Distortion coordinates for simply connected wandering domains: holomorphic dependence beyond J-stability and parametrization beyond attracting Herman-type domains

For a transcendental entire function with a simply connected wandering domain, Ferreira and van Strien (Math. Ann. 392, 2025, 701-732) introduced distortion sequences, a wandering-domain analogue of the multiplier of a periodic orbit, and proved two partial results: a distortion map depending holomorphically on parameters exists whenever the Julia set moves holomorphically in the family (a J-stability hypothesis), and distortion data can genuinely parametrize a natural family, but only for functions with an attracting Herman-type wandering domain. The open problem is to extend both halves of this multiplier-like picture to their natural generality: establish (or rule out) analytic dependence of distortion data in families where J-stability fails or no conjugating holomorphic motion is available, and parametrize natural families by distortion coordinates for broader classes of entire functions with simply connected wandering domains, characterizing the hypotheses under which this is possible.

Background

The iteration of a transcendental entire function $f:\mathbb{C}\to\mathbb{C}$ splits the plane into the Fatou set, where the family of iterates $(f^n)$ is normal (equicontinuous on compact sets, i.e. dynamically tame), and its complement the Julia set, the locus of chaotic dynamics. Connected components of the Fatou set are Fatou components; a component that is never mapped back onto itself or another earlier component is called a wandering domain. Wandering domains do not exist for rational maps (Sullivan's no-wandering-domains theorem) but do exist for transcendental entire functions, and their internal dynamics lack the anchor that periodic components have: they admit no periodic orbit in their closure, so there is no multiplier to organize the dynamics. Benini, Evdoridou, Fagella, Rippon and Stallard classified simply connected wandering domains into contracting, semi-contracting and eventually isometric types, according to the behavior of hyperbolic distances $d_{U_n}(f^n(z),f^n(w))$, where $d_{U_n}$ is the hyperbolic (Poincaré) distance in $U_n=f^n(U)$; the quantitative engine of that classification is the hyperbolic distortion $\|Df\|_{U_{n-1}}^{U_n}$, the operator norm of the derivative measured in the hyperbolic metrics of domain and image.

To discuss how dynamics change with parameters, one needs parameter spaces. The singular values of $f$ (critical values, i.e. images of the critical points where $f'=0$, and asymptotic values, i.e. limits of $f(z)$ along curves escaping to infinity, together with their accumulation points) control the possible Fatou dynamics; $f$ is of finite type if $S(f)$ is finite. Eremenko and Lyubich showed that for finite-type $f$ the quasiconformal equivalence class $M_f=\{\psi\circ f\circ\varphi^{-1}\}$, where $\psi,\varphi$ range over quasiconformal homeomorphisms of $\mathbb{C}$ (homeomorphisms whose Beltrami coefficient $\partial_{\bar z}\varphi/\partial_z\varphi$ is essentially bounded), is a complex manifold of dimension $\#S(f)+2$. A natural family is a subfamily $(f_\lambda=\psi_\lambda\circ f\circ\varphi_\lambda^{-1})_{\lambda\in M}$ in which the quasiconformal conjugacies depend holomorphically on the parameter $\lambda$ in a Banach analytic manifold (a manifold modelled on complex Banach spaces). Ferreira and van Strien extended the Eremenko-Lyubich picture: if $f$ has at least two singular values and $\mathbb{C}\setminus S(f)$ is non-empty, there is a Banach analytic manifold $T_f=T(F)\times(\mathbb{C}^*\times\mathbb{C})^2$ (with $T(F)$ the Teichmüller space, i.e. the space of conformal structures on the complement of $F=S(f)\cup\{\infty\}$ modulo isotopy) and a covering map $\Phi:T_f\to M_f$ through which every natural family containing $f$ lifts holomorphically. Holomorphic motions - families $H(\lambda,z)$, holomorphic in $\lambda$ and injective in $z$, moving a given set pointwise - are the standard bridge between quasiconformal surgery and parameter dependence; a family is J-stable, in the sense used there, when $J(f_\lambda)$ moves holomorphically and the motion conjugates the dynamics on the Julia set.

For a periodic Fatou component, the multiplier $\lambda\mapsto(f_\lambda^p)'(z(\lambda))$ of the attracting/parabolic/Siegel periodic point moves holomorphically and can even serve as a coordinate on the family, a principle underlying the parameter-plane theory of rational dynamics (Mandelbrot-like components parametrized by multipliers). Ferreira and van Strien asked the analogous questions for a simply connected wandering domain $U$ of $f=f_{\lambda_0}$: (1) is there a multiplier-like object attached to $U$ that moves holomorphically with $\lambda$; (2) can it be used to parameterize the family. Their candidate is the distortion sequence: fixing $z_0\in U$ and Riemann maps (conformal bijections) $\psi_n:\mathbb{D}\to U_n$ with $\psi_n(0)=z_n=f^n(z_0)$, the holomorphic self-maps $g_n=\psi_n^{-1}\circ f\circ\psi_{n-1}:\mathbb{D}\to\mathbb{D}$ fix $0$, and $(\alpha_n)_{n\geq 1}=(g_n'(0))_{n\geq 1}\in\ell^\infty$ (the Banach space of bounded sequences) is a distortion sequence of $f$ at $z_0$; its entries are, up to the normalization of the hyperbolic metric, the hyperbolic distortions of the maps $f:U_{n-1}\to U_n$ along the orbit, and their moduli are independent of choices. Their Theorem B assumes J-stability and produces, near $\lambda_0$, a holomorphic map $A:\Lambda'\times\overline{\Lambda'}\to\ell^\infty$ (a distortion map) with $A(\lambda,\lambda)$ a distortion sequence of $f_\lambda$, built by extending the holomorphic motion of the Julia set to the orbit of a point of $U$ while preserving the dynamics, and then controlling the parameter dependence of the Riemann maps of the moving domains $U_{\lambda,n}$ via the Ahlfors-Bers theorem (the quasiconformal maps solving the Beltrami equation depend holomorphically on their Beltrami coefficients). Their Theorem C addresses question (2) only in a special case: for $f$ with an attracting Herman-type wandering domain - meaning $f$ lifts a holomorphic self-map $g$ of $\mathbb{C}^*$ with $\exp\circ f=g\circ\exp$ and $\exp(U)$ an attracting basin of $g$ (parabolic and Siegel Herman-type are defined analogously, with $\exp(U)$ a parabolic domain or a Siegel disc of $g$); the existence of such a lift is equivalent to $f(z)=kz+h(e^z)$ for some nonzero integer $k$ and entire function $h$, a condition on the lift alone that does not by itself make $\exp(U)$ attracting - they construct a natural family over the unit ball of $\ell^\infty$ in which the distortion map is non-constant, indeed $\partial\alpha_n/\partial\lambda_n(0)\neq 0$, via quasiconformal surgery (cutting the map apart and regluing modified pieces with quasiconformal interpolants) on Koenigs linearizing coordinates, i.e. the conjugacy of an attracting map to its linear part near its fixed point, of the attracting dynamics. The paper itself flags what is missing: the constructions route through the J-stability hypothesis and through Herman-type lifts, and it remarks that the dependence of Riemann maps on a holomorphic motion of their domain is a deep and only partially understood problem (see also Pommerenke-Rodin and Zakeri).

Problem Statement

Let $f$ be a transcendental entire function with a simply connected wandering domain $U$, and let $(f_\lambda)_{\lambda\in M}$ be a natural family of entire functions over a Banach analytic manifold with $f=f_{\lambda_0}$, in the sense of Ferreira-van Strien. The overarching objective is to establish the distortion sequence as a full multiplier-like invariant for the wandering domain: an object that (i) depends analytically on the parameter, and (ii) serves as a coordinate system on natural families, thus extending to wandering domains the role the multiplier map plays for periodic orbits in rational dynamics. Two linked directions are open.

(1) Dependence beyond J-stability. Theorem B of Ferreira-van Strien yields a holomorphic distortion map $A:\Lambda'\times\overline{\Lambda'}\to\ell^\infty$ (and hence a real-analytically moving distortion sequence) under the hypothesis that $J(f_\lambda)$ moves holomorphically near $\lambda_0$ with a motion conjugating the dynamics on the Julia set. Establish, or rule out, analytic (holomorphic or real-analytic) dependence of distortion data in families where this hypothesis fails: where the Julia set does not move holomorphically, or where no global conjugating holomorphic motion of the Julia set and the wandering-domain orbit is available. A satisfactory answer should identify natural hypotheses on the family under which some canonically defined distortion data of the wandering domain still move analytically with the parameter, or exhibit a natural family in which the distortion data, under any canonical definition, fail to move analytically.

(2) Parametrization beyond attracting Herman-type domains. Theorem C shows, for $f$ with an attracting Herman-type wandering domain, a natural family over the unit ball of $\ell^\infty$ whose distortion map is non-constant, with $\partial\alpha_n/\partial\lambda_n(0)\neq0$ for the constructed family. Extend this parametrization phenomenon to broader classes of entire functions with simply connected wandering domains - parabolic and Siegel Herman-type wandering domains, and general (non-Herman-type) simply connected wandering domains - and, for each of these classes, characterize under which hypotheses on $f$ and on the family the distortion map gives coordinates on the family (local or global coordinates both count); this part is resolved only by covering all of the enumerated classes. If distortion data are insufficient as coordinates for some class, a complete answer should identify what they fail to capture (for example, shape data of the successive domains, in the spirit of injectivity radii) and prove the resulting non-parametrizability.

A definitive resolution of the stated objective requires answers to both parts (1) and (2), since the two parts are the analytic and the parametric halves of the same multiplier-like picture; an answer to either part alone constitutes partial progress toward the objective. Formulations must respect the known ambiguity that distortion maps are unique only up to the non-uniqueness of holomorphic-motion extensions (two distortion maps agree in the moduli of their entries), and should be stated for simply connected wandering domains, where the distortion sequence is defined.

References

1. G. R. Ferreira and S. van Strien, Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains, Mathematische Annalen 392 (2025) 701-732, DOI 10.1007/s00208-025-03107-8, preprint arXiv:2307.00679, https://arxiv.org/abs/2307.00679
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