Abstract
This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order whose derivative satisfies some growth condition at ∞. This class contains most of the classical families of transcendental functions and goes much beyond. Based on uniform versions of Nevanlinna’s value distribution theory, we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with exponential decay of correlations and a central limit theorem are shown. This part relies on our construction of new positive invariant cones that are adapted to the setting of unbounded phase spaces. This setting rules out the use of Hilbert’s metric along with the usual contraction principle. However, these cones allow us to apply a contraction argument stemming from Bowen’s initial approach.
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References
Ludwig Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
Garrett Birkhoff, Extensions of Jentzsch’s theorem, Trans. Amer. Math. Soc. 85 (1957), 219–227.
Rufus Bowen Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer-Verlag, Berlin, 1975.
William Cherry and Zhuan Ye, Nevanlinna’s Theory of Value Distribution, Springer-Verlag, Berlin, 2001.
Hans Crauel, Random probability Measures on Polish Spaces, Taylor & Francis, London, 2002.
Manfred Denker, Yuri Kifer, and Manuel Stadlbauer, Thermodynamic formalism for random countable Markov shifts Discrete Contin. Dyn. Syst. 22 (2008), 131–164.
Alexandre Eremenko, Ahlfors’ contribution to the theory of meromorphic functions, in Lectures in Memory of Lars Ahlfors (Haifa, 1996), volume 14 of Israel Math. Conf. Proc., Bar-Ilan Univ., Ramat Gan, 2000, pp. 41–63.
John Erik Fornæss and Nessim Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687–708.
Anatoly A. Goldberg and Iossif V. Ostrovskii, Value Distribution of Meromorphic Functions, American Mathematical Society, Providence, RI, 2008.
Mikhail Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739–741.
Hubert Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc. 118 (1993), 627–634.
Einar Hille, Analytic Function Theory, Vol. I, II American Mathematical Society, Providence, RI, 2000.
C. T. Ionescu Tulcea and G. Marinescu. Théorie ergodique pour des classes d’opérations non complétement continues, Ann. of Math. (2) 52 (1950), 52:140–147.
Yuri Kifer, Thermodynamic formalism for random transformations revisited, Stochastics and Dynam. 8 (2008), 77–102.
Yuri Kifer and Pei-Dong Liu, Random dynamics, in Handbook of Dynamical Systems, Vol. 1B, Elsevier,, Amsterdam, 2006, pp. 379–499.
Carlangelo Liverani, Decay of correlations, Ann. of Math. (2) 142 (1995), 239–301.
Volker Mayer, Bartlomiej Skorulski, and Mariusz Urbański, Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Springer, Heidelberg, 2011.
Volker Mayer, Bartlomiej Skorulski, and Mariusz Urbański, Regularity and irregularity of fiber dimensions of non-autonomous dynamical systems, Ann. Acad. Sci. Fenn. Math. 38 (2013), 489–514.
Volker Mayer and Mariusz Urbański, Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order, Ergodic Theory Dynam. Systems 28 (2008), 915–946.
Volker Mayer and Mariusz Urbański Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order, Mem. Amer. Math. Soc. 203 (2010), no. 954.
Rolf Nevanlinna, Analytic Functions, Springer-Verlag, New York, 1970.
Rolf Nevanlinna, Eindeutige analytische Funktionen, Springer-Verlag, Berlin, 1974.
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
Mario Roy and Mariusz Urbański, Random graph directed Markov systems, Discrete Contin. Dyn. Syst. 30 (2011), 261–298.
David Ruelle, Thermodynamic Formalism. TheMathematical Structures of Classical Equilibrium Statistical Mechanics, Addison-Wesley Publishing Co., Reading, Mass., 1978.
[26] Hans Henrik Rugh, On the dimension of conformal repellers, randomness and parameter dependency, Ann. of Math. (2) 168 (2008), 695–748.
[27] Hans Henrik Rugh, Cones and gauges in complex spaces: spectral gaps and complex Perron-Frobenius theory, Ann. of Math. (2) 171 (2010), 1707–1752.
Manuel Stadlbauer, On random topological Markov chains with big images and preimages, Stoch. Dyn. 10 (010), 77–95.
Hiroki Sumi, On dynamics of hyperbolic rational semigroups, J. Math. Kyoto Univ. 37 (1997), 717–733.
Hiroki Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. London Math. Soc. (3) 102 (2011), 50–112.
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Mayer, V., Urbański, M. Random dynamics of transcendental functions. JAMA 134, 201–235 (2018). https://doi.org/10.1007/s11854-018-0007-1
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DOI: https://doi.org/10.1007/s11854-018-0007-1