Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T03:49:45.517Z Has data issue: false hasContentIssue false
  • English
  • Français

Complex asymptotics of Poincaré functions and properties of Julia sets

Published online by Cambridge University Press:  01 November 2008

GREGORY DERFEL ,
PETER J. GRABNER  and
FRITZ VOGL
GREGORY DERFEL
Affiliation:
Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer Sheva 84105, Israel. e-mail: [email protected]
PETER J. GRABNER
Affiliation:
Institut für Analysis und Computational Number Theory (Math A), Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria. e-mail: [email protected]
FRITZ VOGL
Affiliation:
Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraβe 8–10, 1040 Wien, Austria. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The asymptotic behaviour of the solutions of Poincaré's functional equation f(λz) = p(f(z)) (λ > 1) for p a real polynomial of degree ≥ 2 is studied in angular regions W of the complex plain. It is known [9, 10] that f(z) ~ exp(zρF(logλz)), if f(z) → ∞ for z ∞ and z ∈ W, where F denotes a periodic function of period 1 and ρ = logλ deg(p). In this paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function F is characterised in terms of geometric properties of the Julia set of p. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of f is related to the harmonic measure on the Julia set of p.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 3 , November 2008 , pp. 699 - 718
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barlow, M. T. Diffusions on fractals. In Lectures on Probability Theory and Statistics (Saint-Flour, 1995), pages 1–121, (Springer Verlag, 1998).CrossRefGoogle Scholar
[2]Barlow, M. T. and Perkins, E. A.Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields 79 (1988), 543–623.CrossRefGoogle Scholar
[3]Beardon, A.Iteration of Rational Functions (Springer, 1991).CrossRefGoogle Scholar
[4]Bessis, D., Geronimo, J. S. and Moussa, P.Mellin transforms associated with Julia sets and physical applications. J. Statist. Phys. 34 (1984), 75–110.CrossRefGoogle Scholar
[5]Blanchard, P.Complex analytic dynamics on the Riemann sphere. Bull. Amer. Math. Soc. (N.S.) 11 (1984), 85–141.CrossRefGoogle Scholar
[6]Böttcher, L.Beiträge zur Theorie der Iterationsrechnung. Bull. Kasan Math. Soc. 14 (1905), 176.Google Scholar
[7]Brolin, H.Invariant sets under iteration of rational functions. Ark. Mat. 6 (1965), 103–144.CrossRefGoogle Scholar
[8]Buff, X.On the Bieberbach conjecture and holomorphic dynamics. Proc. Amer. Math. Soc. 131 (2003), 755–759 (electronic).CrossRefGoogle Scholar
[9]Derfel, G., Grabner, P. J. and Vogl, F. Asymptotics of the Poincaré functions. In Dawson, D., Jaksic, V. and Vainberg, B., editors, Probability and Mathematical Physics: A Volume in Honor of Stanislav Molchanov. CRM Proceedings and Lecture Notes, vol. 42 pages 113–130 (Centre de Recherches Mathématiques, Montreal, 2007).CrossRefGoogle Scholar
[10]Derfel, G., Grabner, P. J. and Vogl, F.The Zeta function of the Laplacian on certain fractals. Trans. Amer. Math. Soc. 360 (2008), 881–897 (electronic).CrossRefGoogle Scholar
[11]Doetsch, G.Handbuch der Laplace-Transformation. Band I: Theorie der Laplace-Transformation (Birkhäuser Verlag, 1971). Verbesserter Nachdruck der ersten Auflage 1950, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften. Mathematische Reihe, Band 14.Google Scholar
[12]Erëmenko, A. È. and Levin, G. M.Periodic points of polynomials (Russian). Ukrain. Mat. Zh. 41 (1989), 1467–1471, 1581. Translation in Ukrainian Math. J. 41 (1989), 1258–1262.CrossRefGoogle Scholar
[13]Erëmenko, A. È. and Levin, G. M.Estimation of the characteristic exponents of a polynomial (Russian). Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. (1993), pages 30–40. Translation in J. Math. Sci. (New York) 85 (1997), 2164–2171.CrossRefGoogle Scholar
[14]Erëmenko, A. È. and Sodin, M. L.Iterations of rational functions and the distribution of the values of Poincaré functions (Russian). Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 53 (1990), 18–25. Translation in J. Soviet Math. 58 (1992), 504–509.Google Scholar
[15]Falconer, K. J.Fractal Geometry (John Wiley & Sons Inc., 2003). Mathematical foundations and applications.CrossRefGoogle Scholar
[16]Garnett, J. B. and Marshall, D. E.Harmonic Measure. New Mathematical Monographs vol. 2 (Cambridge University Press, 2005).CrossRefGoogle Scholar
[17]Goluzin, G. M.Geometric Theory of Functions of a Complex Variable. Translations of Mathematical Monographs, Vol. 26 (American Mathematical Society, 1969).CrossRefGoogle Scholar
[18]Grabner, P. J.Functional iterations and stopping times for Brownian motion on the Sierpiński gasket. Mathematika 44 (1997), 374–400.CrossRefGoogle Scholar
[19]Hamilton, D. H.Length of Julia curves. Pacific J. Math. 169 (1995), 75–93.CrossRefGoogle Scholar
[20]Harris, T. E.The Theory of Branching Processes (Springer, 1963).CrossRefGoogle Scholar
[21]Ishizaki, K. and Yanagihara, N.Borel and Julia directions of meromorphic Schröder functions. Math. Proc. Camb. Phil. Soc. 139 (2005), 139–147.CrossRefGoogle Scholar
[22]Koenigs, G.Recherches sur les intégrales de certaines équations fonctionelles. Ann. Sci. Ec. Norm. Super. III. Ser. 1 (1884), 3–41.Google Scholar
[23]Koenigs, G.Nouvelles recherches sur les equations fonctionnelles. Ann. Sci. École Norm. Sup. III. Ser. 2 (1885), 385–404.CrossRefGoogle Scholar
[24]Krön, B.Green functions on self-similar graphs and bounds for the spectrum of the Laplacian. Ann. Inst. Fourier (Grenoble) 52 (2002), 1875–1900.CrossRefGoogle Scholar
[25]Krön, B. and Teufl, E.Asymptotics of the transition probabilities of the simple random walk on self-similar graphs. Trans. Amer. Math. Soc. 356 (2004), 393–414 (electronic).CrossRefGoogle Scholar
[26]Kuczma, M., Choczewski, B. and Ger, R.Iterative Functional Equations, volume 32 of Encyclopedia of Mathematics and its Applications (Cambridge University Press, 1990).CrossRefGoogle Scholar
[27]Levin, B. Y.Distribution of Zeros of Entire Functions. Transl. Math. Monog. vol. 5 (American Mathematical Society, 1980), revised edition. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman.Google Scholar
[28]Levin, B. Y.Lectures on Entire Functions. Trans. Math. Monogr. vol. 150 (American Mathematical Society, 1996). In collaboration with, and with a preface by, Yu. Lyubarskii, M. Sodin and V. Tkachenko Translated from the Russian manuscript by V. Tkachenko.Google Scholar
[29]Levin, G. M.On Pommerenke's inequality for the eigenvalues of fixed points. Colloq. Math. 62 (1991), 167–177.CrossRefGoogle Scholar
[30]Lindstrøm, T.Brownian Motion on Nested Fractals. Mem. Amer. Math. Soc. vol. 420 (1990).Google Scholar
[31]Malozemov, L. and Teplyaev, A.Self-similarity, operators and dynamics. Math. Phys. Anal. Geom. 6 (2003) 201–218.CrossRefGoogle Scholar
[32]Mattila, P.Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability. Cambridge Studies in Advanced Math. No. 44 (Cambridge University Press, 1995).CrossRefGoogle Scholar
[33]Milnor, J.Dynamics in one complex variable, volume 160 of Annals of Mathematics Studies (Princeton University Press, Princeton, NJ, 2006), third edition.Google Scholar
[34]Oberhettinger, F.Tables of Mellin Transforms (Springer-Verlag, New York, 1974).CrossRefGoogle Scholar
[35]Paris, R. B. and Kaminski, D.Asymptotics and Mellin-Barnes Integrals. Encyclopedia Math. Appl. vol. 55 (Cambridge University Press, 2001).CrossRefGoogle Scholar
[36]Poincaré, H.Sur une classe étendue de transcendantes uniformes. C. R. Acad. Sci. Paris 103 (1886), 862–864.Google Scholar
[37]Poincaré, H.Sur une classe nouvelle de transcendantes uniformes. J. Math. Pures. Appl. IV. Ser. 6 (1890), 316–365.Google Scholar
[38]Pommerenke, C.On conformal mapping and iteration of rational functions. Complex Variables Theory Appl. 5 (1986), 117–126.Google Scholar
[39]Ransford, T.Potential Theory in the Complex Plane. London Math. Soc. Student Texts vol. 28 (Cambridge University Press, 1995).CrossRefGoogle Scholar
[40]Rivlin, T. J.The Chebyshev Polynomials (Wiley-Interscience [John Wiley & Sons], 1974). Pure and Applied Mathematics.Google Scholar
[41]Romanenko, E. and Sharkovsky, A. Long time properties of solutions of simplest q-difference equations (Russian) (2000). Preprint.Google Scholar
[42]Schröder, E.Über iterierte Funktionen. Math. Ann. 3 (1871), 296–322.CrossRefGoogle Scholar
[43]Teplyaev, A. Spectral zeta function of symmetric fractals. In Bandt, C., Mosco, U. and Zähle, M., editors, Fractal Geometry and Stochastics III. Progr. Probab. vol. 57 (Birkhäuser, 2004), pages 245–262.CrossRefGoogle Scholar
[44]Teplyaev, A.Spectral zeta functions of fractals and the complex dynamics of polynomials. Trans. Amer. Math. Soc. 359 (2007), 4339–4358 (electronic).CrossRefGoogle Scholar
[45]Valiron, G.Lectures on the General Theory of Integral Functions (Private, Toulouse, 1923).Google Scholar
[46]Valiron, G.Fonctions Analytiques (Presses Universitaires de France, 1954).Google Scholar