Multiply connected wandering domains of entire functions (Q2872025)

It is a classical result that for rational functions of the Riemann sphere, all the Fatou components are eventually periodic. Wandering domains in the Fatou set is a specific feature of the dynamics of transcendental functions. Baker gave the first example of an entire function having a wandering domain, and in his example, the corresponding domain is multiply connected [\textit{I. N. Baker}, Math. Z. 81, 206--214 (1963; Zbl 0107.06203); J. Aust. Math. Soc., Ser. A 22, 173--176 (1976; Zbl 0335.30001)].NEWLINENEWLINENEWLINEThis article is about the dynamical, topological and geometric properties of multiply connected wandering domains. \textit{I. N. Baker} was one of the earliest to study such domains and gave many of their properties [Ann. Acad. Sci. Fenn., Ser. A I, Math. 1, 277--283 (1975; Zbl 0329.30019); Proc. Lond. Math. Soc. (3) 49, 563--576 (1984; Zbl 0523.30017)]. This article contains a lot new results about them, which together give quite a detailed picture.NEWLINENEWLINENEWLINEA first important result (Theorem 1.1) is the existence of a harmonic function \(h\), defined in terms of the relative rate of divergence with respect to some arbitrary base point \(z_0\) in the domain: \(h(z)=\lim \frac{\log|f^n(z)|}{\log|f^n(z_0)|}\). They use this to extend a previous result by \textit{J.-H. Zheng} [J. Math. Anal. Appl. 313, No. 1, 24--37 (2006; Zbl 1087.30023)] giving a quantitative description of a sequence of maximal annuli (Theorem 1.2) which acts as an absorbing set for the dynamics (Theorem 1.3).NEWLINENEWLINENEWLINESuch an annulus is of the form \(\{z:| f^n(z_0)|^{a_n}<| z| <| f^n(z_0)|^{b_n}\}\), where \(a_n\) and \(b_n\) are sequences of positive numbers whose limits \(a\) and \(b\) belong respectively to \([0,1)\) and \((1,\infty]\) (Theorem 1.5). Questions about the possible values of the limits give raise to new examples of multiply connected wandering domains with specific limit values. Also, it is possible to express the value of \(h\) in terms of the values of \(a\) and \(b\) and the harmonic measure of the outer boundary of the domain, that is the boundary of the unbounded component of the complement (Theorem 1.6). The function properties of the mapping \(f\) on such inner annuli are also described in Theorems 5.1 and 5.2.NEWLINENEWLINENEWLINE\textit{M. Kisaka} and \textit{M. Shishikura} [Lond. Math. Soc. Lect. Note Ser. 348, 217--250 (2008; Zbl 1167.30011)] have studied the eventual connectivity of multiply connected wandering domains, that is the eventual number of connected components in the complement of the images of the domain, and showed that it should be either 2 or \(\infty\). Moreover there is no critical point in the orbit of any wandering Fatou component whose connectivity is 2.NEWLINENEWLINENEWLINEThe authors of the article greatly extend these results (Theorem 1.7): a multiply connected wandering component has connectivity 2 if and only if there is no critical point in its forward orbit; it has a finite connectivity only if there are at most finitely many such critical point (and there is at least one when the connectivity is greater than 2), which means that the infinite connectivity case is equivalent to having infinitely many critical points in the orbit of the domain. They also transpose those results when one considers the outer and inner connectivity of the domain (Theorem 8.1).NEWLINENEWLINENEWLINETheorem 1.9 is a stability result in the space of entire functions: if the perturbation \(g\) is small enough, then the function \(f+g\) has a multiply connected wandering domain as soon as \(f\) has one. This criterion stands, e.g., when \(g\) is a polynomial.NEWLINENEWLINENEWLINEIt is noteworthy that most of these results can be extended to meromorphic functions with a direct tract, although the authors prefer to give the results for entire functions only.NEWLINENEWLINENEWLINEThis work gives a wealth of properties about multiply connected wandering domains and extends and completes many previous results.