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Kostiantyn Drach

Publications and source records attributed to Kostiantyn Drach.

At least 19 recordsLinked to original sources
arXiv2026

Rigidity of bounded-type Siegel polynomials

We establish rigidity for a class of higher-degree complex polynomials with irrationally indifferent dynamics. Specifically, we consider non-renormalizable (in the sense of Douady and Hubbard) polynomials of degree $d\geqslant 2$ with a Siegel disk whose rotation number is of bounded type. We call such maps atomic Siegel polynomials of bounded type. Our main results are: (A) The Julia set of every atomic Siegel polynomial of bounded type is locally connected; (B) Every atomic Siegel polynomial of bounded type is quasiconformally rigid; equivalently, its Julia set supports no invariant line fields; (C) Any two combinatorially equivalent atomic Siegel polynomials of bounded type are affinely conjugate. In particular, (C) proves the Combinatorial Rigidity Conjecture for atomic Siegel polynomials of bounded type in arbitrary degree. This extends the higher-degree rigidity theory of Avila--Kahn--Lyubich--Shen and Kozlovski--van Strien to the setting of irrationally indifferent dynamics, a setting not previously covered by Yoccoz-type rigidity results.

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arXiv2026

Transcendental correspondences: when Fuchsian groups take over basins of entire maps

In this paper, we initiate a systematic study of $(\infty : \infty)$ holomorphic correspondences that naturally arise as conformal combinations (matings) of transcendental entire maps with Fuchsian groups. This construction parallels the recent theory of finite-degree algebraic correspondences associated with rational maps. Our correspondence combines the dynamics of a transcendental entire function outside a distinguished attracting/parabolic basin with the action of a compatible Fuchsian group within it. We show that the resulting correspondence is the composition of a Möbius involution and the deleted covering correspondence of a meromorphic function having exactly one simple pole. When the transcendental entire function has finitely many singular values, so does this meromorphic function, and its line complex can be described explicitly.

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arXiv2026

Transversality for Interval Translation Maps

An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ of an interval $I$ into $r \ge 2$ subintervals. In contrast to classical interval exchange transformations (IETs), we do not require that the images of these subintervals are disjoint; in particular, ITMs are not assumed to be bijective. Thus, ITMs provide a natural non-invertible generalisation of IETs. In this paper, we prove a transversality theorem for a family of dynamically defined vector subspaces that encode the dynamics of a given ITM. As a consequence, we establish a perturbation result that gives a precise control of the first return dynamics to subintervals in $I$, while preserving the remaining global dynamics of the system. Beyond their independent interest, these results are a key technical ingredient in the proof of the Characterisation of Stability of ITMs in arXiv:2605.00190, and in the establishment of the topological version of the Boshernitzan--Kornfeld Conjecture in arXiv:2605.00186.

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arXiv2026

Topological Prevalence of Finite Type Interval Translation Maps

An interval translation map (ITM) is a map $T \colon I \to I$ defined as a piecewise translation on a finite partition of an interval $I$ into $r \ge 2$ subintervals. Unlike classical interval exchange transformations (IETs), the images of these subintervals are allowed to overlap, making ITMs a natural generalisation of IETs. An ITM $T$ is said to be \textit{of finite type} if its attractor $\bigcap_{n\ge 0} T^n(I)$ is a finite union of intervals; in this case, restricted to this invariant set, $T$ is bijective and hence behaves like an IET. Otherwise, $T$ is of infinite type. In this paper, for every $r \ge 2$, we prove that the set of finite type ITMs contains an open and dense subset in the space of all possible ITMs on $r$ subintervals. This confirms a topological version of a long-standing conjecture due to Boshernitzan and Kornfeld.

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arXiv2026

Characterisation of Stability for Interval Translation Maps

An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ of an interval $I$ into $r \ge 2$ subintervals. In contrast to classical interval exchange transformations (IETs), we do not require that the images of these subintervals are disjoint; in particular, ITMs are not assumed to be bijective. Thus, ITMs provide a natural non-invertible generalisation of IETs. In this paper, we formulate an appropriate notion of stability for general interval translation mappings and prove a characterisation of stability in terms of two dynamically natural properties called the Absence of Critical Connections and Matching. This result can be viewed as the foundational step towards the stability theory of general ITMs.

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arXiv2026

A reverse isoperimetric inequality in three-dimensional space forms

A $λ$-convex body in a three-dimensional space form $M^3(c)$ of constant curvature $c$ is a compact convex set $K$ whose boundary $\partial K$ has normal curvatures bounded below by a constant $λ>0$ (in a weak sense). Within this class, we prove a sharp reverse isoperimetric inequality: among all $λ$-convex bodies in $M^3(c)$, with a fixed surface area, the body of minimal volume is the $λ$-convex lens, i.e., the domain bounded by two totally umbilical caps of curvature $λ$. Moreover, this minimizer is unique. This result confirms Borisenko's Conjecture in the three-dimensional model spaces of constant curvature for $c\neq 0$, and complements recent progress on the conjecture in the Euclidean case $c=0$. As a by-product, our method also yields an alternative proof of the corresponding reverse isoperimetric inequality in two-dimensional hyperbolic space.

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arXiv2026

Stability of reverse isoperimetric inequalities in the plane: area, Cheeger, and inradius

In this paper, we present sharp stability results for various reverse isoperimetric problems in $\mathbb R^2$. Specifically, we prove the stability of the reverse isoperimetric inequality for $λ$-convex bodies -- convex bodies with the property that each of their boundary points $p$ supports a ball of radius $1/λ$ so that the body lies inside the ball in a neighborhood of $p$. For convex bodies with smooth boundaries, $λ$-convexity is equivalent to having the curvature of the boundary bounded below by $λ> 0$. Additionally, within this class of convex bodies, we establish stability for the reverse inradius inequality and the reverse Cheeger inequality. Even without its stability version, the sharp reverse Cheeger inequality is new in dimension $2$.

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arXiv2025

Unmarked spectral rigidity of expanding circle maps

For a smooth expanding map $f$ of the circle, its (unmarked) length spectrum is defined as the set of logarithms of multipliers of periodic orbits of $f$. This spectrum is analogous to the set of lengths of all closed geodesics on negatively curved surfaces -- the classical length spectrum. In the paper, we prove a length spectral rigidity result for expanding circle maps. Namely, we show that a smooth expanding circle map $f$ of degree $d \ge 2$, under certain assumptions on the sparsity of its length spectrum, cannot be perturbed with an arbitrarily small perturbation (depending on $f$) so that its length spectrum stays the same. The proof uses the Whitney extension theorem, a quantitative Livsic-type theorem, and a novel iterative scheme.

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arXiv2025

A solution to Bezdek's conjecture

For a given $λ>0$, a convex body in $\mathbb R^n$ is $λ$-convex if it is the intersection of (finitely or infinitely many) balls of radius $1/λ$. In this note, we show that among all $λ$-convex bodies in $\mathbb R^n$, $n \geqslant 2$, with a given inradius, the $λ$-convex lens (i.e., the intersection of two balls of radius $1/λ$) has the largest mean width. This gives an affirmative answer to the conjecture of K. Bezdek. Under an additional symmetry assumption on $λ$-convex bodies, we resolve the analogous inradius conjecture of Bezdek for arbitrary intrinsic volumes. We also establish an answer to the corresponding conjecture of K. Bezdek about the circumradius. In particular, we prove that the $λ$-convex spindle (i.e., the intersection of all balls of radius $1/λ$ containing a given pair of points) is the unique minimizer of the mean width among all $λ$-convex bodies with a fixed circumradius.

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arXiv2025

Density of Stable Interval Translation Maps

Assume that the interval $I=[0,1)$ is partitioned into finitely many intervals $I_1,\dots,I_r$ and consider a map $T\colon I\to I$ so that $T_{\vert I_s}$ is a translation for each $1 \le s \le r$. We do not assume that the images of these intervals are disjoint. Such maps are called Interval Translation Maps. Let $ITM(r)$ be the space of all such transformations, where we fix $r$ but not the intervals $I_1,\dots,I_r$, nor the translations. The set $X(T):=\bigcap_{n\ge 0} T^n[0,1)$ can be a finite union of intervals (in which case the map is called of finite type), or is a disjoint union of finitely many intervals and a Cantor set (in which case the map is called of infinite type). In this paper we show that there exists an open and dense subset $\mathcal{S}(r)$ of $ITM(r)$ consisting of stable maps, i.e. each $T\in \mathcal{S}(r)$ is of finite type, the first return map to any component of $X(T)$ corresponds to a circle rotation and $\mathcal{S}(r) \ni T \mapsto X(T)$ is continuous in the Hausdorff topology.

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arXiv2023

Reverse isoperimetric problems under curvature constraints

In this paper we solve several reverse isoperimetric problems in the class of $λ$-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some $λ> 0$. We give an affirmative answer in $\mathbb{R}^3$ to a conjecture due to Borisenko which states that the $λ$-convex lens, i.e., the intersection of two balls of radius $1/λ$, is the unique minimizer of volume among all $λ$-convex bodies of given surface area. Also, we prove a reverse inradius inequality: in model spaces of constant curvature and arbitrary dimension, we show that the $λ$-convex lens (properly defined in non-zero curvature spaces) has the smallest inscribed ball among all $λ$-convex bodies of given surface area. This solves a conjecture due to Bezdek on minimal inradius of isoperimetric ball-polyhedra in $\mathbb{R}^n$.

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arXiv2022

The dynamics of complex box mappings

In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is: -To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues. -To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus we call such complex box mappings dynamically natural. -Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques: (*)the Enhanced Nest by Kozlovski-Shen-van Strien; (*)the Covering Lemma by Kahn-Lyubich; (*)the QC-Criterion, the Spreading Principle. The purpose of this paper is to make these tools more accessible so that they can be used as a 'black box', so one does not have to redo the proofs in new settings. -To give an intuitive, but also rather detailed, outline of the proof of the following results by Kozlovski-van Strien for non-renormalizable dynamically natural box mappings: (*)puzzle pieces shrink to points; (*)topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate. -We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings are the analogues of Lattes maps in this setting. -We prove a version of Mane's Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.

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arXiv2020

Rigidity of Newton dynamics

We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points has locally connected Julia set. In the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable ``in the same way''. Our main tool is the concept of complex box mappings due to Kozlovski, Shen, van Strien; we also extend a dynamical rigidity result for such mappings so as to include irrationally indifferent or renormalizable situations.

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arXiv2020

Puzzles and the Fatou-Shishikura injection for rational Newton maps

We establish a principle that we call the Fatou-Shishikura injection for Newton maps of polynomials: there is a dynamically natural injection from the set of non-repelling periodic orbits of any Newton map to the set of its critical orbits. This injection obviously implies the classical Fatou-Shishikura inequality, but it is stronger in the sense that every non-repelling periodic orbit has its own critical orbit. Moreover, for every Newton map we associate a forward invariant graph (a puzzle) which provides a dynamically defined partition of the Riemann sphere into closed topological disks (puzzle pieces). This puzzle construction is for rational Newton maps what Yoccoz puzzles are for polynomials: it provides the foundation for all kinds of rigidity results of Newton maps beyond our Fatou-Shishikura injection. Moreover, it gives necessary structure for a classification of the postcritically finite maps in the spirit of Thurston theory.

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arXiv2019

A Hyperbolic View of the Seven Circles Theorem

In this note, we will explain the connection between the Seven Circles Theorem and hyperbolic geometry, then prove a stronger result about hyperbolic geometry hexagons which implies the Seven Circles Theorem as a special case.

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arXiv2019

A sausage body is a unique solution for a reverse isoperimetric problem

We consider the class of $λ$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/λ$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius $1/λ$ (a sausage body) is a unique volume minimizer among all $λ$-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve the reverse isoperimetric problem by proving a reverse quermassintegral inequality, the second main result of this paper.

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