Spectral structure of transfer operators for expanding circle maps
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 31-43.

We explicitly determine the spectrum of transfer operators (acting on spaces of holomorphic functions) associated to analytic expanding circle maps arising from finite Blaschke products. This is achieved by deriving a convenient natural representation of the respective adjoint operators.

DOI : 10.1016/j.anihpc.2015.08.004
Mots-clés : Spectrum of transfer operators, Analytic circle maps, Mixing rates, Blaschke products, Adjoint operators, Expanding maps 
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     title = {Spectral structure of transfer operators for expanding circle maps},
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Bandtlow, Oscar F.; Just, Wolfram; Slipantschuk, Julia. Spectral structure of transfer operators for expanding circle maps. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 31-43. doi : 10.1016/j.anihpc.2015.08.004. http://www.numdam.org/articles/10.1016/j.anihpc.2015.08.004/

[1] Baladi, V. Smooth Ergodic Theory and Its Applications (1999), pp. 297–325 (Seattle) | MR | Zbl

[2] Baladi, V. Positive Transfer Operators and Decay of Correlations, World Scientific Publishing, Singapore, 2000 | DOI | MR | Zbl

[3] Baladi, V.; Jiang, Y.; Rugh, H.H. Dynamical determinants via dynamical conjugacies for postcritically finite polynomials, J. Stat. Phys., Volume 108 (2002) no. 5–6, pp. 973–993 | MR | Zbl

[4] Bandtlow, O.F. Resolvent estimates for operators belonging to exponential classes, Integral Equ. Oper. Theory, Volume 61 (2008), pp. 21–43 | DOI | MR | Zbl

[5] Bandtlow, O.F.; Jenkinson, O. Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions, Adv. Math., Volume 218 (2008), pp. 902–925 | DOI | MR | Zbl

[6] Bandtlow, O.F.; Jenkinson, O. On the Ruelle eigenvalue sequence, Ergod. Theory Dyn. Syst., Volume 28 (2008) no. 06, pp. 1701–1711 | DOI | MR | Zbl

[7] Boyarsky, A.; Gora, P. Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Probab. Appl., Birkhäuser, 1997 | MR | Zbl

[8] Burckel, R.B. An Introduction to Classical Complex Analysis, vol. 1, Academic Press, Inc., New York–London, 1979 | DOI | MR | Zbl

[9] Cowen, C.C.; MacCluer, B.D. Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995 | MR | Zbl

[10] Duren, P. Theory of Hp-Spaces, Academic Press, New York, 1970 | MR | Zbl

[11] Fried, D. Zeta functions of Ruelle and Selberg I, Ann. Sci. Éc. Norm. Super., Volume 19 (1986), pp. 491–517 | Numdam | MR | Zbl

[12] Keller, G.; Rugh, H.H. Eigenfunctions for smooth expanding circle maps, Nonlinearity, Volume 17 (2004) no. 5, pp. 1723–1730 | DOI | MR | Zbl

[13] Levin, G.M. On Mayer's conjecture and zeros of entire functions, Ergod. Theory Dyn. Syst., Volume 14 (1994) no. 03, pp. 565–574 | DOI | MR | Zbl

[14] Levin, G.M.; Sodin, M.L.; Yuditski, P.M. A Ruelle operator for a real Julia set, Commun. Math. Phys., Volume 141 (1991) no. 1, pp. 119–132 | DOI | MR | Zbl

[15] Levin, G.M.; Sodin, M.L.; Yuditskii, P. Ruelle operators with rational weights for Julia sets, J. Anal. Math., Volume 63 (1994) no. 1, pp. 303–331 | DOI | MR | Zbl

[16] Martin, N.F.G. On finite Blaschke products whose restrictions to the unit circle are exact endomorphisms, Bull. Lond. Math. Soc., Volume 15 (1983) no. 4, pp. 343–348 | MR | Zbl

[17] Mayer, D.H. Continued fractions and related transformations, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, 1991, pp. 175–222 | MR

[18] Pujals, E.R.; Robert, L.; Shub, M. Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family, Ergod. Theory Dyn. Syst., Volume 26 (2006) no. 06, pp. 1931–1937 | DOI | MR | Zbl

[19] Royden, H.L.J. Invariant subspaces of Hp for multiply connected regions, Pac. J. Math., Volume 134 (1988) no. 1 | MR | Zbl

[20] Rudin, W. Real and Complex Analysis, McGraw-Hill Book Co., 1987 | MR | Zbl

[21] Ruelle, D. Zeta-functions for expanding maps and Anosov flows, Invent. Math., Volume 34 (1976) no. 3, pp. 231–242 | DOI | MR | Zbl

[22] Rugh, H.H. The correlation spectrum for hyperbolic analytic maps, Nonlinearity, Volume 5 (1992) no. 6, pp. 1237–1263 | MR | Zbl

[23] Rugh, H.H. Coupled maps and analytic function spaces, Ann. Sci. Éc. Norm. Super., Volume 35 (2002) no. 4, pp. 489–535 | Numdam | MR | Zbl

[24] Sarason, D. The Hp Spaces of an Annulus, Mem. Am. Math. Soc., vol. 56, 1965 | MR | Zbl

[25] Shapiro, J.H. Composition Operators and Classical Function Theory, Springer, 1993 | DOI | MR | Zbl

[26] Slipantschuk, J.; Bandtlow, O.F.; Just, W. Analytic expanding circle maps with explicit spectra, Nonlinearity, Volume 26 (2013) | DOI | MR | Zbl

[27] Slipantschuk, J.; Bandtlow, O.F.; Just, W. On correlation decay in low-dimensional systems, Europhys. Lett., Volume 104 (2013) | DOI

[28] Tischler, D. Blaschke products and expanding maps of the circle, Proc. Am. Math. Soc., Volume 128 (1999) no. 2, pp. 621–622 | DOI | MR | Zbl

[29] Ushiki, S. Fredholm determinant of complex Ruelle operator, Ruelle's dynamical zeta-function, and forward/backward Collet–Eckmann condition, Sūrikaisekikenkyūsho Kōkyūroku, Volume 1153 (2000), pp. 85–102 | MR | Zbl

[30] Walters, P. An Introduction to Ergodic Theory, Springer, 2000 | Zbl

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