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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
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.
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.
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 study the dynamics of a general class of antiholomorphic correspondences; i.e., multi-valued maps with antiholomorphic local branches, on the Riemann sphere. Such correspondences are closely related to a class of single-valued antiholomorphic maps in one complex variable; namely, Schwarz reflection maps of simply connected quadrature domains. Using this connection, we prove that matings of all parabolic antiholomorphic rational maps with connected Julia sets (of arbitrary degree) and antiholomorphic analogues of Hecke groups can be realized as such correspondences. We also draw the same conclusion when parabolic maps are replaced with critically non-recurrent antiholomorphic polynomials with connected Julia sets.
| arXiv:2303.02459 |
Let p be a prime number, let g(x)=xp2+pr+2xp2+1 with r∈ℤ≥0, and let ϕ(x)=x+O(x2) be the Böttcher coordinate satisfying ϕ(g(x))=ϕ(x)p2. Salerno and Silverman conjectured that the radius of convergence of ϕ−1(x) in ℂp is p−p−r/(p−1). In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result.
For a post-critically finite hyperbolic rational map f, we show that the Julia set Jf has Ahlfors-regular conformal dimension one if and only if f is a crochet map, i.e., there is an f-invariant graph G containing the post-critical set such that f|G has topological entropy zero. We use finite subdivision rules to obtain graph virtual endomorphisms, which are 1-dimensional simplifications of post-critically finite rational maps, and approximate the asymptotic conformal energies of the graph virtual endomorphisms to estimate the Ahlfors-regular conformal dimensions. In particular, we develop an idea of reducing finite subdivision rules and prove the monotonicity of asymptotic conformal energies under the decomposition of rational maps.
Submitted 27 September, 2022; originally announced September 2022.
Recent works [MMO1, arXiv:1802.03853, arXiv:1802.04423, arXiv:2101.08956] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifolds M of infinite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of M, giving a complete classification of their closures in M. In particular, we show that the behavior is different depending on whether exotic roofs exist or not. Here an exotic roof is a geodesic plane contained in an end E of M, which limits on the convex core boundary ∂E, but cannot be separated from the core by a support plane of ∂E. A necessary condition for the existence of exotic roofs is the existence of exotic rays for the bending lamination. Here an exotic ray is a geodesic ray that has finite intersection number with a measured lamination L but is not asymptotic to any leaf nor eventually disjoint from L. We establish that exotic rays exist if and only if L is not a multicurve. The proof is constructive, and the ideas involved are important in the construction of exotic roofs. We also show that the existence of geodesic rays satisfying a stronger condition than being exotic, phrased in terms of only the hyperbolic surface ∂E and the bending lamination, is sufficient for the existence of exotic roofs. As a result, we show that geometrically finite ends with exotic roofs exist in every genus. Moreover, in genus 1, when the end is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.
Submitted 15 August, 2024; v1 submitted 8 October, 2022; originally announced October 2022.
This paper studies polynomials with core entropy zero. We give several characterizations of polynomials with core entropy zero. In particular, we show that a degree d post-critically finite polynomial f has core entropy zero if and only if f is in the degree d main molecule. The characterizations define several quantities which measure the complexities of polynomials with core entropy zero. We show that these measures are all comparable.
Let K be an algebraically closed and complete nonarchimedean field with characteristic 0 and let f∈K[z] be a polynomial of degree d≥2. We study the Lyapunov exponent L(f,μ) of f with respect to an f-invariant and ergodic Radon probability measure μ on the Berkovich Julia set of f and the lower Lyapunov exponent L−f(f(c)) of f at a critical value f(c). Under an integrability assumption, we show L(f,μ) has a lower bound only depending on d and K. In particular, if f is tame and has no wandering nonclassical Julia points, then L(f,μ) is nonnegative; moreover, if in addition f possesses a unique Julia critical point c0, we show L−f(f(c0)) is also nonnegative.