arXiv subjects

Oleg Ivrii

Publications and source records attributed to Oleg Ivrii.

At least 19 recordsLinked to original sources

arXiv2026

Critical structures of inner functions II

We study the correspondence proposed in \cite{critical-structures} between inner functions modulo post-compositions with automorphisms of the unit disk and cyclic subspaces of the weighted Bergman space $A^2_1$. The correspondence sends an inner function $I$ to the invariant subspace $[I']$, and in the opposite direction, assigns to a non-zero function $H \in A^2_1$ the Liouville map $I_H$ associated to the canonical solution of the Gauss curvature equation $\Delta u = |H|^2 e^{2u}$. We prove that $I'_H$ generates the same cyclic subspace as $H$. Combined with the results in \cite{critical-structures}, this shows that these two mappings are inverses of one another, and hence the correspondence $I \to [I']$ is a bijection.

math.CV↗

arXiv2026

Dynamics of simply parabolic inner functions

We study the dynamics of Polya-Szeg\"o inner functions and discuss some of their basic properties such as equivalent conditions for simple and double parabolicity. We show that a simply parabolic Polya-Szeg\"o inner function admits forward and backward quotient half-cylinders, which allows one to enrich its dynamics with a Lavaurs map. To proceed, we restrict our attention to simply parabolic inner functions with finite Lyapunov exponent: $\int_{\mathbb{R}} \log |F'| d\ell < \infty$. We define a geodesic flow on the Riemann surface lamination associated to the Lavaurs semigroup and show that it is ergodic. As an application, we establish the Orbit Counting Theorem up to a Ces\`aro average for Lavaurs semigroups. If we additionally assume that $F$ is a parabolic one component inner function, then the geodesic flow is mixing and the full Orbit Counting Theorem holds.

math.DS↗

arXiv2025

Inner Functions, Composition Operators, Symbolic Dynamics and Thermodynamic Formalism

In this paper, we use thermodynamic formalism to study the dynamics of inner functions $F$ acting on the unit disk. If the Denjoy-Wolff point of $F$ is in the open unit disk, then without loss of generality, we can assume that $F(0) = 0$ so that 0 is an attracting fixed point of $F$ and the Lebesgue measure on the unit circle is invariant under $F$. Utilizing the connection between composition operators, Aleksandrov-Clark measures and Perron-Frobenius operators, we develop a rudimentary thermodynamic formalism which allows us to prove the Central Limit Theorem and the Law of Iterated Logarithm for Sobolev multipliers and Hölder continuous observables. Under the more restrictive, but natural hypothesis that $F$ is a one component inner function, we develop a more complete thermodynamic formalism which is sufficient for orbit counting, assuming only the $(1+\varepsilon)$ integrability of $\log|F'|$. As one component inner functions admit countable Markov partitions of the unit circle, we may work in the abstract symbolic setting of countable alphabet subshifts of finite type. Due to the very weak hypotheses on the potential, we need to pay close attention to the regularity of the complex Perron-Frobenius operators $\mathcal L_s$ with $\text{Re }s > 1$ near the boundary. Finally, we discuss inner functions with a Denjoy-Wolff point on the unit circle. We assume a parabolic type behavior of $F$ around this point and we introduce the class of parabolic one component inner functions. By making use of the first return map, we deduce various stochastic laws and orbit counting results from the aforementioned abstract symbolic results.

math.DS↗

arXiv2025

Analytic mappings of the unit disk with bounded compression

In this paper, we study analytic self-maps of the unit disk for which the hyperbolic diameters of the images of hyperbolic balls of radius 1 are uniformly bounded below. We give several characterizations of such maps involving the behaviour along geodesic rays, Aleksandrov-Clark measures, zero sets and critical sets.

math.CV↗

arXiv2024

Analytic mappings of the unit disk which almost preserve hyperbolic area

In this paper, we study analytic self-maps of the unit disk which distort hyperbolic area of large hyperbolic disks by a bounded amount. We give a number of characterizations involving angular derivatives, Lipschitz extensions, Möbius distortion, the distribution of critical points and Aleksandrov-Clark measures. We also study Lyapunov exponents of their Aleksandrov-Clark measures.

math.CV↗

arXiv2024

Inner Functions and Laminations

In this paper, we study orbit counting problems for inner functions using geodesic and horocyclic flows on Riemann surface laminations. For a one component inner function of finite Lyapunov exponent with $F(0) = 0$, other than $z \to z^d$, we show that the number of pre-images of a point $z \in \mathbb{D} \setminus \{ 0\}$ that lie in a ball of hyperbolic radius $R$ centered at the origin satisfies $$ \mathcal{N}(z, R) \, \sim \, \frac{1}{2} \log \frac{1}{|z|} \cdot \frac{1}{\int_{\partial \mathbb{D}} \log |F'| dm}, \quad \text{as }R \to \infty. $$ For a general inner function of finite Lyapunov exponent, we show that the above formula holds up to a Cesàro average. Our main insight is that iteration along almost every inverse orbit is asymptotically linear. We also prove analogues of these results for parabolic inner functions of infinite height.

math.DS↗

arXiv2023

Shapes of infinite conformally balanced trees

Numerical experiments by Werness, Lee and the third author suggested that dessin d'enfants associated to large trivalent trees approximate the developed deltoid introduced by Lee, Lyubich, Makarov and Mukherjee. In this paper, we confirm this conjecture. As a side product of our techniques, we give a new proof of a theorem of Bishop which says that ``true trees are dense.'' We also exhibit a sequence of trees whose conformally natural shapes converge to the cauliflower, the Julia set of $z\mapsto z^2+1/4$.

math.CV↗

arXiv2023

Critical values of inner functions

Let $\mathscr J$ be the space of inner functions of finite entropy endowed with the topology of stable convergence. We prove that an inner function $F \in \mathscr J$ possesses a radial limit (and in fact, a minimal fine limit) in the unit disk at $σ(F')$ a.e. point on the unit circle. We use this to show that the singular value measure $ν(F) = \sum_{c \in \text{crit } F} (1-|c|) \cdot δ_{F(c)} + F_*(σ(F'))$ varies continuously in $F$. Our analysis involves a surprising connection between Beurling-Carleson sets and angular derivatives.

math.CV↗

arXiv2023

Beurling-Carleson sets, inner functions and a semi-linear equation

Beurling-Carleson sets have appeared in a number of areas of complex analysis such as boundary zero sets of analytic functions, inner functions with derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces, Fuchsian groups of Widom-type and the corona problem in quotient Banach algebras. After surveying these developments, we give a general definition of Beurling-Carleson sets and discuss some of their basic properties. We show that the Roberts decomposition characterizes measures that do not charge Beurling-Carleson sets. For a positive singular measure $μ$ on the unit circle, let $S_μ$ denote the singular inner function with singular measure $μ$. In the second part of the paper, we use a corona-type decomposition to relate a number of properties of singular measures on the unit circle such as membership of $S'_μ$ in the Nevanlinna class $\mathcal N$, area conditions on level sets of $S_μ$ and wepability. It was known that each of these properties holds for measures concentrated on Beurling-Carleson sets. We show that each of these properties implies that $μ$ lives on a countable union of Beurling-Carleson sets. We also describe partial relations involving the membership of $S'_μ$ in the Hardy space $H^p$, membership of $S_μ$ in the Besov space $B^p$ and $(1-p)$-Beurling-Carleson sets and give a number of examples which show that our results are optimal. Finally, we show that measures that live on countable unions of $α$-Beurling-Carleson sets are almost in bijection with nearly-maximal solutions of $Δu = u^p \cdot χ_{u > 0}$ when $p > 3$ and $α= \frac{p-3}{p-1}$.

math.CV↗

arXiv2022

On meromorphic functions whose image has finite spherical area

In this paper, we study meromorphic functions on a domain $Ω\subset \mathbb{C}$ whose image has finite spherical area, counted with multiplicity. The paper is composed of two parts. In the first part, we show that the limit of a sequence of meromorphic functions is naturally defined on $Ω$ union a tree of spheres. In the second part, we show that a set $E \subset Ω$ is removable if and only if it is negligible for extremal distance.

math.CV↗

arXiv2020

Critical structures of inner functions

A celebrated theorem of M. Heins says that up to post-composition with a Möbius transformation, a finite Blaschke product is uniquely determined by its critical points. K. Dyakonov suggested that it may interesting to extend this result to infinite degree, however, one needs to be careful since inner functions may have identical critical sets. In this work, we try parametrizing inner functions by 1-generated invariant subspaces of the weighted Bergman space $A^2_1$. Our technique is based on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.

math.CV↗

arXiv2019

Homogenization of random quasiconformal mappings and random Delauney triangulations

In this paper, we solve two problems dealing with the homogenization of random media. We show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map, confirming a conjecture of Stephenson. We also show that on a Riemann surface equipped with a conformal metric, a random Delauney triangulation is close to being circle packed.

math.CV↗

arXiv2019

Simultaneous zero-free approximation and universal optimal polynomial approximants

Let $E$ be a closed subset of the unit circle of measure zero. Recently, Beise and Müller showed the existence of a function in the Hardy space $H^2$ for which the partial sums of its Taylor series approximate any continuous function on $E$. In this paper, we establish an analogue of this result in a non-linear setting where we consider optimal polynomial approximants of reciprocals of functions in $H^2$ instead of Taylor polynomials. The proof uses a new result on simultaneous zero-free approximation of independent interest. Our results extend to Dirichlet-type spaces $\mathcal{D}_α$ for $α\in [0,1]$.

math.CV↗

arXiv2018

Stable convergence of inner functions

Let $\mathscr J$ be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on $\mathscr J$ where $F_n \to F$ if the critical structures of $F_n$ converge to the critical structure of $F$. We show that this occurs precisely when the critical structures of the $F_n$ are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic self-maps of the unit disk and solutions of the Gauss curvature equation. Building on the works of Korenblum and Roberts, we show that this topology also governs the behaviour of invariant subspaces of a weighted Bergman space which are generated by a single inner function.

math.CV↗

arXiv2017

Prescribing inner parts of derivatives of inner functions

Let $\mathscr J$ be the set of inner functions whose derivatives lie in Nevanlinna class. In this note, we show that the natural map $F \to \text{Inn}(F'): \mathscr J/\text{Aut}(\mathbb{D}) \to \text{Inn}/S^1$ is is injective but not surjective. More precisely, we show that that the image consists of all inner functions of the form $BS_μ$ where $B$ is a Blaschke product and $S_μ$ is the singular factor associated to a measure $μ$ whose support is contained in a countable union of Beurling-Carleson sets. Our proof is based on extending the work of D. Kraus and O. Roth on maximal Blaschke products to allow for singular factors. This answers a question raised by K. Dyakonov.

math.CV↗

arXiv2017

Makarov's principle for the Bloch unit ball

Makarov's principle relates three characteristics of Bloch functions that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin. While these quantities need not be equal in general, we show that the universal bounds agree if we take the supremum over the Bloch unit ball. For the supremum (of either of these quantities), we give the estimate $Σ^2_{\mathcal B} < \min(0.9, Σ^2)$, where $Σ^2$ is the analogous quantity associated to the unit ball in the $L^\infty$ norm on the Bloch space. This improves on the upper bound in Pommerenke's estimate $0.685^2 < Σ^2_{\mathcal B} \le 1$.

math.CV↗

arXiv2017

Sparse Beltrami coefficients, integral means of conformal mappings and the Feynman-Kac formula

In this note, we give an estimate for the dimension of the image of the unit circle under a quasiconformal mapping whose dilatation has small support. We also prove an analogous estimate for the rate of growth of a solution of a second-order parabolic equation given by the Feynman-Kac formula (with a sparsely supported potential) and introduce a dictionary between the two settings.

math.CV↗

arXiv2016

The geometry of the Weil-Petersson metric in complex dynamics

In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. Via the Bers embedding, one may view the Weil-Petersson metric as a metric on the main cardioid of the Mandelbrot set. We prove that the metric completion attaches the geometrically finite parameters from the Euclidean boundary of the main cardioid and conjecture that this is the entire completion. For the upper bound, we estimate the intersection of a circle $S_r = \{z : |z| = r\}$, $r \approx 1$, with an invariant subset $\mathcal G \subset \mathbb{D}$ called a half-flower garden, defined in this work. For the lower bound, we use gradients of multipliers of repelling periodic orbits on the unit circle. Finally, utilizing the convergence of Blaschke products to vector fields, we compute the rate at which the Weil-Petersson metric decays along radial degenerations.

math.DS↗