Dynamical rigidity of transcendental meromorphic functions
This interesting paper is devoted to the study of the dynamics of iterates for transcendental meromorphic functions in the complex plane~\(\mathbb C\). The Fatou set \({\mathcal F}(f)\) of a meromorphic function \(f:{\mathbb C}\to\hat{\mathbb C}\) is the maximal open set on which the iterates \(f^{\circ n}\), \(n\in\mathbb N\), are all well-defined and form a normal family. The Julia set \({\mathcal J}(f):=\hat{\mathbb C}\setminus{\mathcal F}(f)\) is defined as its spherical complement. A meromorphic function is said to be tame, if its postcritical set NEWLINE\[NEWLINEPS(f):=\overline{\bigcup_{n=1}^{+\infty}f^{\circ n}(\text{sing} f^{-1})},NEWLINE\]NEWLINE where \(\text{sing} f^{-1}\) stands for the set of all singularities of~\(f^{-1}\), does not contain \({\mathcal J}(f)\). The main result of the paper, Theorem~5.1, is as follows.NEWLINENEWLINEIf two tame transcendental meromorphic functions \(f,\,g: {\mathbb C}\to\hat{\mathbb C}\) are conjugate on their Julia sets by a bi-Lipschitz homeomorphism NEWLINE\[NEWLINE{H:{\mathcal J}(f)\to{\mathcal J}(g)},\quad {\text{ i.e. }} H(f(z))=g(H(z)) \tag{*} NEWLINE\]NEWLINE for all~\(z\in{\mathcal J}(f)\setminus\{\infty\}\), then \(H\) extends to an affine map \(H:{\mathbb C}\to{\mathbb C}\); \(z\mapsto az+b\), for which~ (*) holds for all~\(z\in \mathbb C\).NEWLINENEWLINEThe methods used in the proof are interesting on their own. Continuing the study made in~[\textit{D.\ Mauldin, F. Przytycki} and the second author, Compos. Math. 129, No. 3, 273--299 (2001; Zbl 1106.37306)], the authors consider conformal iterated function systems (for the definition see~[\textit{D. Mauldin} and the second author, Proc. Lond. Math. Soc., III. Ser. 73, No. 1, 105--154 (1996; Zbl 0852.28005); Graph directed Markov systems. Geometry and dynamics of limit sets. Cambridge: Cambridge University Press (2003; Zbl 1033.37025)]) associated to the so-called nice sets of a tame meromorphic function, see [\textit{J. Rivera-Letelier}, Ergodic Theory Dyn. Syst. 27, No. 2, 595--636 (2007; Zbl 1110.37037)] and~[\textit{N.\,Dobbs}, Math. Proc. Camb. Philos. Soc. 150, No. 1, 157--165 (2011; Zbl 1210.37029)]. They show (Proposition~3.3) that such an iterated function system satisfies the open set condition and is not essentially affine (i.e., roughly speaking, there is no holomorphic change of variable simultaneously making all functions affine). For conformal iterated function systems satisfying these two conditions they prove, in particular, that any bi-Lipschtz conjugacy on the limit sets extends to a conformal map between neighbourhoods of the limit sets (Theorem~4.2). This fact is a cornerstone in the proof of the main result.
scientific article; zbMATH DE number 6071986