# Sergiy MerenkovCity College of New York | CCNY · Department of Mathematics

Sergiy Merenkov

Ph.D.

## About

29

Publications

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254

Citations

Introduction

Current research interests: Fractal geometry and dynamics

Additional affiliations

September 2016 - present

January 2015 - present

August 2013 - August 2016

Education

January 1999 - August 2003

September 1991 - June 1996

## Publications

Publications (29)

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...

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 $\mathcal C^1$-diffeomorphisms of hyperbolic surfaces. Besides the bounded geometry of wandering domains, the assumptions are either anal...

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 produce matings of anti-polynomials and necklace reflection groups, show conformal removability...

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 (Bonk and Merenkov in Ann Math (2) 177:591–643, 2013, Theorem 1.2). We also show that a similar conclusion holds for quasisymmetries of the double of \(S_p\) acro...

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 $\Lambda_H$. We design a surgery that relates $H$ to a rational map $g$ whose Julia set $\mathcal{J}_g$ is (non-qu...

We give a description of the group of all quasisymmetric self-maps of the Julia set of f(z) = z²−1 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 qua...

We prove that every quasisymmetric homeomorphism of a standard square Sierpi\'nski carpet $S_p$, $p\ge 3$ odd, is an isometry. This strengthens and completes earlier work by the authors~\cite[Theo\-rem~1.2]{BM}. We also show that a similar conclusion holds for quasisymmetries of the double of $S_p$ across the outer peripheral circle. Finally, as an...

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 $\xi$ is a quasisymmetric homeomorphism between Sierpi\'nski
carpets that are the Julia sets of postcritically-finite rational maps, then
$\xi$ is the restriction of a M\"obius transformation to the Julia set. This
implies that the group of quasisymmetric homeomorphisms of a Sierpi\'nski
carpet Julia set of a postcritically-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...

Let $G$ and $\tilde G$ be Kleinian groups whose limit sets $S$ and $\tilde
S$, respectively, are homeomorphic to the standard Sierpi\'nski 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...

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 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.

The problem of describing the range of a Bergman space B_2(G) under the
Cauchy transform K for a Jordan domain G was solved by Napalkov (Jr) and
Yulmukhametov. It turned out that K(B_2(G))=B_2^1(C\bar G) if and only if G is
a quasidisk; here B_2^1(C\bar G) is the Dirichlet space of the complement of
\bar G. The description of K(B_2(G)) for an integ...

A relative Schottky set in a planar domain Ω is a subset of Ω obtained by removing from Ω open geometric disks whose closures
are in Ω and are pairwise disjoint. In this paper, we study quasisymmetric and related maps between relative Schottky sets
of measure zero. We prove, in particular, that quasisymmetric maps between such sets in Jordan domain...

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 [Special characters omitted.] . 1 . We give a new proof of a theorem due to P. Doyle. The probl...

We prove that the Hausdorff dimension of the set of three-period orbits in
classical billiards is at most one. Moreover, if the set of three-period orbits
has Hausdorff dimension one, then it has a tangent line at almost every point.

We study a quasisymmetric version of the classical Koebe uniformization
theorem in the context of Ahlfors regular metric surfaces. In particular, we
prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely
connected domain in the standard 2-sphere is quasisymmetrically equivalent to a
circle domain if and only if X is linearly lo...

We call a complement of a union of at least three disjoint (round) open balls
in the unit sphere S^n a Schottky set. We prove that every quasisymmetric
homeomorphism of a Schottky set of spherical measure zero to another Schottky
set is the restriction of a Mobius transformation on S^n. In the other
direction we show that every Schottky set in S^2...

We prove that every quasisymmetric self-homeomorphism of the standard
1/3-Sierpi\'nski carpet $S_3$ is a Euclidean isometry. For carpets in a more
general family, the standard $1/p$-Sierpi\'nski carpets $S_p$, $p\ge 3$ odd, we
show that the groups of quasisymmetric self-maps are finite dihedral. We also
establish that $S_p$ and $S_q$ are quasisymme...

Motivated by questions in geometric group theory we define a quasisymmetric co-Hopfian property for metric spaces and provide
an example of a metric Sierpiński carpet with this property. As an application we obtain a quasi-isometrically co-Hopfian
Gromov hyperbolic space with a Sierpiński carpet boundary at infinity. In addition, we give a complete...

We settle the problem of finding an entire function with three singular values whose Nevanlinna characteristic domi-nates an arbitrarily prescribed function.

We prove the existence of a hyperbolic surface spread over the sphere for which the projection map has all its singular values on the extended real line, and such that the preimage of the extended real line under the projection map is homeomorphic to the square grid in the plane. This answers a question raised by \`E. B. Vinberg.

Consider a simply-connected Riemann surface represented by a Speiser graph. Nevanlinna asked if the type of the surface is determined by the mean excess of the graph: whether mean excess zero implies that the surface is parabolic, and negative mean excess implies that the surface is hyperbolic. Teichmüller gave an example of a hyperbolic simply-con...

If f is a bijection from C^n onto a complex manifold M, which conjugates every holomorphic map in C^n to an endomorphism in M, then we prove that f is necessarily biholomorphic or antibiholomorphic. This extends a result of A. Hinkkanen to higher dimensions. As a corollary, we prove that if there is an epimorphism from the semigroup of all holomorp...

Consider a simply connected Riemann surface represented by a Speiser graph. Nevanlinna asked if the type of the surface is determined by the mean excess of the graph: whether mean excess zero implies that the surface is parabolic and negative mean excess implies that the surface is hyperbolic. Teichmuller gave an example of a hyperbolic simply conn...

For two bounded domains Ω 1 , Ω 2 \Omega _1,\ \Omega _2 in C \mathbb {C} whose semigroups of analytic endomorphisms E ( Ω 1 ) , E ( Ω 2 ) E(\Omega _1), \ E(\Omega _2) are isomorphic with an isomorphism φ : E ( Ω 1 ) → E ( Ω 2 ) \varphi :\ E(\Omega _1)\rightarrow E(\Omega _2) , Eremenko proved in 1993 that there exists a conformal or anticonformal m...

We show that for every non-negative integer d, there exist differential equations w''+Pw=0, where P is a polynomial of degree d, such that some non-trivial solution w has all zeros real. Comment: 20 pages, 5 figures. Some misprints corrected