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Abstract: We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it is regular. On the other hand, we prove that a planar continuum is a Julia component of some meromorphic function if and only if it has empty interior. We do so by constructing meromorphic functions with wandering continua using approximation theory.
In this paper, we use iterations of skinning maps on Teichmüller spaces to study circle packings. This allows us to develop a renormalization theory for circle packings whose nerves satisfy certain subdivision rules. We characterize when the skinning map has bounded image. Under the corresponding condition, we prove that the renormalization operator is uniformly contracting. This allows us to give complete answers for the existence and moduli problems for such circle packings. The exponential contraction has many consequences. In particular, we prove that homeomorphisms between any two such circle packings are asymptotically conformal.
We prove {\em a priori} bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials fc:z↦z2+c of bounded type. It implies local connectivity of the corresponding Julia sets J(fc) and MLC (local connectivity of the Mandelbrot set $\Mandel$) at the corresponding parameters c. It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg-Khanin-Sinai in the early 1980s.
The goal of this survey is to present intimate interactions between four branches of conformal dynamics: iterations of anti-rational maps, actions of Kleinian reflection groups, dynamics generated by Schwarz reflections in quadrature domains, and algebraic correspondences. We start with several examples of Schwarz reflections as well as algebraic correspondences obtained by matings between anti-rational maps and reflection groups, and examples of Julia set realizations for limit sets of reflection groups (including classical Apollonian-like gaskets). We follow up these examples with dynamical relations between explicit Schwarz reflection parameter spaces and parameter spaces of anti-rational maps and of reflection groups. These are complemented by a number of general results and illustrations of important technical tools, such as David surgery and straightening techniques. We also collect several analytic applications of the above theory
For a hyperbolic polynomial automorphism of C^2 with a disconnected Julia set, and under a mild dissipativity condition, we give a topological description of the components of the Julia set. Namely, there are finitely many "quasi-solenoids" that govern the asymptotic behavior of the orbits of all non-trivial components. This can be viewed as a refined Spectral Decomposition for a hyperbolic map, as well as a two-dimensional version of the (generalized) Branner-Hubbard theory in one-dimensional polynomial dynamics. An important geometric ingredient of the theory is a John-like property of the Julia set in the unstable leaves.
A twisted rational map over a non-archimedean field K is the composition of a rational function over K and a continuous automorphism of K. We explore the dynamics of some twisted rational maps on the Berkovich projective line.
On a Riemann surface, periods of a meromorphic differential along closed loops define a period character from the absolute homology group into the additive group of complex numbers. Fixing the period character in strata of meromorphic differentials defines the isoperiodic foliation where the remaining degrees of freedom are the relative periods between the zeroes of the differential. In strata of meromorphic differentials with exactly two zeroes, leaves have a natural structure of translation surface. In this paper, we give a complete description of the isoperiodic leaves in marked stratum H(1,1,−2) of meromorphic 1-forms with two simple zeroes and a pole of order two on an elliptic curve. For each character, the corresponding leaf is a connected Loch Ness Monster. The translation structures of generic leaves feature a ramified cover of infinite degree over the flat torus defined by the lattice of absolute periods. By comparison, isoperiodic leaves of the unmarked stratum are complex disks endowed with a half-translation structure having infinitely many singular points. Finally, we give a description of the large-scale conformal geometry of the wall-and-chamber decomposition of the leaves.
Submitted 24 January, 2024; v1 submitted 11 May, 2023; originally announced May 2023.
We provide a natural canonical decomposition of postcritically finite rational maps with non-empty Fatou sets based on the topological structure of their Julia sets. The building blocks of this decomposition are maps where all Fatou components are Jordan disks with disjoint closures (Sierpiński maps), as well as those where any two Fatou components can be connected through a countable chain of Fatou components with common boundary points (crochet or Newton-like maps). We provide several alternative characterizations for our decomposition, as well as an algorithm for its effective computation. We also show that postcritically finite rational maps have dynamically natural quotients in which all crochet maps are collapsed to points, while all Sierpiński maps become small spheres; the quotient is a maximal expanding cactoid. The constructions work in the more general setup of Böttcher expanding maps, which are metric models of postcritically finite rational maps.
In a previous paper, we constructed an explicit dynamical correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps on the Riemann sphere. In this paper, we show that their deformation spaces share many striking similarities. We establish an analogue of Thurston's compactness theorem for critically fixed anti-rational maps. We also characterize how deformation spaces interact with each other and study the monodromy representations of the union of all deformation spaces.
We prove uniform a priori bounds for Siegel disks of bounded type that give a uniform control of oscillations of their boundaries in all scales. As a consequence, we construct the Mother Hedgehog for any quadratic polynomial with a neutral periodic point.