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On the Ruelle eigenvalue sequence

Published online by Cambridge University Press:  15 September 2008

OSCAR F. BANDTLOW  and
OLIVER JENKINSON
OSCAR F. BANDTLOW
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK (email: o.bandtlow@qmul.ac.uk, omj@maths.qmul.ac.uk)
OLIVER JENKINSON
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK (email: o.bandtlow@qmul.ac.uk, omj@maths.qmul.ac.uk)
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Abstract

For certain analytic data, we show that the eigenvalue sequence of the associated transfer operator ℒ is insensitive to the holomorphic function space on which ℒ acts. Explicit bounds on this eigenvalue sequence are established.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1701 - 1711
Copyright
Copyright © 2008 Cambridge University Press

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References

[1] Artuso, R., Aurell, E. and Cvitanović, P.. Recycling of strange sets. II. Applications. Nonlinearity 3 (1990), 361–386.Google Scholar
[2] Avez, A.. Differential Calculus. Wiley, Chichester, 1986.Google Scholar
[3] Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, Singapore, 2000.CrossRefGoogle Scholar
[4] Bandtlow, O. F. and Jenkinson, O.. Explicit a priori bounds on transfer operator eigenvalues. Comm. Math. Phys. 276 (2007), 901–905.CrossRefGoogle Scholar
[5] Bandtlow, O. F. and Jenkinson, O.. Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions. Adv. Math. 218 (2008), 902–925.CrossRefGoogle Scholar
[6] Christiansen, F., Cvitanović, P. and Rugh, H.-H.. The spectrum of the period-doubling operator in terms of cycles. J. Phys. A: Math. Gen. 23 (1990), L713–L717.Google Scholar
[7] Christiansen, F., Paladin, G. and Rugh, H.-H.. Determination of correlation spectra in chaotic systems. Phys. Rev. Lett. 65 (1990), 2087–2090.Google Scholar
[8] Earle, C. J. and Hamilton, R. S.. A fixed point theorem for holomorphic mappings. Global Analysis (Proceedings Symposia in Pure Mathematics, XVI). Eds. S. Chern and S. Smale. American Mathematical Society, Providence, RI, 1970, pp. 61–65.CrossRefGoogle Scholar
[9] Farkov, Yu. A.. The N-widths of Hardy–Sobolev spaces of several complex variables. J. Approx. Theory 75 (1993), 183–197.CrossRefGoogle Scholar
[10] Grabiner, S.. Spectral consequences of the existence of intertwining operators. Ann. Soc. Math. Polonae Ser. 1 22 (1970), 227–238.Google Scholar
[11] Grothendieck, A.. Produits Tensoriels Topologiques et Espaces Nucléaires (Memoirs of the American Mathematical Society, 16). American Mathematical Society, Providence, RI, 1955.Google Scholar
[12] Guillopé, L., Lin, K. and Zworski, M.. The Selberg zeta function for convex co-compact Schottky groups. Comm. Math. Phys. 245 (2004), 149–176.CrossRefGoogle Scholar
[13] Hinrichs, A.. Optimal Weyl inequality in Banach spaces. Proc. Amer. Math. Soc. 134 (2006), 731–735.Google Scholar
[14] Jiang, Y., Morita, T. and Sullivan, D.. Expanding direction of the period-doubling operator. Comm. Math. Phys. 144 (1992), 509–520.CrossRefGoogle Scholar
[15] Jenkinson, O. and Pollicott, M.. Orthonormal expansions of invariant densities for expanding maps. Adv. Math. 192 (2005), 1–34.CrossRefGoogle Scholar
[16] Krantz, S. G.. Function Theory of Several Complex Variables, 2nd edn. AMS Chelsea, New York, 1992.Google Scholar
[17] Mayer, D. H.. On a ζ function related to the continued fraction transformation. Bull. Soc. Math. France 104 (1976), 195–203.Google Scholar
[18] Mayer, D. H.. On composition operators on Banach spaces of holomorphic functions. J. Funct. Anal. 35 (1980), 191–206.CrossRefGoogle Scholar
[19] Mayer, D. H.. Continued fractions and related transformations. Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces. Eds. T. Bedford, M. Keane and C. Series. Oxford University Press, Oxford, 1991.Google Scholar
[20] Pietsch, A.. Eigenvalues and s-Numbers. Cambridge University Press, Cambridge, 1987.Google Scholar
[21] Pollicott, M.. Some applications of thermodynamic formalism to manifolds of constant negative curvature. Adv. Math. 85 (1991), 161–192.Google Scholar
[22] Pollicott, M.. A note on the Artuso–Aurell–Cvitanovic approach to the Feigenbaum tangent operator. J. Stat. Phys. 62 (1991), 257–267.CrossRefGoogle Scholar
[23] Pollicott, M. and Rocha, A.. A remarkable formula for the determinant of the Laplacian. Invent. Math. 130 (1997), 399–414.Google Scholar
[24] Range, R. M.. Holomorphic Functions and Integral Representations in Several Complex Variables. Springer, New York, 1986.Google Scholar
[25] Rudin, W.. Function Theory in the Unit Ball of C n (Grundlehren der Mathematischen Wissenschaften, 241). Springer, Berlin, 1980.Google Scholar
[26] Ruelle, D.. Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34 (1976), 231–242.CrossRefGoogle Scholar
[27] Ruelle, D.. Resonances for Axiom A flows. J. Differential Geom. 25 (1987), 117–137.Google Scholar
[28] Schaefer, H. H.. Topological Vector Spaces. Springer, New York, 1971.Google Scholar
[29] Weyl, H.. Inequalities between two kinds of eigenvalues of a linear transformation. Proc. Natl Acad. Sci. USA 35 (1949), 408–411.Google Scholar