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Polina Vytnova

Publications and source records attributed to Polina Vytnova.

10 recordsLinked to original sources
arXiv2026

On the graph of the dimension function of the Lagrange and Markov spectra

We study the graph of the function $d(t)$ encoding the Hausdorff dimensions of the classical Lagrange and Markov spectra with half-infinite lines of the form $(-\infty, t)$. For this sake, we use the fact that the Hausdorff dimension of dynamically Cantor sets drop after erasing an element of its Markov partition to determine twelve nontrivial plateaux of $d(t)$. Next, we employ rigorous numerical methods (from our recent joint paper with Pollicott) to produce approximations of the graph of $d(t)$ between these twelve plateaux. As a corollary, we prove that the largest ten non-trivial plateaux of $d(t)$ are exactly those plateaux with lengths $> 0.005$.

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arXiv2025

The Bowen$\unicode{x2013}$Series coding and zeros of zeta functions

We give a discussion of the classical Bowen$\unicode{x2013}$Series coding and, in particular, its application to the study of zeta functions associated to geodesic flows and their zeros. In the case of compact surfaces of constant negative curvature $-1$ the analytic extension of the Selberg zeta function to the entire complex plane is classical, and can be achieved using the Selberg trace formula. However, an alternative dynamical approach is to use the Bowen$\unicode{x2013}$Series coding on the boundary at infinity to obtain a piecewise analytic expanding map from which the extension of the zeta function can be obtained using properties of the associated transfer operator. This latter method has the advantage that it also applies in the case of infinite area surfaces provided they do not have cusps. For such examples the location of the zeros is somewhat more mysterious. However, in particularly simple examples there is a striking structure to the zeros when we take appropriate limits. We will try to give some insight into this phenomenon. The newer version, in addition to the study of pair of pants, also includes heuristic analysis of the zeta function associated to the geodesic flow on symmetric one-funneled tori; in particular, there is a number of pure numerical results on location of its small zeros.

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arXiv2024

Hausdorff dimension of the Apollonian gasket

The Apollonian gasket is a well-studied circle packing. Important properties of the packing, including the distribution of the circle radii, are governed by its Hausdorff dimension. No closed form is currently known for the Hausdorff dimension, and its computation is a special case of a more general and hard problem: effective, rigorous estimates of dimension of a parabolic limit set. In this paper we develop an efficient method for solving this problem which allows us to compute the dimension of the gasket to 128 decimal places and rigorously justify the error bounds. We expect our approach to generalise easily to other parabolic fractals.

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arXiv2022

Hausdorff dimension of Gauss--Cantor sets and two applications to classical Lagrange and Markov spectra

This paper is dedicated to the study of two famous subsets of the real line, namely Lagrange spectrum $L$ and Markov spectrum $M$. Our first result, Theorem 2.1, provides a rigorous estimate on the smallest value $t_1$ such that the portion of the Markov spectrum $(-\infty,t_1)\cap M$ has Hausdorff dimension $1$. Our second result, Theorem 3.1, gives a new upper bound on the Hausdorff dimension of the set difference $M\setminus L$. Our method combines new facts about the structure of the classical spectra together with finer estimates on the Hausdorff dimension of Gauss--Cantor sets of continued fraction expansions whose entries satisfy appropriate restrictions.

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arXiv2022

Groups, drift and harmonic measures

In this short note we will describe an old problem and a new approach which casts light upon it. The old problem is to understand the nature of harmonic measures for cocompact Fuchsian groups. The new approach is to compute numerically the value of the drift and, in particular, get new results on the dimension of the measure in some new examples.

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arXiv2022

Zeros of the Selberg zeta function for symmetric infinite area hyperbolic surfaces

In the present paper we give a simple mathematical foundation for describing the zeros of the Selberg zeta functions $Z_X$ for certain very symmetric infinite area surfaces $X$. For definiteness, we consider the case of three funneled surfaces. We show that the zeta function is a complex almost periodic function which can be approximated by complex trigonometric polynomials on large domains (in Theorem 4.2). As our main application, we provide an explanation of the striking empirical results of Borthwick (arXiv:1305.4850) (in Theorem 1.5) in terms of convergence of the affinely scaled zero sets to standard curves $\mathcal C$.

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arXiv2022

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas red of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira arXiv:1803.01230. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain-Kontorovich arXiv:1107.3776v2, Huang arXiv:1310.3772v4 and Kan arXiv:1604.04884. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen. In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in section 3 in a way that is straightforward to implement.

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arXiv2022

Uniform lower bounds on the dimension of Bernoulli convolutions

In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure $μ_λ$ is the probability measure corresponding to the law of the random variable $ξ= \sum_{k=0}^\infty ξ_kλ^k$, where $ξ_k$ are i.i.d. random variables assuming values $-1$ and $1$ with equal probability and $\frac12 < λ< 1$. In particular, for Bernoulli convolutions we give a uniform lower bound $\dim_H(μ_λ) \geq 0.96399$ for all $\frac12<λ<1$.

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arXiv2020

How many inflections are there in the Lyapunov spectrum?

Iommi & Kiwi showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture of Iommi & Kiwi by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question of Iommi & Kiwi by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

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