arXiv subjects
Publications and source records attributed to Dzmitry Dudko.
In this Note, we present recent developments in the Renormalization Theory of quadratic polynomials and discuss their applications, with an emphasis on the MLC conjecture, the problem of local connectivity of the Mandelbrot set, and on its geometric counterparts.
We extend uniform pseudo-Siegel bounds for neutral quadratic polynomials to $\psi^\bullet$-quadratic-like Siegel maps. In this form, the bounds are compatible with the $\psi$-quadratic-like renormalization theory and are easily transferable to various families of rational maps. The main theorem states that the degeneration of a Siegel disk is equidistributed among combinatorial intervals. This provides a precise description of how the $\psi^\bullet$-quadratic-like structure degenerates around the Siegel disk on all geometric scales except on the ``transitional scales'' between two specific combinatorial levels.
We consider certain correspondences on a Riemann surface, and show that they admit a weak form of hyperbolicity: sufficiently long loops get shorter under lifting at a fixed point and closing. In terms of their algebraic encoding by bisets, this translates to contraction of fundamental group elements along sequences arising from iterated lifting. As an application, we show that apart from the usual Latt\`es counterexamples, for any rational map on $\mathbb P^1$ with $4$ post-critical points, there is a finite invariant collection of isotopy classes of curves into which every curve is attracted under iterated lifting. More generally, among graphs of given complexity, there exists a finite invariant collect ion of isotopy classes of graphs into which every graph is attracted. Applied to sufficiently rich graphs, the graph attr actor provides a finite set of topological normal forms for the rational map. We also present a strategy towards proving the same statements for maps with more than $4$ post-critical points.
We prove a priori bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local connectivity of the Mandelbrot set) 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.
We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; b) The Julia sets J_c are also locally connected and have positive area; c) M is self-similar near Siegel parameters of constant type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed in [DLS] as a global transcendental family.
We prove uniform ``pseudo-Siegel'' 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 controlling the postcritical set for any quadratic polynomial with a neutral periodic point and show that this hedgehog has a star-like structure. Pseudo-Siegel bounds imply uniform a priori bounds of the Sector Renormalization, which gives an opportunity to extend Siegel/Pacman Renormalization Theory and Near-Parabolic Renormalization Theory to all near-neutral quadratic polynomials. Various applications beyond quadratic polynomials are also underway.
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 the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of "Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. Then we prove that these periodic points are hyperbolic with one-dimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters.
We consider "Thurston maps": branched self-coverings of the sphere with ultimately periodic critical points, and prove that the Thurston equivalence problem between them (continuous deformation of maps along with their critical orbits) is decidable. More precisely, we consider the action of mapping class groups, by pre- and post-composition, on branched coverings, and encode them algebraically as "mapping class bisets". We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called "small mapping class bisets". We phrase the decision problem of Thurston equivalence between branched self-coverings of the sphere in terms of the conjugacy and centralizer problems in a mapping class biset. Our decomposition results on mapping class bisets reduce these decision problems to small mapping class bisets; they correspond to rational maps, homeomorphisms and maps double covered by a torus endomorphism, and their conjugacy and centralizer problems are solvable respectively in terms of complex analysis, group theory and linear algebra. Branched coverings themselves are also encoded into bisets, with actions of the fundamental groups. We characterize those bisets that arise from branched coverings between topological spheres, and extend this correspondence to maps between spheres with multicurves, whose algebraic counterparts are "sphere trees of bisets". To illustrate the difference between Thurston maps and homeomorphisms, we produce a Thurston map with infinitely generated centralizer --- while centralizers of homeomorphisms are always finitely generated.
Let $\tilde f\colon(S^2,\tilde A)\to(S^2,\tilde A)$ be a Thurston map and let $M(\tilde f)$ be its mapping class biset: isotopy classes rel $\tilde A$ of maps obtained by pre- and post-composing $\tilde f$ by the mapping class group of $(S^2,\tilde A)$. Let $A\subseteq\tilde A$ be an $\tilde f$-invariant subset, and let $f\colon(S^2,A)\to(S^2,A)$ be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group $\mathrm{Mod}(S^2,\tilde A)$ is an iterated extension of $\mathrm{Mod}(S^2,A)$ by fundamental groups of punctured spheres, $M(\tilde f)$ is an iterated extension of $M(f)$ by the dynamical biset of $f$. Thurston equivalence of Thurston maps classically reduces to a conjugacy problem in mapping class bisets. Our short exact sequence of mapping class bisets allows us to reduce in polynomial time the conjugacy problem in $M(\tilde f)$ to that in $M(f)$. In case $\tilde f$ is geometric (either expanding or doubly covered by a hyperbolic torus endomorphism) we show that the dynamical biset $B(f)$ together with a "portrait of bisets" induced by $\tilde A$ is a complete conjugacy invariant of $\tilde f$. Along the way, we give a complete description of bisets of $(2,2,2,2)$-maps as a crossed product of bisets of torus endomorphisms by the cyclic group of order $2$, and we show that non-cyclic orbisphere bisets have no automorphism. We finally give explicit, efficient algorithms that solve the conjugacy and centralizer problems for bisets of expanding or torus maps.
This is the announcement, and the long summary, of a series of articles on the algorithmic study of Thurston maps. We describe branched coverings of the sphere in terms of group-theoretical objects called bisets, and develop a theory of decompositions of bisets. We introduce a canonical "Levy" decomposition of an arbitrary Thurston map into homeomorphisms, metrically-expanding maps and maps doubly covered by torus endomorphisms. The homeomorphisms decompose themselves into finite-order and pseudo-Anosov maps, and the expanding maps decompose themselves into rational maps. As an outcome, we prove that it is decidable when two Thurston maps are equivalent. We also show that the decompositions above are computable, both in theory and in practice.
Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as "Levy-free" maps and as maps with "contracting biset". We prove that every Thurston map decomposes along a unique minimal multicurve into Levy-free and finite-order pieces, and this decomposition is algorithmically computable. Each of these pieces admits a geometric structure. We apply these results to matings of post-critically finite polynomials, extending a criterion by Mary Rees and Tan Lei: they are expanding if and only if they do not admit a cycle of periodic rays.
We give a combinatorial definition of "core entropy" for quadratic polynomials as the growth exponent of the number of certain precritical points in the Julia set (those that separate the $α$ fixed point from its negative). This notion extends known definitions that work in cases when the polynomial is postcritically finite or when the topology of the Julia set has good properties, and it applies to all quadratic polynomials in the Mandelbrot set. We prove that core entropy is continuous as a function of the complex parameter. In fact, we model the Julia set as an invariant quadratic lamination in the sense of Thurston: this depends on the external angle of a parameter in the boundary of the Mandelbrot set, and one can define core entropy directly from the angle in combinatorial terms. As such, core entropy is continuous as a function of the external angle. Moreover, we prove a conjecture of Giulio Tiozzo about local and global maxima of core entropy as a function of external angles: local maxima are exactly dyadic angles, and the unique global maximum within any wake occurs at the dyadic angle of lowest denominator. We also describe where local minima occur. An appendix by Wolf Jung relates different concepts of core entropy and biaccessibility dimension and thus shows that biaccessibility dimension is continuous as well.
We develop a general theory of "bisets": sets with two commuting group actions. They naturally encode topological correspondences. Just as van Kampen's theorem decomposes into a graph of groups the fundamental group of a space given with a cover, we prove analogously that the biset of a correspondence decomposes into a "graph of bisets": a graph with bisets at its vertices, given with some natural maps. The "fundamental biset" of the graph of bisets recovers the original biset. We apply these results to decompose the biset of a Thurston map (a branched self-covering of the sphere whose critical points have finite orbits) into a graph of bisets. This graph closely parallels the theory of Hubbard trees. This is the first part of a series of five articles, whose main goal is to prove algorithmic decidability of combinatorial equivalence of Thurston maps.
We give a topological description of the space of quadratic rational maps with superattractive two-cycles: its "non-escape locus" M2 (the analog of the Mandelbrot set M) is locally connected, it is the continuous image of M under a canonical map, and it can be described as M (minus the 1/2-limb), mated with the lamination of the basilica. The latter statement is a refined version of a conjecture of Ben Wittner, which in its original version requires local connectivity of M to even be stated. Our methods of mating with a lamination also apply to dynamical matings of certain non-locally connected Julia sets.
We prove that for every hyperbolic component of the Mandelbrot set, any two limbs with equal denominators are homeomorphic so that the homeomorphism preserves periods of hyperbolic components. This settles a conjecture on the Mandelbrot set that goes back to 1994.
We prove the decoration theorem for the Mandelbrot set (and Multibrot sets) which says that when a "little Mandelbrot set" is removed from the Mandelbrot set, then most of the resulting connected components have small diameters.