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Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order
About this Title
Volker Mayer, Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d’Ascq Cedex, France and Mariusz Urbański, Department of Mathematics, University of North Texas, Denton, Texas 76203-1430
Publication: Memoirs of the American Mathematical Society
Publication Year:
2010; Volume 203, Number 954
ISBNs: 978-0-8218-4659-9 (print); 978-1-4704-0568-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-09-00577-8
Published electronically: August 26, 2009
Keywords: Holomorphic dynamics,
thermodynamical formalism,
transcendental functions,
fractal geometry
MSC: Primary 30D05, 37F10
Table of Contents
Chapters
- 1. Introduction
- 2. Balanced functions
- 3. Transfer operator and Nevanlinna Theory
- 4. Preliminaries, Hyperbolicity and Distortion Properties
- 5. Perron–Frobenius Operators and Generalized Conformal Measures
- 6. Finer properties of Gibbs States
- 7. Regularity of Perron-Frobenius Operators and Topological Pressure
- 8. Multifractal analysis
- 9. Multifractal Analysis of Analytic Families of Dynamically Regular Functions
Abstract
The thermodynamical formalism has been developed by the authors for a very general class of transcendental meromorphic functions. A function $f:\mathbb {C}\to \hat {{\mathbb C}}$ of this class is called dynamically (semi-) regular. The key point in our earlier paper (2008) was that one worked with a well chosen Riemannian metric space $(\hat {{\mathbb C}} , \sigma )$ and that the Nevanlinna theory was employed.
In the present manuscript we first improve upon our earlier paper in providing a systematic account of the thermodynamical formalism for such a meromorphic function $f$ and all potentials that are Hölder perturbations of $-t\log |f’|_\sigma$. In this general setting, we prove the variational principle, we show the existence and uniqueness of Gibbs states (with the definition appropriately adapted for the transcendental case) and equilibrium states of such potentials, and we demonstrate that they coincide. There is also given a detailed description of spectral and asymptotic properties (spectral gap, Ionescu-Tulcea and Marinescu Inequality) of Perron-Frobenius operators, and their stochastic consequences such as the Central Limit Theorem, K-mixing, and exponential decay of correlations.
Then we provide various, mainly geometric, applications of this theory. Indeed, we examine the finer fractal structure of the radial (in fact non-escaping) Julia set by developing the multifractal analysis of Gibbs states. In particular, the Bowen’s formula for the Hausdorff dimension of the radial Julia set from our earlier paper is reproved. Moreover, the multifractal spectrum function is proved to be convex, real-analytic and to be the Legendre transform conjugate to the temperature function. In the last chapter we went even further by showing that, for a analytic family satisfying a symmetric version of the growth condition (1.1) in a uniform way, the multifractal spectrum function is real-analytic also with respect to the parameter. Such a fact, up to our knowledge, has not been so far proved even for hyperbolic rational functions nor even for the quadratic family $z\mapsto z^2+c$. As a by-product of our considerations we obtain real analyticity of the Hausdorff dimension function.
- Jan M. Aarts and Lex G. Oversteegen, The geometry of Julia sets, Trans. Amer. Math. Soc. 338 (1993), no. 2, 897–918. MR 1182980, DOI 10.1090/S0002-9947-1993-1182980-3
- Krzysztof Barański, Hausdorff dimension and measures on Julia sets of some meromorphic maps, Fund. Math. 147 (1995), no. 3, 239–260. MR 1348721
- Ranjit Bhattacharjee and Robert L. Devaney, Tying hairs for structurally stable exponentials, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1603–1617. MR 1804947, DOI 10.1017/S0143385700000882
- Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188. MR 1216719, DOI 10.1090/S0273-0979-1993-00432-4
- Walter Bergweiler, Non-real periodic points of entire functions, Canad. Math. Bull. 40 (1997), no. 3, 271–275. MR 1464835, DOI 10.4153/CMB-1997-033-x
- François Berteloot and Volker Mayer, Rudiments de dynamique holomorphe, Cours Spécialisés [Specialized Courses], vol. 7, Société Mathématique de France, Paris; EDP Sciences, Les Ulis, 2001 (French, with English and French summaries). MR 1973050
- Emile Borel, Sur les zéros des fonctions entières, Acta Math. 20 (1897), no. 1, 357–396 (French). MR 1554885, DOI 10.1007/BF02418037
- Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
- Rufus Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11–25. MR 556580
- William Cherry and Zhuan Ye, Nevanlinna’s theory of value distribution, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. The second main theorem and its error terms. MR 1831783, DOI 10.1007/978-3-662-12590-8
- Ion Coiculescu and Bartłomiej Skorulski, Thermodynamic formalism of transcendental entire maps of finite singular type, Monatsh. Math. 152 (2007), no. 2, 105–123. MR 2346428, DOI 10.1007/s00605-007-0497-x
- Ion Coiculescu and Bartłomiej Skorulski, Perturbations in the Speiser class, Rocky Mountain J. Math. 37 (2007), no. 3, 763–800. MR 2351291, DOI 10.1216/rmjm/1182536163
- Manfred Denker, Feliks Przytycki, and Mariusz Urbański, On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 255–266. MR 1389624, DOI 10.1017/S0143385700008804
- Manfred Denker and Mariusz Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), no. 2, 563–587. MR 1014246, DOI 10.1090/S0002-9947-1991-1014246-4
- M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), no. 1, 103–134. MR 1092887
- Robert L. Devaney, Cantor bouquets, explosions, and Knaster continua: dynamics of complex exponentials, Publ. Mat. 43 (1999), no. 1, 27–54. MR 1697515, DOI 10.5565/PUBLMAT_{4}3199_{0}2
- Robert L. Devaney and MichałKrych, Dynamics of $\textrm {exp}(z)$, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 35–52. MR 758892, DOI 10.1017/S014338570000225X
- Elfving, G., Über eine Klasse von Riemannschen Flächen und ihre Uniformisierung, Acta Soc. Sci. Fenn. 2 (1934).
- Alexandre Eremenko, Ahlfors’ contribution to the theory of meromorphic functions, Lectures in memory of Lars Ahlfors (Haifa, 1996) Israel Math. Conf. Proc., vol. 14, Bar-Ilan Univ., Ramat Gan, 2000, pp. 41–63. MR 1786560
- A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020 (English, with English and French summaries). MR 1196102
- M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739–741 (Russian). MR 0251785
- W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
- J. M. Hemke, Measurable dynamics of meromorphic maps, thesis, Kiel (2005).
- Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass.-New York-Toronto, 1962. MR 0201608
- Einar Hille, Ordinary differential equations in the complex domain, Dover Publications, Inc., Mineola, NY, 1997. Reprint of the 1976 original. MR 1452105
- E. Hille On the zeroes of the Functions of the Parabolic Cylinder, Ark. Mat. Astron. Fys., Vol. 18, No. 26 (1924).
- A. Hinkkanen, A sharp form of Nevanlinna’s second fundamental theorem, Invent. Math. 108 (1992), no. 3, 549–574. MR 1163238, DOI 10.1007/BF02100617
- I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR 0322926
- C. T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d’opérations non complètement continues, Ann. of Math. (2) 52 (1950), 140–147 (French). MR 37469, DOI 10.2307/1969514
- F. Iversen Recherches sur les fonctions inverses dess fonctions méromorphes, Thèse de Helsingfors (1914).
- G. Jank, L. Volkmann, Meromorphe Funktionen und Differentialgleichungen,
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
- Janina Kotus and Mariusz Urbański, Conformal, geometric and invariant measures for transcendental expanding functions, Math. Ann. 324 (2002), no. 3, 619–656. MR 1938460, DOI 10.1007/s00208-002-0356-y
- Janina Kotus and Mariusz Urbański, Geometry and ergodic theory of non-recurrent elliptic functions, J. Anal. Math. 93 (2004), 35–102. MR 2110325, DOI 10.1007/BF02789304
- Janina Kotus and Mariusz Urbański, The dynamics and geometry of the Fatou functions, Discrete Contin. Dyn. Syst. 13 (2005), no. 2, 291–338. MR 2152392, DOI 10.3934/dcds.2005.13.291
- Janina Kotus and Mariusz Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions, Transcendental dynamics and complex analysis, London Math. Soc. Lecture Note Ser., vol. 348, Cambridge Univ. Press, Cambridge, 2008, pp. 251–316. MR 2458807, DOI 10.1017/CBO9780511735233.013
- I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433, DOI 10.1007/978-1-4615-6927-5
- J.K. Langley, Postgraduate notes on complex analysis, preprint.
- Carlangelo Liverani, Central limit theorem for deterministic systems, International Conference on Dynamical Systems (Montevideo, 1995) Pitman Res. Notes Math. Ser., vol. 362, Longman, Harlow, 1996, pp. 56–75. MR 1460797
- M. Yu. Lyubich, Some typical properties of the dynamics of rational maps, Russian Math. Surveys, 8, 5 (1983) 154-155.
- M. Yu. Lyubich, Dynamics of rational transformations: topological picture, Uspekhi Mat. Nauk 41 (1986), no. 4(250), 35–95, 239 (Russian). MR 863874
- R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343
- Curt McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), no. 1, 329–342. MR 871679, DOI 10.1090/S0002-9947-1987-0871679-3
- Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- R. Daniel Mauldin and Mariusz Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105–154. MR 1387085, DOI 10.1112/plms/s3-73.1.105
- R. Daniel Mauldin and Mariusz Urbański, Graph directed Markov systems, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003. Geometry and dynamics of limit sets. MR 2003772, DOI 10.1017/CBO9780511543050
- Volker Mayer, Comparing measures and invariant line fields, Ergodic Theory Dynam. Systems 22 (2002), no. 2, 555–570. MR 1898805, DOI 10.1017/S0143385702000275
- V. Mayer, Rational functions without conformal measures on the conical set, Preprint 2002.
- V. Mayer, The size of the Julia set of meromorphic functions, Math. Nachrichten (to appear).
- Volker Mayer and Mariusz Urbański, Gibbs and equilibrium measures for elliptic functions, Math. Z. 250 (2005), no. 3, 657–683. MR 2179616, DOI 10.1007/s00209-005-0770-4
- Volker Mayer and Mariusz Urbański, Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 915–946. MR 2422021, DOI 10.1017/S0143385707000648
- Volker Mayer and Mariusz Urbański, Fractal measures for meromorphic functions of finite order, Dyn. Syst. 22 (2007), no. 2, 169–178. MR 2327991, DOI 10.1080/14689360600893736
- Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330
- Rolf Nevanlinna, Analytic functions, Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig. MR 0279280
- Rolf Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math. 58 (1932), no. 1, 295–373 (German). MR 1555350, DOI 10.1007/BF02547780
- William Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0262464
- S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241–273. MR 450547, DOI 10.1007/BF02392046
- F. Przytycki, M. Urbański, Fractals in the Plane - the Ergodic Theory Methods, available on Urbański’s webpage, to appear Cambridge Univ. Press.
- Feliks Przytycki, Mariusz Urbański, and Anna Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I, Ann. of Math. (2) 130 (1989), no. 1, 1–40. MR 1005606, DOI 10.2307/1971475
- Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
- Lasse Rempe, Hyperbolic dimension and radial Julia sets of transcendental functions, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1411–1420. MR 2465667, DOI 10.1090/S0002-9939-08-09650-0
- P. J. Rippon and G. M. Stallard, Iteration of a class of hyperbolic meromorphic functions, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3251–3258. MR 1610785, DOI 10.1090/S0002-9939-99-04942-4
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
- David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
- David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 99–107. MR 684247, DOI 10.1017/s0143385700009603
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
- Zbigniew Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), no. 2, 347–355. MR 1037218, DOI 10.1090/S0002-9939-1991-1037218-8
- Gwyneth M. Stallard, The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions, Math. Proc. Cambridge Philos. Soc. 127 (1999), no. 2, 271–288. MR 1705459, DOI 10.1017/S0305004199003813
- Gwyneth M. Stallard, Dimensions of Julia sets of transcendental meromorphic functions, Transcendental dynamics and complex analysis, London Math. Soc. Lecture Note Ser., vol. 348, Cambridge Univ. Press, Cambridge, 2008, pp. 425–446. MR 2458811, DOI 10.1017/CBO9780511735233.017
- D. Sullivan, Seminar on conformal and hyperbolic geometry, Preprint IHES (1982).
- Mariusz Urbański, Measures and dimensions in conformal dynamics, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 281–321. MR 1978566, DOI 10.1090/S0273-0979-03-00985-6
- Mariusz Urbański, Recurrence rates for loosely Markov dynamical systems, J. Aust. Math. Soc. 82 (2007), no. 1, 39–57. MR 2301970, DOI 10.1017/S1446788700017468
- Mariusz Urbański and Anna Zdunik, The finer geometry and dynamics of the hyperbolic exponential family, Michigan Math. J. 51 (2003), no. 2, 227–250. MR 1992945, DOI 10.1307/mmj/1060013195
- Mariusz Urbański and Anna Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Ergodic Theory Dynam. Systems 24 (2004), no. 1, 279–315. MR 2041272, DOI 10.1017/S0143385703000208
- Mariusz Urbański and Anna Zdunik, Geometry and ergodic theory of non-hyperbolic exponential maps, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3973–3997. MR 2302520, DOI 10.1090/S0002-9947-07-04151-7
- M. Urbański, A. Zdunik, Maximizing Measures on Metrizable Non-Compact Spaces, Preprint 2004, to appear Math. Proc. Edinburgh Math. Soc.
- Mariusz Urbański and Michel Zinsmeister, Geometry of hyperbolic Julia-Lavaurs sets, Indag. Math. (N.S.) 12 (2001), no. 2, 273–292. MR 1913648, DOI 10.1016/S0019-3577(01)80032-X
- Anna Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, 627–649. MR 1032883, DOI 10.1007/BF01234434
- Michel Zinsmeister, Formalisme thermodynamique et systèmes dynamiques holomorphes, Panoramas et Synthèses [Panoramas and Syntheses], vol. 4, Société Mathématique de France, Paris, 1996 (French, with English and French summaries). MR 1462079