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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

Oscar F. Bandtlow and Oliver Jenkinson, On the Ruelle eigenvalue sequence, Ergodic Theory Dynam. Systems, 28 (2008), 1701-1711. doi: 10.1017/S0143385708000059.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

David Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. École Norm. Sup. (4), 19 (1986), 491-517.  Google Scholar

[7]

Israel Gohberg, Seymour Goldberg and Nahum Krupnik, "Traces and Determinants of Linear Operators," Operator Theory: Advances and Applications, 116, Birkhäuser Verlag, Basel, 2000.  Google Scholar

[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.  Google Scholar

[9]

Thomas Ransford, "Potential Theory in the Complex Plane," London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995.  Google Scholar

[10]

David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231-242.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

Oscar F. Bandtlow and Oliver Jenkinson, On the Ruelle eigenvalue sequence, Ergodic Theory Dynam. Systems, 28 (2008), 1701-1711. doi: 10.1017/S0143385708000059.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

David Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. École Norm. Sup. (4), 19 (1986), 491-517.  Google Scholar

[7]

Israel Gohberg, Seymour Goldberg and Nahum Krupnik, "Traces and Determinants of Linear Operators," Operator Theory: Advances and Applications, 116, Birkhäuser Verlag, Basel, 2000.  Google Scholar

[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.  Google Scholar

[9]

Thomas Ransford, "Potential Theory in the Complex Plane," London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995.  Google Scholar

[10]

David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231-242.  Google Scholar

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