Abstract
This chapter provides a brief overview of the subject of the iteration of rational maps of the sphere with a focus on dynamical properties. By a rational map we mean an analytic map of the Riemann sphere, \(\widehat {\mathbb {C}}= \mathbb C \cup \{\infty \}\); it is well-known that each such map can be written as the quotient of two polynomials. We use meromorphic functions as a tool in our study, as they are analytic maps except at isolated poles, and some provide dynamically significant maps from \(\mathbb C\) to \(\widehat {\mathbb {C}}\). The subject of iterated rational maps is a classical one that dates back to the turn of the twentieth century and is still very active with many exciting and elusive problems remaining.
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References
L. Ahlfors, Complex analysis, in An Introduction to the Theory of Analytic Functions of One Complex Variable. International Series in Pure and Applied Mathematics, 3rd edn. (McGraw-Hill, New York, 1978)
M. Aspenberg, The Collet-Eckmann condition for rational functions on the Riemann sphere. Math. Z. 273(3–4), 935–980 (2013)
J. Barnes, Conservative exact rational maps of the sphere. J. Math. Anal. Appl. 230, 350–374 (1999)
A. Beardon, Iterations of Rational Functions. Graduate Texts in Mathematics, vol. 132 (Springer, Berlin, 1991)
H. Brolin, Invariant sets under iteration of rational functions. Ark. Mat. 6, 103–144 (1965)
R. Brooks, P. Matelski, The dynamics of 2-generator subgroups of PSL(2,C), in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. Annals of Mathematics Studies, vol. 97 (Princeton University Press, Princeton, 1981), pp. 65 – 71
X. Buff, A. Cheritat, Quadratic Julia sets with positive area. Ann. Math. 176(2), 673–746 (2012)
L. Carleson, T. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics (Springer, New York, 1993)
A. Chéritat, A. Epstein, Bounded type Siegel disks of finite type maps with few singular values. Sci. China Math. 61(12), 2139–2156 (2018)
J.B. Conway, Functions of One Complex Variable II. Graduate Texts in Mathematics, vol. 159 (Springer, Berlin, 1995)
R. Devaney, Complex dynamics and entire functions, in Complex Dynamical Systems (Cincinnati, OH, 1994). Proceedings of Symposia in Applied Mathematics, vol. 49 (American Mathematical Society, Providence, 1994), pp.181–206
A. Douady, J. Hubbard, On the dynamics of polynomial-like mappings. Ann. Sci. École Norm. Sup. (4) 18, 287–343 (1985)
P. DuVal, Elliptic Functions and Elliptic Curves (Cambridge University Press, Cambridge, 1973)
E. Eremenko, M. Lyubich, The dynamics of analytical transformations, Leningrad Math. 1, 563–634 (1990)
A. Freire, A. Lopes, R. Mañé. An invariant measure for rational maps. Bol. Soc. Brasil Mat. 14, 45–62 (1983)
P. Grzegorczyk, F. Przytycki, W. Szlenk, On iterations of Misiurewicz’s rational maps on the Riemann sphere, in Hyperbolic Behaviour of Dynamical Systems (Paris, 1990). Annales Henri Poincaré Physics ThŽor, vol. 53, no. 4 (1990), pp. 431–444
M. Herman, Are there critical points on the boundaries of singular domains? Comm. Math. Phys. 99(4), 593– 612 (1985)
M. Herman, Notes inachevées – sélectionnées par Jean-Christophe Yoccoz, Documents Mathématiques (Paris), vol. 16, (Société Mathématique de France, Paris, 2018)
G. Jones, D. Singerman, Complex Functions: An Algebraic and Geometric Viewpoint (Cambridge University Press, Cambridge, 1997)
L. Koss, Ergodic and Bernoulli Properties of analytic maps of complex projective space. Trans. Amer. Math. Soc. 354(6), 2417–2459 (2002)
F. Ledrappier, Quelques propriétés ergodiques des applications rationnelles. C. R. Acad. Sci. Paris Sér. I Math. 299(1), 37–40 (1984)
M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere. Erg. Th. and Dyn. Sys. 3, 351–385 (1983)
B. Mandelbrot, Fractal aspects of the iteration of z↦λz(1 − z) for complex λ, z. Ann. NY Acad. Sci. 35, 249–259 (1980)
R. Mañé, On the uniqueness of the maximizing measure for rational maps. Bol. Soc. Bras. Math. 14(1), 27–43 (1983)
R. Mañé, On a theorem of Fatou. Bol. Soc. Brasil. Mat. (N.S.) 24, 1–11 (1993)
C. McMullen, Complex Dynamics and Renormalization (Princeton University Press, Princeton, 1994)
J. Milnor, Dynamics in One Complex Variable (Friedrich Vieweg & Sohn, Verlagsgesellschaft mbH, 2000)
J. Milnor, On Lattès maps, in Dynamics on the Riemann sphere: A Bodil Branner Festschrift. (Eur. Math. Soc., Zurich, 2006), pp. 9–43
M. Rees, Positive measure sets of ergodic rational maps. Ann. scient. Ec. Norm. Sup. 19, 383–407 (1986)
M. Shishikura, On the quasiconformal surgery of rational functions. Ann. Sci. École Norm. Sup. 20(1), 1– 29 (1987)
N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems (Walter de Gruyter, Berlin, 1993)
D. Sullivan, Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains. Ann. Math. 122, 401–418 (1985)
M. Taylor, Introduction to Complex Analysis, Graduate Studies in Mathematics,vol. 202 (American Mathematical Society, New York, 2019)
J.-C. Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Petits diviseurs en dimension 1. Astérisque 231, 3–88 (1995)
A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math. 99, 627–649 (1990)
G. Zhang, All bounded type Siegel disks of rational maps are quasi-disks. Invent. Math. 185(2), 421–466 (2011)
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Hawkins, J. (2021). Complex Dynamics. In: Ergodic Dynamics. Graduate Texts in Mathematics, vol 289. Springer, Cham. https://doi.org/10.1007/978-3-030-59242-4_12
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