The Ruelle spectrum of generic transfer operators

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July 2012, 32(7): 2521-2531. doi: 10.3934/dcds.2012.32.2521

The Ruelle spectrum of generic transfer operators

Frédéric Naud ^1,

1.	Université d’Avignon, Laboratoire d’Analyse non linéraire et Géométrie, 33, rue Louis Pasteur, 84000, France

Received December 2009 Revised July 2010 Published March 2012

Abstract
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We define a natural space of transfer operators related to holomorphic contraction systems. We show that the classical upper bounds on the Ruelle eigenvalue sequence are optimal for a dense set of transfer operators. A similar statement is derived for Perron-Frobenius operators related to uniformly expanding piecewise real analytic interval maps. The proof is based on potential theory.

Keywords: decay of correlation, Perron-Frobenius operators, potential theory., Ruelle eigenvalues, holomorphic function spaces, Transfer operators.

Mathematics Subject Classification: 37C30, 37D2.

Citation: Frédéric Naud. The Ruelle spectrum of generic transfer operators. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2521-2531. doi: 10.3934/dcds.2012.32.2521

References:

[1]	Viviane Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co, Inc., River Edge, NJ, 2000.
[2]	Oscar F. Bandtlow and Oliver Jenkinson, Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions, Adv. Math., 218 (2008), 902-925. doi: 10.1016/j.aim.2008.02.005.
[3]	Oscar F. Bandtlow and Oliver Jenkinson, On the Ruelle eigenvalue sequence, Ergodic Theory Dynam. Systems, 28 (2008), 1701-1711. doi: 10.1017/S0143385708000059.
[4]	T. Christiansen, Several complex variables and the distribution of resonances in potential scattering, Comm. Math. Phys., 259 (2005), 711-728. doi: 10.1007/s00220-005-1381-y.
[5]	T. J. Christiansen, Several complex variables and the order of growth of the resonance counting function in Euclidean scattering, Int. Math. Res. Not., 2006, Art. ID 43160, 36 pp.
[6]	David Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. École Norm. Sup. (4), 19 (1986), 491-517.
[7]	Israel Gohberg, Seymour Goldberg and Nahum Krupnik, "Traces and Determinants of Linear Operators," Operator Theory: Advances and Applications, 116, Birkhäuser Verlag, Basel, 2000.
[8]	Pierre Lelong and Lawrence Gruman, "Entire Functions of Several Complex Variables," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 282, Springer-Verlag, Berlin, 1986.
[9]	Thomas Ransford, "Potential Theory in the Complex Plane," London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995.
[10]	David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231-242.

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References:

[1]	Viviane Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co, Inc., River Edge, NJ, 2000.
[2]	Oscar F. Bandtlow and Oliver Jenkinson, Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions, Adv. Math., 218 (2008), 902-925. doi: 10.1016/j.aim.2008.02.005.
[3]	Oscar F. Bandtlow and Oliver Jenkinson, On the Ruelle eigenvalue sequence, Ergodic Theory Dynam. Systems, 28 (2008), 1701-1711. doi: 10.1017/S0143385708000059.
[4]	T. Christiansen, Several complex variables and the distribution of resonances in potential scattering, Comm. Math. Phys., 259 (2005), 711-728. doi: 10.1007/s00220-005-1381-y.
[5]	T. J. Christiansen, Several complex variables and the order of growth of the resonance counting function in Euclidean scattering, Int. Math. Res. Not., 2006, Art. ID 43160, 36 pp.
[6]	David Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. École Norm. Sup. (4), 19 (1986), 491-517.
[7]	Israel Gohberg, Seymour Goldberg and Nahum Krupnik, "Traces and Determinants of Linear Operators," Operator Theory: Advances and Applications, 116, Birkhäuser Verlag, Basel, 2000.
[8]	Pierre Lelong and Lawrence Gruman, "Entire Functions of Several Complex Variables," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 282, Springer-Verlag, Berlin, 1986.
[9]	Thomas Ransford, "Potential Theory in the Complex Plane," London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995.
[10]	David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231-242.

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