Dynamical zeta functions and dynamical determinants ...

Dynamical zeta functions and dynamical determinants for hyperbolic maps – a functional approach –

Viviane Baladi Abstract. In this series of four 90 minutes lectures given at the University of Porto, June 14-17 2010, we present some techniques introduced in the past 5 years to study the spectrum of transfer operators associated to differentiable hyperbolic dynamical systems, in particular through dynamical zeta functions or dynamical determinants (which are power series constructed from the periodic orbit data and some weights). The relevance of this spectral data for the statistical analysis of dynamics is briefly discussed in the first lecture. The other lectures will be of a functional analytic nature, as the main step consists in defining a suitable Banach space on which the transfer operator acts. We will describe some anisotropic Sobolev spaces which turn out to be appropriate. To make the course accessible to students, we will start by revisiting the case of smooth uniformly expanding dynamics, where “ordinary” (isotropic) Sobolev spaces can be used.

References This bibliography is not complete, in particular does not cover the analytic setting or the continuoustime case. Surveys and books [Ba1] [BBB] [BB] [BT0]

[Bo] [Cv] [DS] [E]

V. Baladi, Periodic orbits and dynamical spectra, Cf www.math.jussieu.fr/∼baladi/etds.ps, Ergodic Theory Dynamical Systems 18 (1998), 255–292. V. Baladi, Positive transfer operators and the decay of correlations, vol. 16, World Scientific (Advanced Series in Nonlinear Dynamics), Singapore, 2000. V. Baladi, Decay of correlations, 1999 AMS Summer Institute on Smooth ergodic theory and applications, Seattle,, AMS (Proc. Symposia in Pure Math. Vol. 69), 2001, pp. 297-325. V. Baladi and M. Tsujii, Spectra of differentiable hyperbolic maps, (see also more detailed preliminary version on www.dma.ens.fr/∼baladi/ecole.html), Traces in number theory, geometry and quantum fields (S. Albeverio, M. Marcolli, S. Paycha, ed.), Proceedings MPIM Bonn 2005, Vieweg Verlag, Aspects of Mathematics E38, 2008, pp. 1–21. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer (Lecture Notes in Math., Vol. 470), Berlin, 1975. P. Cvitanovi´ c, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quantum http://ChaosBook.org, Niels Bohr Institute, Copenhagen, 2009. N. Dunford and J.T. Schwartz, Linear Operators, Part I, General Theory, Wiley-Interscience (Wiley Classics Library), New York, 1988. J.-P. Eckmann, Resonances in dynamical systems, IXth International Congress on Mathematical Physics (Swansea, 1988), Hilger, Bristol, 1989, pp. 192–207. Typeset by AMS-TEX

1

[ER] [Ka] [KH]

[Li00] [Li0]

[PP1] [Pie] [Ru1] [Ru2] [Ta] [Wa] [Z]

J.-P. Eckmann, and D. Ruelle, Ergodic theory of chaos and strange attractors., Rev. Modern Phys. 57 (1985), 617–656. T. Kato, Perturbation Theory for Linear Operators, Second Corrected Printing of the Second Edition, Springer-Verlag, Berlin, 1984. A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems., Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. C. Liverani, Computing the rate of decay of correlations in expanding and hyperbolic systems, Markov Processes and Related Fields 8 (2002), 155-162. C. Liverani, Invariant measures and their properties. A functional analytic point of view, Dynamical systems. Part II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, pp. 185–237. W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Soci´ et´ e Math´ ematique de France (Ast´ erisque, vol. 187-188), Paris, 1990. A. Pietsch, Eigenvalues and s-numbers, Cambridge University Press, Cambridge, 1987. D. Ruelle, Resonances of chaotic dynamical systems, Phys. Rev. Lett 56 (1986), 405–407. D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc 49 (2002), 887–895. M.E. Taylor, Pseudodifferential operators and nonlinear PDE, Birkh¨ auser, Boston, 1991. P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79, SpringerVerlag, New York-Berlin, 1982. M. Zinsmeister, Thermodynamic formalism and holomorphic dynamical systems, Translated from the 1996 French original by C. Greg Anderson. SMF/AMS Texts and Monographs, 2., American Mathematical Society, Providence, RI, 2000.

The Gauss map [BVa] [Ma] [Ma2] [Ma3]

V. Baladi and B. Vall´ ee, Euclidean algorithms are Gaussian, J. Number Theory 110 (2005), 331–386. D. Mayer, On a ζ-function related to the continued fraction transformation, Bull. Soc. Math. France 104 (1976), 195–203. D.H. Mayer, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys. 130 (1990), 311–333. D.H. Mayer, The thermodynamic formalism approach to Selberg’s zeta function for P SL(2, Z), Bull. Amer. Math. Soc. (N.S.) 25 (1991), 55–60.

Differentiable framework: smooth (locally) expanding maps [BB] [CE] [CI] [Fr] [GL]

[KR] [Ru3]

V. Baladi et M. Baillif, Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case, Ergodic Theory Dynam. Systems 25 (2005), 1437–1470. P. Collet, and J.-P. Eckmann, Liapunov Multipliers and Decay of Correlations in Dynamical Systems, J. Statist. Phys. 115 (2004), 217–254. P. Collet and S. Isola, On the essential spectrum of the transfer operator for expanding Markov maps, Comm. Math. Phys. 139 (1991), 551–557. D. Fried, The flat-trace asymptotics of a uniform system of contractions, Ergodic Theory Dynamical Systems 15 (1995), 1061–1073. V.M. Gundlach and Y. Latushkin, A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Holder spaces, Ergodic Theory Dynam. Systems 23 (2003), 175–191. G. Keller and H.H. Rugh, Eigenfunctions for smooth expanding circle maps, Nonlinearity 17 (2004), 1723–1730. D. Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys 125 (1989), 239–262.

2

[Ru4]

D. Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes Etudes Sci. Publ. Math. 72 (1991), 175–193.

Differentiable framework: smooth hyperbolic maps (anisotropic approach) [B00]

[BT1] [BT2]

[BKL] [FRS] [GL1] [GL2] [Ki] [Li] [LT]

V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: C ∞ foliations, Algebraic and Topological Dynamics (S. Kolyada, Y. Manin & T. Ward, ed.), Contemporary Mathematics, Amer. Math. Society, 2005, pp. 123–136. V. Baladi and M. Tsujii, Anisotropic H¨ older and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier 57 (2007), 127–154. V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, Probabilistic and Geometric Structures in Dynamics (K. Burns, D. Dolgopyat and Ya. Pesin, ed.), Volume in honour of M. Brin’s 60th birthday, Amer. Math. Soc. (Contemp. Math. 469), 2008, pp. 29–68. M. Blank, G. Keller, and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity 15 (2002), 1905–1973. F. Faure, N. Roy, J. Sj¨ ostrand, Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances, The Open Mathematics Journal 1 (2008), 35-81. S. Gou¨ ezel and C. Liverani, Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties, J. Diff. Geom. 79 (2008), 433-477. S. Gou¨ ezel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory and Dynamical Systems 26 (2006), 189–217. A. Kitaev, Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness, (see also Corrigendum, 1717–1719), Nonlinearity 12 (1999), 141–179. C. Liverani, Fredholm determinants, Anosov maps and Ruelle resonances, Discrete Contin. Dyn. Syst. 13 (2005), 1203–1215. C. Liverani and M. Tsujii, Zeta functions and Dynamical Systems, Nonlinearity 19 (2006), 2467–2473.

Piecewise differentiable framework: analysis)

piecewise expanding maps (without harmonic

[BaKe] V. Baladi and G. Keller, Zeta functions and transfer operators for piecewise monotone transformations, Comm. Math. Phys. 127 (1990), 459–479. [BuKe] J. Buzzi and G. Keller, Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps, Ergodic Theory Dynamical Systems 21 (2001), 689–716. [RRu] D. Ruelle, Dynamical zeta functions for piecewise monotone maps of the interval., CRM Monograph Series, 4., American Mathematical Society, Providence, RI, 1994. Piecewise differentiable framework: piecewise hyperbolic maps (anisotropic approach) [BG1] [BG2] [DL]

V. Baladi and S. Gou¨ ezel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Annales de l’Institut Henri Poincar´ e / Analyse non lin´ eaire 26 (2009), 1453-1481. V. Baladi and S. Gou¨ ezel, Banach spaces for piecewise cone hyperbolic maps, J. Modern Dyn. 4 (2010), 91–137. M.F. Demers and C. Liverani, Stability of Statistical Properties in Two-dimensional Piecewise Hyperbolic Maps, Transactions of the American Mathematical Society 360 (2008), 4777–4814.

3