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Absolutely continuous invariant measures for non-uniformly expanding maps

Published online by Cambridge University Press:  01 August 2009

HUYI HU  and
SANDRO VAIENTI
HUYI HU
Affiliation:
Mathematics Department, Michigan State University, East Lansing, MI 48824, USA (email: [email protected])
SANDRO VAIENTI
Affiliation:
UMR-6207 Centre de Physique Théorique, CNRS, Universités d’Aix-Marseille I, II, Université du Sud Toulon-Var and FRUMAM, Fédéderation de Recherche des Unitiés de Mathématiques de Marseille, France (email: [email protected])
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Abstract

For a large class of non-uniformly expanding maps of ℝm, with indifferent fixed points and unbounded distortion and that are non-necessarily Markovian, we construct an absolutely continuous invariant measure. We extend previously used techniques for expanding maps on quasi-Hölder spaces to our case. We give general conditions and provide examples to which our results apply.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1185 - 1215
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Adl-Zarabi, K.. Absolutely continuous invariant measures for piecewise expanding C 2 transformations in ℝn on domains with cusps on the boundaries. Ergod. Th. & Dynam. Sys. 16 (1996), 1–18.CrossRefGoogle Scholar
[2]Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction in mostly expanding. Invent. Math. 140 (2000), 351–398.CrossRefGoogle Scholar
[3]Blank, M.. Stochastic properties of deterministic dynamical systems. Sov. Sci. Rev. C Math. Phys. 6 (1987), 243–271.Google Scholar
[4]Boyarsky, A. and Gora, P.. Absolutely continuous invariant measures for piecewise expanding C 2 transformations in ℝn. Israel J. Math. 67 (1987), 272–286.Google Scholar
[5]Buzzi, J.. Absolutely continuous invariant probability measures for arbitrary expanding piecewise R-analytic mappings of the plane. Ergod. Th. & Dynam. Sys. 20 (2000), 697–708.CrossRefGoogle Scholar
[6]Buzzi, J. and Keller, G.. Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps. Ergod. Th. & Dynam. Sys. 21 (2001), 689–716.CrossRefGoogle Scholar
[7]Buzzi, J., Paccaut, F. and Schmitt, B.. Conformal measures for multidimensional piecewise invertible maps. Ergod. Th. & Dynam. Sys. 21 (2001), 1035–1049.CrossRefGoogle Scholar
[8]Cowienson, W. J.. Absolutely continuous invariant measures for most piecewise smooth expanding maps. Ergod. Th. & Dynam. Sys. 22 (2002), 1061–1078.Google Scholar
[9]Fan, A. and Jiang, Y.. On Ruelle–Perron–Frobenius operators. I. Ruelle theorem. Comm. Math. Phys. 223 (2001), 125–141.CrossRefGoogle Scholar
[10]Hu, H.. Equilibriums of some non-Hölder potentials. Trans. Amer. Math. Soc. 390 (2008), 2153–2190.Google Scholar
[11]Ionescu-Tulcea, C. and Marinescu, G.. Théorie ergodique pour des classes d’opérations non complétement continues. Ann. of Math. (2) 52 (1950), 140–147.CrossRefGoogle Scholar
[12]Keller, G.. Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahr. verw. Geb. 69 (1985), 461–478.CrossRefGoogle Scholar
[13]Lasota, A. and Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.CrossRefGoogle Scholar
[14]Poppe, H., Gora, P. and Boyarsky, A.. Inadequacy of the bounded variation technique in the ergodic theory of higher-dimensional transformations in ℝn. Israel J. Math. 3 (1990), 1081–1087.Google Scholar
[15]Pianigiani, G.. First return map and invariant measures. Israel J. Math. 35 (1980), 32–48.CrossRefGoogle Scholar
[16]Saussol, B.. Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math. 116 (2000), 223–248.CrossRefGoogle Scholar
[17]Shilov, G. and Gurevich, B.. Integral, Measure, and Derivative; a Unified Approach. Prentice-Hall, Englewood Cliffs, NJ, 1966.Google Scholar
[18]Thaler, M.. Estimates of the invariant densities of endomorphisms with indifferent periodic points. Israel J. Math. 37 (1980), 303–314.CrossRefGoogle Scholar
[19]Tsujii, M.. Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143 (2001), 349–373.CrossRefGoogle Scholar
[20]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1950), 585–650.CrossRefGoogle Scholar
[21]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar
[22]Yuri, M.. Invariant measures for certain multi-dimensional maps. Nonlinearity 7 (1994), 1093–1124.CrossRefGoogle Scholar
[23]Yuri, M.. On the speed of convergence to equilibrium states for multi-dimensional maps with indifferent periodic points. Nonlinearity 15 (2002), 429–445.CrossRefGoogle Scholar