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Lower bounds for the Ruelle spectrum of analytic expanding circle maps

Published online by Cambridge University Press:  04 May 2017

OSCAR F. BANDTLOW  and
FRÉDÉRIC NAUD
OSCAR F. BANDTLOW
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK email o.bandtlow@qmul.ac.uk
FRÉDÉRIC NAUD
Affiliation:
Laboratoire de Mathématiques d’Avignon, Université d’Avignon, Campus Jean-Henri Fabre, 301 rue de Baruch de Spinoza, 84916 Avignon cedex 9, France email frederic.naud@univ-avignon.fr
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Abstract

We prove that there exists a dense set of analytic expanding maps of the circle for which the Ruelle eigenvalues enjoy exponential lower bounds. The proof combines potential theoretic techniques and explicit calculations for the spectrum of expanding Blaschke products.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 2 , February 2019 , pp. 289 - 310
Copyright
© Cambridge University Press, 2017 

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References

Bandtlow, O. F.. Resolvent estimates for operators belonging to exponential classes. Integr. Equ. Oper. Theory 61(1) (2008), 21–43.Google Scholar
Bandtlow, O. F. and Chu, C.-H.. Eigenvalue decay of operators on harmonic function spaces. Bull. Lond. Math. Soc. 41(5) (2009), 903–915.Google Scholar
Bandtlow, O. F. and Jenkinson, O.. Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions. Adv. Math. 218(3) (2008), 902–925.Google Scholar
Bandtlow, O. F. and Jenkinson, O.. On the Ruelle eigenvalue sequence. Ergod. Th. & Dynam. Sys. 28(6) (2008), 1701–1711.Google Scholar
Bandtlow, O. F., Just, W. and Slipantschuk, J.. Spectral structure of transfer operators for expanding circle maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 34(1) (2017), 31–43.Google Scholar
Burckel, R. B.. An Introduction to Classical Complex Analysis (Pure and Applied Mathematics, 82) . Vol. 1. Academic Press, Harcourt Brace Jovanovich, New York, London, 1979.Google Scholar
Duren, P. L.. Theory of H p Spaces (Pure and Applied Mathematics, 38) . Academic Press, New York, 1970.Google Scholar
Lelong, P. and Gruman, L.. Entire Functions of Several Complex Variables (Grundlehren der Mathematischen Wissenschaften, 282 [Fundamental Principles of Mathematical Sciences]) . Springer, Berlin, 1986.Google Scholar
Levin, G. M., Sodin, M. L. and Yuditski, P. M.. A Ruelle operator for a real Julia set. Comm. Math. Phys. 141(1) (1991), 119–132.Google Scholar
Martin, N. F. G.. On finite Blaschke products whose restrictions to the unit circle are exact endomorphisms. Bull. Lond. Math. Soc. 15(4) (1983), 343–348.Google Scholar
Pietsch, A.. Eigenvalues and s-numbers (Cambridge Studies in Advanced Mathematics, 13) . Cambridge University Press, Cambridge, 1987.Google Scholar
Pujals, E. R., Robert, L. and Shub, M.. Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family. Ergod. Th. & Dynam. Sys. 26(6) (2006), 1931–1937.Google Scholar
Ransford, T.. Potential Theory in the Complex Plane (London Mathematical Society Student Texts, 28) . Cambridge University Press, Cambridge, 1995.Google Scholar
Royden, H. L.. Invariant subspaces of H p for multiply connected regions. Pacific J. Math. 134(1) (1988), 151–172.Google Scholar
Rudin, W.. Real and Complex Analysis, 3rd edn. McGraw-Hill, New York, 1987.Google Scholar
Ruelle, D.. Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34(3) (1976), 231–242.Google Scholar
Sarason, D.. The H p spaces of an annulus. Mem. Amer. Math. Soc. 56 (1965), 78pp.Google Scholar
Slipantschuk, J., Bandtlow, O. F. and Just, W.. Analytic expanding circle maps with explicit spectra. Nonlinearity 26(12) (2013), 3231–3245.Google Scholar
Tischler, D.. Blaschke products and expanding maps of the circle. Proc. Amer. Math. Soc. 128(2) (2000), 621–622.Google Scholar