August  2013, 33(8): 3741-3751. doi: 10.3934/dcds.2013.33.3741

A footnote on expanding maps

1. 

Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma

Received  June 2012 Revised  October 2012 Published  January 2013

I introduce Banach spaces on which it is possible to precisely characterize the spectrum of the transfer operator associated to a piecewise expanding map with Hölder weight.
Citation: Carlangelo Liverani. A footnote on expanding maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3741-3751. doi: 10.3934/dcds.2013.33.3741
References:
[1]

V. Araujo, S. Galatolo and M.-J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors,, Preprint , ().   Google Scholar

[2]

V. Baladi, "Positive Transfer Operators & Decay of Correlation,", 16 of Advanced Series in Nonlinear Dynamics. World Scientific, 16 (2000).  doi: 10.1142/9789812813633.  Google Scholar

[3]

V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations,, Algebraic and Topological Dynamics, 385 (2005), 123.  doi: 10.1090/conm/385/07194.  Google Scholar

[4]

V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation,, Ann. Inst. Henri Poincaré, 26 (2009), 1453.  doi: 10.1016/j.anihpc.2009.01.001.  Google Scholar

[5]

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

[6]

V. Baladi and C. Liverani, Exponential decay of correlations for piecewise cone hyperbolic contact flows,, Communications in Mathematical Physics, 314 (2012), 689.  doi: 10.1007/s00220-012-1538-4.  Google Scholar

[7]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms,, Annales de L'Institut Fourier, 57 (2007), 127.   Google Scholar

[8]

V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms,, in, 469 (2008), 29.  doi: 10.1090/conm/469/09160.  Google Scholar

[9]

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

[10]

O. Butterley, An alternative approach to generalised BV and the application to expanding interval maps,, Discrete Contin. Dyn. Syst., 33 (2013), 3355.   Google Scholar

[11]

, O. Butterey,, Private Communication., ().   Google Scholar

[12]

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

[13]

M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps,, Trans. Amer. Math. Soc., 360 (2008), 4777.  doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar

[14]

V. M. Gundlach and Y. Latushkin, A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces,, Ergodic Theory Dynam. Systems, 23 (2003), 175.  doi: 10.1017/S0143385702000962.  Google Scholar

[15]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergodic Theory Dynam. Systems, 26 (2006), 189.  doi: 10.1017/S0143385705000374.  Google Scholar

[16]

S. Gouëzel and C. Liverani, Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties,, J. Diff. Geom., 79 (2008), 433.   Google Scholar

[17]

M. Gromov, "Metric Structures for Riemannian and Non-Riemannian Spaces,", 152 of Progress in Mathematics, 152 (1999).   Google Scholar

[18]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations,, Probability Theory and Related Fields, 69 (1985), 461.  doi: 10.1007/BF00532744.  Google Scholar

[19]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141.   Google Scholar

[20]

C. Liverani, Invariant measures and their properties. A functional analytic point of view,, Dynamical Systems. Part II, (2003), 185.   Google Scholar

[21]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach,, Ergodic Theory and Dynamical Systems, 33 (2013), 168.  doi: 10.1017/S0143385711000939.  Google Scholar

[22]

C. Liverani, On contact Anosov flows,, Ann. of Math. (2), 159 (2004), 1275.  doi: 10.4007/annals.2004.159.1275.  Google Scholar

[23]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223.  doi: 10.1007/BF02773219.  Google Scholar

[24]

M. Rychlik, Bounded variation and invariant measures,, Studia Math., 76 (1983), 69.   Google Scholar

[25]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps,, Discrete Contin. Dyn. Syst., 30 (2011), 917.  doi: 10.3934/dcds.2011.30.917.  Google Scholar

[26]

T. Masato, Quasi-compactness of transfer operators for contact Anosov flows,, Nonlinearity, 23 (2010), 1495.  doi: 10.1088/0951-7715/23/7/001.  Google Scholar

[27]

Z. Roger, Integration of Hölder forms and currents in snowflake spaces,, Calc. Var. Partial Differential Equations, 40 (2011), 99.  doi: 10.1007/s00526-010-0335-1.  Google Scholar

show all references

References:
[1]

V. Araujo, S. Galatolo and M.-J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors,, Preprint , ().   Google Scholar

[2]

V. Baladi, "Positive Transfer Operators & Decay of Correlation,", 16 of Advanced Series in Nonlinear Dynamics. World Scientific, 16 (2000).  doi: 10.1142/9789812813633.  Google Scholar

[3]

V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations,, Algebraic and Topological Dynamics, 385 (2005), 123.  doi: 10.1090/conm/385/07194.  Google Scholar

[4]

V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation,, Ann. Inst. Henri Poincaré, 26 (2009), 1453.  doi: 10.1016/j.anihpc.2009.01.001.  Google Scholar

[5]

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

[6]

V. Baladi and C. Liverani, Exponential decay of correlations for piecewise cone hyperbolic contact flows,, Communications in Mathematical Physics, 314 (2012), 689.  doi: 10.1007/s00220-012-1538-4.  Google Scholar

[7]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms,, Annales de L'Institut Fourier, 57 (2007), 127.   Google Scholar

[8]

V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms,, in, 469 (2008), 29.  doi: 10.1090/conm/469/09160.  Google Scholar

[9]

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

[10]

O. Butterley, An alternative approach to generalised BV and the application to expanding interval maps,, Discrete Contin. Dyn. Syst., 33 (2013), 3355.   Google Scholar

[11]

, O. Butterey,, Private Communication., ().   Google Scholar

[12]

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

[13]

M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps,, Trans. Amer. Math. Soc., 360 (2008), 4777.  doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar

[14]

V. M. Gundlach and Y. Latushkin, A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces,, Ergodic Theory Dynam. Systems, 23 (2003), 175.  doi: 10.1017/S0143385702000962.  Google Scholar

[15]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergodic Theory Dynam. Systems, 26 (2006), 189.  doi: 10.1017/S0143385705000374.  Google Scholar

[16]

S. Gouëzel and C. Liverani, Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties,, J. Diff. Geom., 79 (2008), 433.   Google Scholar

[17]

M. Gromov, "Metric Structures for Riemannian and Non-Riemannian Spaces,", 152 of Progress in Mathematics, 152 (1999).   Google Scholar

[18]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations,, Probability Theory and Related Fields, 69 (1985), 461.  doi: 10.1007/BF00532744.  Google Scholar

[19]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141.   Google Scholar

[20]

C. Liverani, Invariant measures and their properties. A functional analytic point of view,, Dynamical Systems. Part II, (2003), 185.   Google Scholar

[21]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach,, Ergodic Theory and Dynamical Systems, 33 (2013), 168.  doi: 10.1017/S0143385711000939.  Google Scholar

[22]

C. Liverani, On contact Anosov flows,, Ann. of Math. (2), 159 (2004), 1275.  doi: 10.4007/annals.2004.159.1275.  Google Scholar

[23]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223.  doi: 10.1007/BF02773219.  Google Scholar

[24]

M. Rychlik, Bounded variation and invariant measures,, Studia Math., 76 (1983), 69.   Google Scholar

[25]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps,, Discrete Contin. Dyn. Syst., 30 (2011), 917.  doi: 10.3934/dcds.2011.30.917.  Google Scholar

[26]

T. Masato, Quasi-compactness of transfer operators for contact Anosov flows,, Nonlinearity, 23 (2010), 1495.  doi: 10.1088/0951-7715/23/7/001.  Google Scholar

[27]

Z. Roger, Integration of Hölder forms and currents in snowflake spaces,, Calc. Var. Partial Differential Equations, 40 (2011), 99.  doi: 10.1007/s00526-010-0335-1.  Google Scholar

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