Anosov flows and dynamical zeta functions

Abstract

We study the Ruelle and Selberg zeta functions for $C^r$ Anosov flows, $r > 2$, on a compact smooth manifold. We prove several results, the most remarkable being (a) for $C^\infty$ flows the zeta function is meromorphic on the entire complex plane; (b) for contact flows satisfying a bunching condition (e.g., geodesic flows on manifolds of negative curvature better than $\frac 19$-pinched), the zeta function has a pole at the topological entropy and is analytic in a strip to its left; (c) under the same hypotheses as in (b) we obtain sharp results on the number of periodic orbits. Our arguments are based on the study of the spectral properties of a transfer operator acting on suitable Banach spaces of anisotropic currents.

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    @article {GouezelLiverani06, MRKEY = {2201945},
      AUTHOR = {Gou{ë}zel, S{é}bastien and Liverani, Carlangelo},
      TITLE = {Banach spaces adapted to {A}nosov systems},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {26},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {189--217},
      ISSN = {0143-3857},
      MRCLASS = {37D20 (37C30 46F05 47B38)},
      MRNUMBER = {2201945},
      MRREVIEWER = {Masato Tsujii},
      DOI = {10.1017/S0143385705000374},
      ZBLNUMBER = {1088.37010},
      }
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    @article {GouezelLiverani08, MRKEY = {2433929},
      AUTHOR = {Gou{ë}zel, S{é}bastien and Liverani, Carlangelo},
      TITLE = {Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {79},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {433--477},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {37D20 (37C30)},
      MRNUMBER = {2433929},
      MRREVIEWER = {Nikolai Chernov},
      URL = {http://projecteuclid.org/euclid.jdg/1213798184},
      ZBLNUMBER = {1166.37010},
      }
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    @article {Hasselblatt94, MRKEY = {1258914},
      AUTHOR = {Hasselblatt, Boris},
      TITLE = {Horospheric foliations and relative pinching},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {39},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {57--63},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {58F17 (53C20 58F15 58F18)},
      MRNUMBER = {1258914},
      MRREVIEWER = {Garrett Stuck},
      URL = {http://projecteuclid.org/euclid.jdg/1214454676},
      ZBLNUMBER = {0795.53026},
      }
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    @article {Has1, MRKEY = {1304137},
      AUTHOR = {Hasselblatt, Boris},
      TITLE = {Regularity of the {A}nosov splitting and of horospheric foliations},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {14},
      YEAR = {1994},
      NUMBER = {4},
      PAGES = {645--666},
      ISSN = {0143-3857},
      MRCLASS = {58F15 (58F17 58F18)},
      MRNUMBER = {1304137},
      MRREVIEWER = {Edoh Amiran},
      DOI = {10.1017/S0143385700008105},
      ZBLNUMBER = {0821.58032},
      }
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    @article {Hasselblatt97, MRKEY = {1440773},
      AUTHOR = {Hasselblatt, Boris},
      TITLE = {Regularity of the {A}nosov splitting. {II}},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {17},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {169--172},
      ISSN = {0143-3857},
      MRCLASS = {58F15 (58F17)},
      MRNUMBER = {1440773},
      MRREVIEWER = {Edoh Amiran},
      DOI = {10.1017/S0143385797069757},
      ZBLNUMBER = {0872.58045},
      }
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    @article {Hestenes41, MRKEY = {0003434},
      AUTHOR = {Hestenes, M. R.},
      TITLE = {Extension of the range of a differentiable function},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {8},
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      PAGES = {183--192},
      ISSN = {0012-7094},
      MRCLASS = {27.0X},
      MRNUMBER = {0003434},
      MRREVIEWER = {W. T. Reid},
      DOI = {10.1215/S0012-7094-41-00812-8},
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      SERIES = {Universitext},
      EDITION = {Fifth},
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      ADDRESS = {New York},
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      ISBN = {978-3-540-77340-5},
      MRCLASS = {53Cxx (53-02 58E05 58E10 58J05)},
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      }
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      AUTHOR = {Kato, Tosio},
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      SERIES = {Classics in Math.},
      NOTE = {reprint of the 1980 edition},
      PUBLISHER = {Springer-Verlag},
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      YEAR = {1995},
      PAGES = {xxii+619},
      ISBN = {3-540-58661-X},
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      AUTHOR = {Katok, Anatole},
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      NOTE = {with the collaboration of Keith Burns},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
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      }
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      AUTHOR = {Katok, Anatole and Hasselblatt, Boris},
      TITLE = {Introduction to the Modern Theory of Dynamical Systems},
      SERIES = {Encyclopedia Math. Appl.},
      VOLUME = {54},
      PUBLISHER = {Cambridge Univ. Press},
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      YEAR = {1995},
      PAGES = {xviii+802},
      ISBN = {0-521-34187-6},
      MRCLASS = {58Fxx (34Cxx 34Dxx 58-01 58F11 58F15)},
      MRNUMBER = {1326374},
      MRREVIEWER = {Edoh Amiran},
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      AUTHOR = {Kitaev, A. Yu.},
      TITLE = {Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness},
      JOURNAL = {Nonlinearity},
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      VOLUME = {12},
      YEAR = {1999},
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      PAGES = {141--179},
      ISSN = {0951-7715},
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      MRCLASS = {37D20 (37C30)},
      MRNUMBER = {1668543},
      MRREVIEWER = {Hans Henrik Rugh},
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      AUTHOR = {Lichtenstein, Leon},
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      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
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      YEAR = {1929},
      NUMBER = {1},
      PAGES = {794--795},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
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      MRNUMBER = {1545093},
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      AUTHOR = {Liverani, Carlangelo},
      TITLE = {Decay of correlations},
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      VOLUME = {142},
      YEAR = {1995},
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      CODEN = {ANMAAH},
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      MRNUMBER = {1343323},
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    @article {Liverani04, MRKEY = {2113022},
      AUTHOR = {Liverani, Carlangelo},
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      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {159},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {1275--1312},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37D20 (37A99 37D40 37J60)},
      MRNUMBER = {2113022},
      MRREVIEWER = {M. L. Blank},
      DOI = {10.4007/annals.2004.159.1275},
      ZBLNUMBER = {1067.37031},
      }
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    @article {Liverani05, MRKEY = {2166265},
      AUTHOR = {Liverani, Carlangelo},
      TITLE = {Fredholm determinants, {A}nosov maps and {R}uelle resonances},
      JOURNAL = {Discrete Contin. Dyn. Syst.},
      FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A},
      VOLUME = {13},
      YEAR = {2005},
      NUMBER = {5},
      PAGES = {1203--1215},
      ISSN = {1078-0947},
      MRCLASS = {37C30 (37D20)},
      MRNUMBER = {2166265},
      MRREVIEWER = {J{é}r{ô}me Buzzi},
      DOI = {10.3934/dcds.2005.13.1203},
      ZBLNUMBER = {1160.37330},
      }
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    @article {LiveraniTsujii06, MRKEY = {2260273},
      AUTHOR = {Liverani, Carlangelo and Tsujii, Masato},
      TITLE = {Zeta functions and dynamical systems},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {19},
      YEAR = {2006},
      NUMBER = {10},
      PAGES = {2467--2473},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {37D20 (37C30)},
      MRNUMBER = {2260273},
      MRREVIEWER = {Giacomo Monti Bragadin},
      DOI = {10.1088/0951-7715/19/10/011},
      ZBLNUMBER = {1190.37025},
      }
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    @article {LW3, MRKEY = {1313459},
      AUTHOR = {Liverani, Carlangelo and Wojtkowski, Maciej P.},
      TITLE = {Generalization of the {H}ilbert metric to the space of positive definite matrices},
      JOURNAL = {Pacific J. Math.},
      FJOURNAL = {Pacific Journal of Mathematics},
      VOLUME = {166},
      YEAR = {1994},
      NUMBER = {2},
      PAGES = {339--355},
      ISSN = {0030-8730},
      CODEN = {PJMAAI},
      MRCLASS = {53C60 (15A48)},
      MRNUMBER = {1313459},
      MRREVIEWER = {Sorin Dragomir},
      DOI = {10.2140/pjm.1994.166.339},
      ZBLNUMBER = {0824.53019},
      }
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      AUTHOR = {Liverani, Carlangelo and Wojtkowski, Maciej P.},
      TITLE = {Ergodicity in {H}amiltonian systems},
      BOOKTITLE = {Dynamics Reported},
      SERIES = {Dynam. Report. Expositions Dynam. Systems},
      VOLUME = {4},
      PAGES = {130--202},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1995},
      MRCLASS = {58F15 (58F05 58F11 70H05)},
      MRNUMBER = {1346498},
      MRREVIEWER = {Nikolai Chernov},
      ZBLNUMBER = {0824.58033},
      DOI = {10.1007/978-3-642-61215-2_3},
     }
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      AUTHOR = {Margulis, Grigoriy A.},
      TITLE = {On Some Aspects of the Theory of {A}nosov Systems},
      SERIES = {Springer Monogr. Math.},
      NOTE = {with a survey by Richard Sharp: Periodic orbits of hyperbolic flows; translated from the Russian by Valentina Vladimirovna Szulikowska},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2004},
      PAGES = {vi+139},
      ISBN = {3-540-40121-0},
      MRCLASS = {37D20 (37C27 37C30 37C40)},
      MRNUMBER = {2035655},
      MRREVIEWER = {Boris Hasselblatt},
      ZBLNUMBER = {1140.37010},
      DOI = {10.1007/978-3-662-09070-1},
     }
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      AUTHOR = {Marklof, Jens},
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      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {397},
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      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2011},
      EDITOR = {J. Bolte and F. Steiner},
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      {II}},
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      PUBLISHER = {Math. Soc. Japan},
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      JOURNAL = {Astérisque},
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      AUTHOR = {Pilyugin, Sergei Yu.},
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      MRCLASS = {37C50 (37C05 37C10 37C20 37D05 37L30)},
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      JOURNAL = {Ergodic Theory Dynam. Systems},
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      MRCLASS = {58F11 (28D99 58F20)},
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      MRREVIEWER = {Radu Nicolae Gologan},
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    @article {Pollicott85, MRKEY = {0807065},
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      TITLE = {On the rate of mixing of {A}xiom {A} flows},
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      ISSN = {0020-9910},
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      MRNUMBER = {0807065},
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      }
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      AUTHOR = {Pollicott, Mark},
      TITLE = {Analytic extensions of the zeta functions for surfaces of variable negative curvature},
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      FJOURNAL = {Journal of Differential Geometry},
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Authors

Paolo Giulietti

Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

Carlangelo Liverani

Università di Roma Tor Vergata 00133 Roma, Italy

Mark Pollicott

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom