# American Institute of Mathematical Sciences

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, 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, Singapore, 2000. doi: 10.1142/9789812813633.  Google Scholar [3] V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations, Algebraic and Topological Dynamics, 123-135, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, (2005). 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, Anal. Non. Lin., 26 (2009), 1453-1481. 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), 91-137. 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), 689-773. 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), 127-154.  Google Scholar [8] V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, in "Probabilistic and Geometric Structures in Dynamics" (eds. K. Burns, D. Dolgopyat and Ya. Pesin ), Contemp. Math., Amer. Math. Soc., 469 (2008), 29-68. 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), 1905-1973. 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), 3355-3363. 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), 1061-1078. 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), 4777-4814. 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), 175-191. doi: 10.1017/S0143385702000962.  Google Scholar [15] S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-218. 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), 433-477.  Google Scholar [17] M. Gromov, "Metric Structures for Riemannian and Non-Riemannian Spaces," 152 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1999. With Appendices by M. Katz, P. Pansu and S. Semmes.  Google Scholar [18] G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Probability Theory and Related Fields, 69 (1985), 461-478. 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), 141-152.  Google Scholar [20] C. Liverani, Invariant measures and their properties. A functional analytic point of view, Dynamical Systems. Part II, 185-237, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, (2003).  Google Scholar [21] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182. doi: 10.1017/S0143385711000939.  Google Scholar [22] C. Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312. 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), 223-248. doi: 10.1007/BF02773219.  Google Scholar [24] M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80.  Google Scholar [25] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917-944. doi: 10.3934/dcds.2011.30.917.  Google Scholar [26] T. Masato, Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, 23 (2010), 1495-1545. 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), 99-124. 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, Singapore, 2000. doi: 10.1142/9789812813633.  Google Scholar [3] V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations, Algebraic and Topological Dynamics, 123-135, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, (2005). 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, Anal. Non. Lin., 26 (2009), 1453-1481. 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), 91-137. 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), 689-773. 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), 127-154.  Google Scholar [8] V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, in "Probabilistic and Geometric Structures in Dynamics" (eds. K. Burns, D. Dolgopyat and Ya. Pesin ), Contemp. Math., Amer. Math. Soc., 469 (2008), 29-68. 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), 1905-1973. 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), 3355-3363. 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), 1061-1078. 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), 4777-4814. 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), 175-191. doi: 10.1017/S0143385702000962.  Google Scholar [15] S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-218. 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), 433-477.  Google Scholar [17] M. Gromov, "Metric Structures for Riemannian and Non-Riemannian Spaces," 152 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1999. With Appendices by M. Katz, P. Pansu and S. Semmes.  Google Scholar [18] G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Probability Theory and Related Fields, 69 (1985), 461-478. 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), 141-152.  Google Scholar [20] C. Liverani, Invariant measures and their properties. A functional analytic point of view, Dynamical Systems. Part II, 185-237, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, (2003).  Google Scholar [21] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182. doi: 10.1017/S0143385711000939.  Google Scholar [22] C. Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312. 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), 223-248. doi: 10.1007/BF02773219.  Google Scholar [24] M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80.  Google Scholar [25] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917-944. doi: 10.3934/dcds.2011.30.917.  Google Scholar [26] T. Masato, Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, 23 (2010), 1495-1545. 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), 99-124. doi: 10.1007/s00526-010-0335-1.  Google Scholar
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