossRefGoogle Scholar
[3]Baladi, V. and Gouëzel, S.. Good Banach spaces for piecewise hyperbolic maps via interpolation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1453–1481.CrossRefGoogle Scholar
[4]Baladi, V. and Gouëzel, S.. Banach spaces for piecewise cone hyperbolic maps. J. Mod. Dyn. 4 (2010), 91–137.CrossRefGoogle Scholar
[5]Baladi, V. and Liverani, C.. Exponential decay of correlations for piecewise cone hyperbolic contact flows. Preprint, arXiv:1105.0567v1.Google Scholar
[6]Baladi, V. and Tsujii, M.. Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier 57 (2007), 127–154.CrossRefGoogle Scholar
[7]Baladi, V. and Tsujii, M.. Dynamical determinants and spectrum for hyperbolic diffeomorphisms. Geometric and Probabilistic Structures in Dynamics (Contemporary Mathematics, 469). Eds. Burns, K., Dolgopyat, D. and Pesin, Ya.. American Mathematical Society, Providence, RI, 2008, pp. 29–68.CrossRefGoogle Scholar
[8]Blank, M., Keller, G. and Liverani, C.. Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15(6) (2001), 1905–1973.CrossRefGoogle Scholar
[9]Butterley, O. and Liverani, C.. Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 2 (2007), 301–322.CrossRefGoogle Scholar
[10]Buzzi, J.. Absolutely continuous invariant probability measures for arbitrary expanding piecewise ℝ-analytic mappings of the plane. Ergod. Th. & Dynam. Sys. 20(3) (2000), 697–708.CrossRefGoogle Scholar
[11]Buzzi, J.. No or infinitely many a.c.i.p. for piecewise expanding C r maps in higher dimensions. Comm. Math. Phys. 222(3) (2001), 495–501.CrossRefGoogle Scholar
[12]Buzzi, J.. Absolutely continuous invariant measures for generic multi-dimensional piecewise affine expanding maps. Discrete dynamical systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9(9) (1999), 1743–1750.CrossRefGoogle Scholar
[13]Cowieson, W. J.. Stochastic stability for piecewise expanding maps in R d. Nonlinearity 13(5) (2000), 1745–1760.CrossRefGoogle Scholar
[14]Cowieson, W. J.. Absolutely continuous invariant measures for most piecewise smooth expanding maps. Ergod. Th. & Dynam. Sys. 22(4) (2002), 1061–1078.CrossRefGoogle Scholar
[15]Demers, M. and Liverani, C.. Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360 (2008), 4777–4814.CrossRefGoogle Scholar
[16]Evans, L. C. and Gariepy, R. F.. Measure Theory and Fine Properties of Functions (Studies in Advanced Mathematics). CRC Press, Boca Raton, FL, 1992, viii+268 pp.Google Scholar
[17]Góra, P. and Boyarsky, A.. Absolutely continuous invariant measures for piecewise expanding C 2 transformation in R N. Israel J. Math. 67(3) (1989), 272–286.CrossRefGoogle Scholar
[18]Gouëzel, S.. Almost sure invariance principle for dynamical systems by spectral methods. Ann. Probab. 38(4) (2010), 1639–1671.CrossRefGoogle Scholar
[19]Gouëzel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26(1) (2006), 189–217.CrossRefGoogle Scholar
[20]Gouëzel, S. and Liverani, C.. Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differential Geom. 79 (2008), 433–477.CrossRefGoogle Scholar
[21]Hennion, H.. Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118(2) (1993), 627–634.Google Scholar
[22]Keller, G.. Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d’une région bornée du plan. C. R. Acad. Sci. Paris Sér. A–B 289(12) (1979), A625–A627.Google Scholar
[23]Keller, G.. Mixing for finite systems of coupled tent maps. Tr. Mat. Inst. Steklova 216 (1997), 320–326 Din. Sist. i Smezhnye Vopr., translation in Proc. Steklov Inst. Math. 216(1) (1997), 315–332.Google Scholar
[24]Keller, G. and Liverani, C.. 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 Physics, 671). Eds. Chazottes, J. and Fernandez, B.. Springer, Berlin, 2005, pp. 115–151.CrossRefGoogle Scholar
[25]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
[26]Keller, G. and Liverani, C.. Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135(3) (2009), 519–534.CrossRefGoogle Scholar
[27]Liverani, C.. Decay of correlations. Ann. of Math. (2) 142 (1995), 239–301.CrossRefGoogle Scholar
[28]Liverani, C.. Central limit theorem for deterministic systems. International Conference on Dynamical Systems (Montevideo, 1995) (Pitman Research Notes in Mathematics Series, 362). Longman, Harlow, 1996, pp. 56–75.Google Scholar
[29]Liverani, C.. On contact Anosov flows. Ann. of Math. (2) 159(3) (2004), 1275–1312.CrossRefGoogle Scholar
[30]Liverani, C.. Fredholm determinants, Anosov maps and Ruelle resonances. Discrete Contin. Dyn. Syst. 13(5) (2005), 1203–1215.CrossRefGoogle Scholar
[31]Liverani, C. and Tsujii, M.. Zeta functions and dynamical systems. Nonlinearity 19(10) (2006), 2467–2473.CrossRefGoogle Scholar
[32]Murray, R.. Existence, mixing and approximation of invariant densities for expanding maps on ℝr. Nonlinear Anal. 45(1) (2001), 37–72 Ser. A: Theory Methods.CrossRefGoogle Scholar
[33]Proppe, H., Góra, P. and Boyarsky, A.. Inadequacy of the bounded variation technique in the ergodic theory of higher-dimensional transformations. Nonlinearity 3(4) (1990), 1081–1087.CrossRefGoogle Scholar
[34]Rychlik, M.. Bounded variation and invariant measures. Studia Math. 76(1) (1983), 69–80.CrossRefGoogle Scholar
[35]Saussol, B.. Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math. 116 (2000), 223–248.CrossRefGoogle Scholar
[36]Thomine, D.. A spectral gap for transfer operators of piecewise expanding maps. Discrete Contin. Dyn. Syst. 30(3) (2011), 917–944.CrossRefGoogle Scholar
[37]Tsujii, M.. Piecewise expanding maps on the plane with singular ergodic properties. Ergod. Th. & Dynam. Sys. 20(6) (2000), 1851–1857.CrossRefGoogle Scholar
[38]Tsujii, M.. Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143(2) (2001), 349–373.CrossRefGoogle Scholar