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August  2011, 30(3): 917-944. doi: 10.3934/dcds.2011.30.917

A spectral gap for transfer operators of piecewise expanding maps

Damien Thomine 1,

1. 

45 rue d'Ulm, 75005 Paris, France

Received  May 2010 Revised  November 2010 Published  March 2011

We consider piecewise $\C^{1+\alpha}$ uniformly expanding maps on a Riemannian manifold, and study their invariant physical measures. We study the Perron-Frobenius operator on Sobolev spaces and bounded variation spaces, and prove that it is quasicompact if some conditions on the Lyapunov exponent and the combinatorial complexities are satisfied. Then, we get strong results concerning the existence of physical ergodic measures, and the exponential mixing of smooth observables.
Keywords: Ergodic theory, mixing rate., expanding maps, Perron-Frobenius operators.
Mathematics Subject Classification: Primary: 37A25; Secondary: 47A3.
Citation: Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
References:
[1]

V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ Foliations, in "Algebraic and Topological Dynamics" (eds. Sergiy Kolyada, Yuri Manin and Thomas Ward), Contemporary Mathematics (2005), 123-136.  Google Scholar

[2]

V. Baladi, "Positive Transfer Operators and Decay Of Correlations," World scientific, 2000. doi: 10.1142/9789812813633.  Google Scholar

[3]

V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Annales de l'Institut Henri Poincaré, Analyse non linéaire, 26 (2009), 1453-1481.  Google Scholar

[4]

V. Baladi and S. Gouëzel, Banach spaces for piecewise cone hyperbolic maps, Journal of Modern Dynamics, 4 (2010), 91-137. doi: 10.3934/jmd.2010.4.91.  Google Scholar

[5]

M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973. doi: 10.1088/0951-7715/15/6/309.  Google Scholar

[6]

J. Buzzi, Intrisic ergodicity of affine maps in $[0,1]^d$, Monatshefte für Mathematik, 124 (1997), 97-118.  Google Scholar

[7]

J. Buzzi, No or infinitely many A.C.I.P. for piecewise expanding $C^r$ maps in higher dimensions, Communications in Mathematical Physics, 222 (2001), 495-501. doi: 10.1007/s002200100509.  Google Scholar

[8]

W. J. Cowieson, Stochastic stability for piecewise expanding maps in $\R^d$, Nonlinearity, 13 (2000), 1745-1760. doi: 10.1088/0951-7715/13/5/316.  Google Scholar

[9]

W. J. Cowieson, Absolutely continuous invariant measures for most piecewise smooth expanding maps, Ergodic Theory and Dynamical Systems, 22 (2002), 1061-1078. doi: 10.1017/S0143385702000627.  Google Scholar

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Birkhaüser, Boston, 1984.  Google Scholar

[11]

P. Góra and A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding $C^2$ transformations in $\R^n$, Israël Journal of Mathematics, 67 (1989), 272-290. doi: 10.1007/BF02764946.  Google Scholar

[12]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, (French) Proceedings of the American Mathematical Society, 118 (1993), 627-639.  Google Scholar

[13]

G. Keller, Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d'une région bornée du plan, (French) Comptes-rendus de l'Académie des Sciences de Paris, 289 (1979), 625-627.  Google Scholar

[14]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Zeitschrift für Wahrscheinlichkeitheorie und verwandte Geliete, 69 (1985), 461-478.  Google Scholar

[15]

G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps, Nonlinearity, 17 (2004), 1723-1730.  Google Scholar

[16]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of the American Mathematical Society, 186 (1973), 481-488.  Google Scholar

[17]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israël Journal of Mathematics, 116 (2000), 223-248.  Google Scholar

[18]

R. S. Strichartz, Multipliers on fractional Sobolev spaces, Journal of Mathematics and Mechanics, 16 (1967), 1031-1060.  Google Scholar

[19]

H. Triebel, General function spaces. III. (Spaces $B_{p,q}^{g(x)}$ and $F_{p,q}^{g(x)}$, $1: Basic properties), Analysis Mathematica, 3 (1977), 221-249.  Google Scholar

[20]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar

[21]

H. Triebel, "Theory of Function Spaces. II," Birkhäuser, Basel, 1992.  Google Scholar

[22]

M. Tsujii, Piecewise expanding maps on the plane with singular ergodic properties, Ergodic Theory and Dynamical Systems, 20 (2000), 1851-1857.  Google Scholar

show all references

References:
[1]

V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ Foliations, in "Algebraic and Topological Dynamics" (eds. Sergiy Kolyada, Yuri Manin and Thomas Ward), Contemporary Mathematics (2005), 123-136.  Google Scholar

[2]

V. Baladi, "Positive Transfer Operators and Decay Of Correlations," World scientific, 2000. doi: 10.1142/9789812813633.  Google Scholar

[3]

V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Annales de l'Institut Henri Poincaré, Analyse non linéaire, 26 (2009), 1453-1481.  Google Scholar

[4]

V. Baladi and S. Gouëzel, Banach spaces for piecewise cone hyperbolic maps, Journal of Modern Dynamics, 4 (2010), 91-137. doi: 10.3934/jmd.2010.4.91.  Google Scholar

[5]

M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973. doi: 10.1088/0951-7715/15/6/309.  Google Scholar

[6]

J. Buzzi, Intrisic ergodicity of affine maps in $[0,1]^d$, Monatshefte für Mathematik, 124 (1997), 97-118.  Google Scholar

[7]

J. Buzzi, No or infinitely many A.C.I.P. for piecewise expanding $C^r$ maps in higher dimensions, Communications in Mathematical Physics, 222 (2001), 495-501. doi: 10.1007/s002200100509.  Google Scholar

[8]

W. J. Cowieson, Stochastic stability for piecewise expanding maps in $\R^d$, Nonlinearity, 13 (2000), 1745-1760. doi: 10.1088/0951-7715/13/5/316.  Google Scholar

[9]

W. J. Cowieson, Absolutely continuous invariant measures for most piecewise smooth expanding maps, Ergodic Theory and Dynamical Systems, 22 (2002), 1061-1078. doi: 10.1017/S0143385702000627.  Google Scholar

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Birkhaüser, Boston, 1984.  Google Scholar

[11]

P. Góra and A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding $C^2$ transformations in $\R^n$, Israël Journal of Mathematics, 67 (1989), 272-290. doi: 10.1007/BF02764946.  Google Scholar

[12]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, (French) Proceedings of the American Mathematical Society, 118 (1993), 627-639.  Google Scholar

[13]

G. Keller, Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d'une région bornée du plan, (French) Comptes-rendus de l'Académie des Sciences de Paris, 289 (1979), 625-627.  Google Scholar

[14]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Zeitschrift für Wahrscheinlichkeitheorie und verwandte Geliete, 69 (1985), 461-478.  Google Scholar

[15]

G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps, Nonlinearity, 17 (2004), 1723-1730.  Google Scholar

[16]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of the American Mathematical Society, 186 (1973), 481-488.  Google Scholar

[17]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israël Journal of Mathematics, 116 (2000), 223-248.  Google Scholar

[18]

R. S. Strichartz, Multipliers on fractional Sobolev spaces, Journal of Mathematics and Mechanics, 16 (1967), 1031-1060.  Google Scholar

[19]

H. Triebel, General function spaces. III. (Spaces $B_{p,q}^{g(x)}$ and $F_{p,q}^{g(x)}$, $1: Basic properties), Analysis Mathematica, 3 (1977), 221-249.  Google Scholar

[20]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar

[21]

H. Triebel, "Theory of Function Spaces. II," Birkhäuser, Basel, 1992.  Google Scholar

[22]

M. Tsujii, Piecewise expanding maps on the plane with singular ergodic properties, Ergodic Theory and Dynamical Systems, 20 (2000), 1851-1857.  Google Scholar

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