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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.
Let ℂv be a characteristic zero algebraically closed field which is complete with respect to a non-Archimedean absolute value. We provide a necessary and sufficient condition for two tame polynomials in ℂv[z] of degree d≥2 to be analytically conjugate on their basin of infinity. In the space of monic centered polynomials, tame polynomials with all their critical points in the basin of infinity form the tame shift locus. We show that a tame map f∈ℂv[z] is in the closure of the tame shift locus if and only if the Fatou set of f coincides with the basin of infinity.
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.
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.
In this paper we initiate the study of birational Kleinian groups, i.e.\ groups of birational transformations of complex projective varieties acting in a free, properly discontinuous and cocompact way on an open set of the variety with respect to the usual topology. We obtain a classification in dimension two.
| arXiv:2103.09350 |
In this paper, we develop a theory on the degenerations of Blaschke products d to study the boundaries of hyperbolic components. We give a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic componentd containing zd. We show the closure d⎯⎯⎯⎯⎯⎯⎯⎯ is not a topological manifold with boundary for d≥4 by constructing self-bumps on its boundary.
In this paper, we study quasi post-critically finite degenerations for rational maps. We construct limits for such degenerations as geometrically finite rational maps on a finite tree of Riemann spheres. We prove the boundedness for such degenerations of hyperbolic rational maps with Sierpinski carpet Julia set and give criteria for the convergence for quasi-Blaschke products d, making progress towards the analogues of Thurston's compactness theorem for acylindrical 3-manifold and the double limit theorem for quasi-Fuchsian groups in complex dynamics. In the appendix, we apply such convergence results to show the existence of certain polynomial matings.
Abstract: We develop a general technique for realising full closed subsets of the complex plane as wandering sets of entire functions. Using this construction, we solve a number of open problems. (1) We construct a counterexample to Eremenko's conjecture, a central problem in transcendental dynamics that asks whether every connected component of the set of escaping points of a transcendental entire function is unbounded. (2) We prove that there is a transcendental entire function for which infinitely many Fatou components share the same boundary. This resolves the long-standing problem whether "Lakes of Wada continua" can arise in complex dynamics, and answers the analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. (3) We answer a question of Rippon concerning the existence of non-escaping points on the boundary of a bounded escaping wandering domain, that is, a wandering Fatou component contained in the escaping set. In fact we show that the set ofsuch points can have positive Lebesgue measure. (4) We give the first example of an entire function having a simply connected Fatou component whose closure has a disconnected complement, answering a question of Boc Thaler. In view of (3), we introduce the concept of "maverick points": points on the boundary of a wandering domain whose accumulation behaviour differs from that of internal points. We prove that the set of such points has harmonic measure zero, but that both escaping and oscillating wandering domains can contain large sets of maverick points.