arXiv subjects
Publications and source records attributed to Sergei Merenkov.
We prove that there is no local quasiconformal map between the limit set of a convex-cocompact Kleinian group and the Julia set of a postcritically finite rational map, provided that both are Sierpiński carpets. This contrasts with the recent results by Y. Luo, D. Ntalampekos and Y. Luo, M. Mj, S. Mukherjee for gasket and tree-like spaces, respectively.
We prove that if $f$ and $g$ are postcritically finite rational maps whose Julia sets $\mathcal{J}(f), \mathcal{J}(g)$, respectively, are Sierpiński carpets, and if $ξ$ is a quasiregular map of the Riemann sphere $\widehat{\mathbb{C}}$ with $ξ^{-1}(\mathcal{J}(g))=\mathcal{J}(f)$, then $ξ$ is the restriction of a rational map to the Julia set $\mathcal{J}(f)$. Moreover, when $g=f$ we prove that, for some positive integers $k$ and $l$, $f^k\circ ξ^l=f^{2k}$. These conclusions extend the main results of M. Bonk, M. Lyubich, S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, Adv. Math, 301 (2016), 383-422. Finally, we demonstrate that when Julia sets of postcritically finite rational maps are not Sierpiński carpets, say they are tree-like or gaskets, the above conclusions no longer hold.
We provide a David extension result for circle homeomorphisms conjugating two dynamical systems such that parabolic periodic points go to parabolic periodic points, but hyperbolic points can go to parabolics as well. We use this result, in particular, to prove the existence of a new class of welding homeomorphisms, to establish an explicit dynamical connection between critically fixed anti-rational maps and kissing reflection groups, to show conformal removability of the Julia sets of geometrically finite polynomials and of the limit sets of necklace reflection groups, to produce matings of anti-polynomials and necklace reflection groups, and to give a new proof of the existence of Suffridge polynomials (extremal points in certain spaces of univalent maps).
A question of F. Kwakkel and V. Markovic on existence of C^1-diffeomorphisms of closed surfaces that permute a dense collection of domains with bounded geometry is answered in the negative. In fact, it is proved that for closed surfaces of genus at least one such diffeomorphisms do not exist regardless of whether they have positive or zero topological entropy.
We study Carathéodory convergence for open, simply connected surfaces spread over the sphere and, in particular, provide examples demonstrating that in the Speiser class the conformal type can change when two singular values collide.
According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group $H$ whose limit set is an Apollonian-like gasket $Λ_H$. We design a surgery that relates $H$ to a rational map $g$ whose Julia set $\mathcal{J}_g$ is (non-quasiconformally) homeomorphic to $Λ_H$. We show for a large class of triangulations, however, the groups of quasisymmetries of $Λ_H$ and $\mathcal{J}_g$ are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of $H$, this group is equal to the group of Möbius symmetries of $Λ_H$, which is the semi-direct product of $H$ itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when $Λ_ H$ is the classical Apollonian gasket), we give a piecewise affine model for the above actions which is quasiconformally equivalent to $g$ and produces $H$ by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle.
We extend the results of T. Giordano, I. F. Putnam, C. F. Skau contained in ``$\mathbb Z^d$-odometers and cohomology", Groups Geom. Dyn. 13 (2019), no. 3, P. 909-938, on characterization of conjugacy, isomorphism, and continuous orbit equivalence of $\mathbb Z^d$-odometers to dimensions $d>2$. We then apply these extensions to the case of odometers defined by matrices with integer coefficients.
A question whether sufficiently regular manifold automorphisms may have wandering domains with controlled geometry is answered in the negative for quasiconformal or smooth homeomorphisms of $n$-tori, $n\ge2$, and hyperbolic surfaces. Besides control on geometry of wandering domains, the assumptions are either analytic, e.g., minimal sets having measure zero or supporting invariant conformal structures, or geometric, such as uniform relative separation of wandering domains.
We give a description of the group of all quasisymmetric self-maps of the Julia set of $f(z)=z^2-1$ that have orientation preserving homeomorphic extensions to the whole plane. More precisely, we prove that this group is the uniform closure of the group generated by the Thompson group of the unit circle and an inversion. Moreover, this result is quantitative in the sense that distortions of the approximating maps are uniformly controlled by the distortion of the given map.
We prove that every quasisymmetric homeomorphism of a standard square Sierpiński carpet $S_p$, $p\ge 3$ odd, is an isometry. This strengthens and completes earlier work by the authors. We also show that a similar conclusion holds for quasisymmetries of the double of $S_p$ across the outer peripheral circle. Finally, as an application of the techniques developed in this paper, we prove that no standard square carpet $S_p$ is quasisymmetrically equivalent to the Julia set of a postcritically-finite rational map.
We prove that if $n\geq 2$, then there is no $C^1$-diffeomorphism $f$ of $n$-torus, such that $f$ is semi-conjugate to a minimal translation and its wandering domains are geometric balls. This improves a recent result of A. Navas, who proved it assuming $C^{n+1}$ regularity of $f$.
We prove that if $ξ$ is a quasisymmetric homeomorphism between Sierpiński carpets that are the Julia sets of postcritically-finite rational maps, then $ξ$ is the restriction of a Möbius transformation to the Julia set. This implies that the group of quasisymmetric homeomorphisms of a Sierpiński carpet Julia set of a postcritically-finite rational map is finite.
We extend fundamental results concerning Apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and geometric group theory. In particular, we give an analogue of D. W. Boyd's theorem (relating the curvature distribution function of an Apollonian packing to its exponent and the Hausdorff dimension of the residual set) for Sierpiński carpets that are Julia sets of hyperbolic rational maps.
Let $G$ and $\tilde G$ be Kleinian groups whose limit sets $S$ and $\tilde S$, respectively, are homeomorphic to the standard Sierpiński carpet, and such that every complementary component of each of $S$ and $\tilde S$ is a round disc. We assume that the groups $G$ and $\tilde G$ act cocompactly on triples on their respective limit sets. The main theorem of the paper states that any quasiregular map (in a suitably defined sense) from an open connected subset of $S$ to $\tilde S$ is the restriction of a Möbius transformation that takes $S$ onto $\tilde S$, in particular it has no branching. This theorem applies to the fundamental groups of compact hyperbolic 3-manifolds with non-empty totally geodesic boundaries. One consequence of the main theorem is the following result. Assume that $G$ is a torsion-free hyperbolic group whose boundary at infinity $\dee_\infty G$ is a Sierpiński carpet that embeds quasisymmetrically into the standard 2-sphere. Then there exists a group $H$ that contains $G$ as a finite index subgroup and such that any quasisymmetric map $f$ between open connected subsets of $\dee_\infty G$ is the restriction of the induced boundary map of an element $h\in H$.
We settle two problems of reconstructing a biholomorphic type of a manifold. In the first problem we use graphs associated to Riemann surfaces of a particular class. In the second one we use the semigroup structure of analytic endomorphisms of domains in $\C^n$.
We introduce Schottky maps-conformal maps between relative Schottky sets, and study their local rigidity properties. This continues the investigations of relative Schottky sets initiated in [S. Merenkov, "Planar relative Schottky sets and quasisymmetric maps", Proc. London Math. Soc. (3) 104 (2012), 455-485]. Besides being of independent interest, the latter and current works provide key ingredients in the forthcoming proof of quasisymmetric rigidity of Sierpiński carpet Julia sets of rational functions.
The range of the Bergman space B_2(G) under the Cauchy transform K is described for a large class of domains. For a quasidisk G the relation K(B_2^*(G))=B_2^1(\mathbb C\setminus\bar{G}) is proved.
For two bounded domains in the complex plane whose semigroups of analytic endomorphisms are isomorphic, Eremenko proved in 1993 that the isomorphism is given as a conjugation by a conformal or anticonformal map. In the present paper we prove an analogue of this result for the case of bounded domains in \mathbb C^n.