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On Ruelle’s property

Part of:Riemann surfaces

Published online by Cambridge University Press:  02 February 2021

SHENGJIN HUOOpen the ORCID record for SHENGJIN HUO[Opens in a new window] and
MICHEL ZINSMEISTER
SHENGJIN HUO*
Affiliation:
Department of Mathematics, Tiangong University, Tianjin 300387, China
MICHEL ZINSMEISTER
Affiliation:
Institut Denis Poisson, COST, Université d’Orléans BP 6749, 45067 Orléans Cedex 2, France (e-mail: zins@unvi-orleans.fr)
*
e-mail: huoshengjin@tiangong.edu.cn
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Abstract

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In this paper we investigate the range of validity of Ruelle’s property. First, we show that every finitely generated Fuchsian group has Ruelle’s property. We also prove the existence of an infinitely generated Fuchsian group satisfying Ruelle’s property. Concerning the negative results, we first generalize Astala and Zinsmeister’s results [Mostow rigidity and Fuchsian groups. C. R. Math. Acad. Sci. Paris 311 (1990), 301–306; Teichmüller spaces and BMOA. Math. Ann. 289 (1991), 613–625] by proving that all convergence-type Fuchsian groups of the first kind fail to have Ruelle’s property. Finally, we give some results about second-kind Fuchsian groups. [-3.2pc]

Keywords

iterated function systemMarkov mapRuelle’s property

MSC classification

Primary:30F35: Fuchsian groups and automorphic functions
Secondary:30F60: Teichmüller theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 4 , April 2022, pp. 1474 - 1486
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Aaronson, J. and Denker, M.. The Poincaré series of $\mathbb{C}\backslash \mathbb{Z}$ . Ergod. Th. & Dynam. Sys. 19 (1999), 1–20.CrossRefGoogle Scholar
Anderson, J. W. and Rocha, A. C.. Analyticity of Hausdorff dimension of limit sets of Kleinian groups. Ann. Acad. Sci. Fenn. Math. 22 (1997), 349–364.Google Scholar
Astala, K. and Zinsmeister, M.. Mostow rigidity and Fuchsian groups. C. R. Math. Acad. Sci. Paris 311 (1990),301–306.Google Scholar
Astala, K. and Zinsmeister, M.. Teichmüller spaces and BMOA. Math. Ann. 289 (1991), 613–625.CrossRefGoogle Scholar
Astala, K. and Zinsmeister, M.. Holomorphic families of quasi-Fuchsian groups. Ergod. Th. & Dynam. Sys. 14 (1994), 207–212.CrossRefGoogle Scholar
Astala, K. and Zinsmeister, M.. Abelian coverings, Poincare exponent of convergence and holomorphic deformations. Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 81–86.Google Scholar
Beardon, A. F.. The exponent of convergence of Poincare series. Proc. Lond. Math. Soc. 18 (1968), 461–483.CrossRefGoogle Scholar
Bishop, C. J.. Divergence groups have the Bowen property. Ann. of Math. 154 (2001), 205–217.CrossRefGoogle Scholar
Bishop, C. J.. Big deformations near infinity. Illinois J. Math. 47 (2003), 977–996.CrossRefGoogle Scholar
Bishop, C. J.. A criterion for failure of Rueller’s property. Ergod. Th. & Dynam. Sys. 26 (2006), 1733–1748.CrossRefGoogle Scholar
Bishop, C. J. and Jones, P. W.. Hausdorff dimension and Kleinian groups. Acta. Math. 179 (1997), 1–39.CrossRefGoogle Scholar
Bishop, C. J. and Jones, P. W.. Compact deformations of Fuchsian group. J. Anal. Math. 87 (2002), 5–36.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
Bowen, R.. Hausdorff dimension of quasicircles. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–25.CrossRefGoogle Scholar
Bowen, R. and Series, C.. Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 153–170.CrossRefGoogle Scholar
Garnett, J. B.. Bounded Analytic Functions, revised 1st edn. Springer, New York, 2010.Google Scholar
Garnett, J., Gehring, F. and Jones, P.. Quasiconformal groups and the conocal limit set. Holomorphic Functions and Moduli II (MSRI Publications, 11). Springer, New York, 1988, pp. 59–67.CrossRefGoogle Scholar
Fernandez, J. L. and Melian, M. V.. Escaping geodesics of Riemann surfaces. Acta Math. 187 (2001), 213–236.CrossRefGoogle Scholar
Huo, S.. On Carleson measures induced by Beltrami coefficients being compatible with Fuchsian groups. Ann. Acad. Sci. Fenn. Math. 46 (2021), to appear, arXiv:1908.05174v1.Google Scholar
Mauldin, R. D. and Urbanski, M.. Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc. 73(3) (1996), 105–154.CrossRefGoogle Scholar
Pommerenke, C.. Schlichte Funktionen und BMOA. Comment. Math. Helv. 52 (1977), 591–602.CrossRefGoogle Scholar
Rubel, L. A. and Ryff, J. V.. The bounded weak-star topology and the bounded analytic functions. J. Funct. Anal. 5 (1970), 167–183.CrossRefGoogle Scholar
Ruelle, D.. Thermodynamic Formalism. Addison-Wesley, Reading, 1978.Google Scholar
Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–107.CrossRefGoogle Scholar
Semmes, S.. Quasiconformal mappings and chord-arc curves. Trans. Amer. Math. Soc. 306 (1988), 233–263.CrossRefGoogle Scholar
Sullivan, D.. Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc. 6 (1982), 57–73.CrossRefGoogle Scholar
Sullivan, D.. Entropy, Hausdorff measures old and new, and limit of geometrically finite Kleinian groups. Acta Math. 259 (1984), 259–277.CrossRefGoogle Scholar