arXiv subjects

Mariusz Urbański

Publications and source records attributed to Mariusz Urbański.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotics of the Hausdorff measure for the Gauss map and its linearized analogue

Let $G(x):=\{1/x\}$ be the Gauss map. By $g_n(x)=\frac{1}{x+n}$ we denote its continuous/real analytic inverse branches. We define iterated function system (IFS) $G_n$ by limiting the collection of functions $g_k$, $k\in\mathbb N$, to the first $n$ elements, meaning that $G_n = \{g_k \}_{k=1}^n$. We are interested in the asymptotics of the Hausdorff measure of the limit set $J_n$ i. e. set consisting of irrational elements of $[0,1]$ having continued fraction expansion with entries at most $n$. In the first part of the paper, we deal with the piecewise-linear analogue of the Gauss map and resulting IFSs. We prove that \[ \lim \limits_{n \to \infty } \frac{1-H_n(J_n)}{1-h_n} \cdot \frac{1}{\ln n} = 1, \] where $J_n$ is the limit set of the piecewise-linear analogue of $G_n$, $h_n$ is its Hausdorff dimension and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$. In the second part, we focus on the IFS generated by the first $n$ branches of Gauss map and prove, as our main result, that $$ \lim_{n\to\infty} \frac{1-H_n}{(1-h_n)\ln n}= 1 $$ and equivalently, due to Hensley's result, $$ \lim_{n\to\infty} \frac{n(1-H_n)}{\ln n}= \frac{6}{π^2}, $$ where $J_n$ is the limit set of the system $G_n$, i.e. the set consisting of irrational numbers in $[0,1]$ that continued fraction expansion with entries not exceeding $n$. Similarly as for the piecewise linear map, $h_n$ is the Hausdorff dimension of $J_n$ and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$.

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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.

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arXiv2025

Rigorous Hausdorff dimension estimates for conformal fractals

We develop a versatile framework which allows us to rigorously estimate the Hausdorff dimension of maximal conformal graph directed Markov systems in $\mathbb{R}^n$ for $n \geq 2$. Our method is based on piecewise linear approximations of the eigenfunctions of the Perron-Frobenius operator via a finite element framework for discretization and iterative mesh schemes. One key element in our approach is obtaining bounds for the derivatives of these eigenfunctions, which, besides being essential for the implementation of our method, are of independent interest.

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arXiv2025

The dimension spectrum of the infinitely generated Apollonian gasket

We prove that the infinitely generated Apollonian gasket has full Hausdorff dimension spectrum. Our proof, which is computer assisted, relies on an iterative technique introduced by the first three authors in [3] and on a flexible method for rigorously estimating Hausdorff dimensions of limit sets of conformal iterated function systems, which we recently developed in [5]. Another key ingredient in our proof is obtaining reasonably sized distortion constants for the (infinite) Apollonian iterated function system.

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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.

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arXiv2023

A variational principle in the parametric geometry of numbers

We extend the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of $m$ linear forms in $n$ variables, and establish a new connection to the metric theory via a variational principle that computes fractal dimensions of a variety of sets of number-theoretic interest. The proof relies on two novel ingredients: a variant of Schmidt's game capable of computing the Hausdorff and packing dimensions of any set, and the notion of templates, which generalize Roy's rigid systems. In particular, we compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of Dani's correspondence principle, the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Other applications include quantitative strengthenings of theorems due to Cheung and Moshchevitin, which originally resolved conjectures due to Starkov and Schmidt respectively; as well as dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing partial results due to Baker, Bugeaud, Cheung, Chevallier, Dodson, Laurent and Rynne.

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arXiv2023

On hyperbolic dimension gap for entire functions

Polynomials and entire functions whose hyperbolic dimension is strictly smaller than the Hausdorff dimension of their Julia set are known to exist but in all these examples the latter dimension is maximal, i.e. equal to two. In this paper we show that there exist hyperbolic entire functions $f$ having Hausdorff dimension of the Julia set $\HD (\J _f)<2$ and hyperbolic dimension $\HypDim(f)<\HD(\J_f)$.

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arXiv2023

Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism

We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism à la Sinai--Ruelle--Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarník and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all $\leq N$; as well as its counterpart for reals whose partial quotients are all $\geq N$ due to Good from 1941.

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arXiv2022

Estimating Hausdorff measure for Denjoy maps

By employing the recurrence method worked out in `Estimating the Hausdorff measure by recurrence', we provide effective lower estimates of the proper--dimensional Hausdorff measure of minimal sets of circle homeomorphisms that are not conjugate to any rotation.

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arXiv2022

Equilateral Triangulations and The Postcritical Dynamics of Meromorphic Functions

We show that any dynamics on any planar set $S$ discrete in some domain $D$ can be realized by the postcritical dynamics of a function holomorphic in $D$, up to a small perturbation. A key step in the proof, and a result of independent interest, is that any planar domain $D$ can be equilaterally triangulated with triangles whose diameters $\rightarrow0$ (at any prescribed rate) near $\partial D$.

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arXiv2022

The Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups

We study skew-product dynamics for a large class of finitely-generated semi--hyperbolic semigroups of rational maps acting on the Riemann sphere, which generalizes both the theory of iteration of a single rational map of a single complex variable complex/holomorphic dynamics) and the theory of countable alphabet conformal iterated function systems (CIFSs). We construct the thermodynamic formalism for such dynamical systems and geometric potentials by developing the notion of nice families that extend to the case of our highly disconnected skew product phase space the powerful notion of nice sets due to Rivera--Letelier and Przytycki, and the allied earlier notion of $K(V)$ sets due to Denker and the last named author. We leverage out techniques to prove the existence and uniqueness of equilibrium states for a wide class of Hölder potentials, and concomitant statistical laws: central limit theorem, law of iterated logarithm, and exponential decay of correlations. We devote lots of space and effort to control (non-recurrent) critical points which is a notoriously challenging task even for a single rational function; more generators add qualitatively new challenges. Beyond dynamics, but still with dynamical methods, we advance the study of finer fractal geometrical properties of the intricate Julia sets associated to such systems and, in particular, via equilibrium states, we perform a multifractal analysis of Lyapunov exponents. We use the Nice Open Set Condition (NOSC) introduced by the last two authors, and apply our new techniques to settle a long-standing problem in the theory of rational semigroups by proving that for our class of semigroups the Hausdorff dimension of each fiber Julia set is strictly smaller than the Hausdorff dimension of the global Julia set of the semigroup.

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arXiv2021

The exact value of Hausdorff dimension of escaping sets of class B meromorphic functions

We consider the subclass of class ${\mathcal B} $ consisting of meromorphic functions $f:{\mathbb C}\to\hat{\mathbb C}$ for which infinity is not an asymptotic value and whose all poles have orders uniformly bounded from above. This class was introduced in \cite{BwKo2012} and the Hausdorff dimension ${\rm HD}({{\mathcal I} (f)})$ of the set ${{\mathcal I} (f)}$ of all points escaping to infinity under forward iteration of $f$ was estimated therein. In this paper we provide a closed formula for the exact value of ${\rm HD}({{\mathcal I} (f)})$ identifying it with the critical exponent of the natural series introduced in \cite{BwKo2012}. This exponent is very easy to calculate for many concrete functions. In particular, we construct a function from this class which is of infinite order and for which ${\rm HD}({{\mathcal I} (f)})=0$.

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arXiv2020

Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [{\it Invent. Math.} {\bf 138}(3) (1999), 451--494] resolving Sprind\v zuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. {\it Selecta Math.} {\bf 10} (2004), 479--523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson--Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW's sufficient conditions for extremality. In the first of this series of papers [{\it Selecta Math.} {\bf 24}(3) (2018), 2165--2206], we introduce and develop a systematic account of two classes of measures, which we call {\it quasi-decaying} and {\it weakly quasi-decaying}. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ``inherited exponent of irrationality'' version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson--Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.

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arXiv2019

Thermodynamic formalism and integral means spectrum of asymptotic tracts for transcendental entire functions

We provide the full theory of thermodynamic formalism for a very general collection of entire functions in class $\mathcal B$. This class overlaps with the collection of all entire functions for which thermodynamic formalism has been so far established and contains many new functions. The key point is that we introduce an integral means spectrum for logarithmic tracts which takes care of the fractal behavior of the boundary of the tract near infinity. It turns out that this spectrum behaves well as soon as the tracts have some sufficiently nice geometry which, for example, is the case for quasicircle, John or Hölder tracts. In this case we get a good control of the corresponding transfer operators, leading to full thermodynamic formalism along with its applications such as exponential decay of correlations, central limit theorem and a Bowen's formula for the Hausdorff dimension of radial Julia sets. Our approach applies in particular to every hyperbolic function from any Eremenko-Lyubich analytic family of Speiser class $\mathcal S$ provided this family contains at least one function with Hölder tracts. The latter is, for example, the case if the family contains a Poincaré linearizer.

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arXiv2019

Porosity in conformal dynamical systems

In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that, under some natural assumptions, their limit sets are porous in large (in the sense of category and dimension) subsets, and they are mean porous almost everywhere. On the other hand, we prove that if the limit set of a GDMS is not porous then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction. We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterization of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are mean porous almost everywhere. We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.

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arXiv2019

Long hitting times for expanding systems

We prove a new result in the area of hitting time statistics. Currently, there is a lot of papers showing that the first entry times into cylinders or balls are often faster than the Birkhoff's Ergodic Theorem would suggest. We provide an opposite counterpart to these results by proving that the hitting times into shrinking balls are also often much larger than these theorems would suggest, by showing that for many dynamical systems $$ \displaystyle \limsup_{r\to 0} τ_{B(y,r)}(x)μ(B(y,r))=+\infty, $$ for an appropriately large, at least of full measure, set of points $y$ and $x$. We first do this for all transitive open distance expanding maps and Gibbs/equilibrium states of Hölder continuous potentials; in particular for all irreducible subshifts of finite type with a finite alphabet. Then we prove such result for all finitely irreducible subshifts of finite type with a countable alphabet and Gibbs/equilibrium states for Hölder continuous summable potentials. Next, we show that the \emph{limsup} result holds for all graph directed Markov systems (far going natural generalizations of iterated function systems) and projections of aforementioned Gibbs states on their limit sets. By utilizing the first return map techniques, we then prove the \emph{limsup} result for all tame topological Collect--Eckmann multimodal maps of an interval, all tame topological Collect--Eckmann rational functions of the Riemann sphere, and all dynamically semi--regular transcendental meromorphic functions from $\mathbb{C}$ to $\widehat{\mathbb{C}}$.

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arXiv2019

Skew product Smale endomorphisms over countable shifts of finite type

We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible two-sided topological Markov shifts with countable alphabets. We describe then the thermodynamic formalism for Smale skew products over countable-to-1 endomorphisms, and give several applications to measures on natural extensions of endomorphisms. We show that the exact dimensionality of conditional measures on fibers, implies the global exact dimensionality of the measure, in certain cases. We then study equilibrium states for skew products over endomorphisms generated by graph directed Markov systems, in particular for skew products over expanding Markov-Renyi(EMR) maps, and we settle the question of the exact dimensionality of such measures. In particular, this applies to skew products over the continued fractions transformation, and over parabolic maps. We prove next two results related to Diophantine approximation, which make the renowned Doeblin-Lenstra Conjecture more general and more precise, for a different class of measures than in the classical case. In the end, we prove exact dimensionality and find a computable formula for the dimension of equilibrium measures, for induced maps of natural extensions $\mathcal T_β$ of beta-maps, for arbitrary β> 1.

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arXiv2019

Thin annuli property and exponential distribution of return times for Weakly Markov systems

We deal with the problem of asymptotic distribution of first return times to shrinking balls under iteration generated by a large general class of dynamical systems called weakly Markov. Our ultimate main result is that these distributions converge to the exponential law when the balls shrink to points. We apply this result to many classes of smooth dynamical systems that include conformal iterated function systems, rational functions on the Riemann sphere $\widehat{\mathbb C}$, and transcendental meromorphic functions on the complex plane $\mathbb{C}$. We also apply them to expanding repellers and holomorphic endomorphisms of complex projective spaces. One of the key ingredients in our approach is to solve the well known, in this field of mathematics, problem of appropriately estimating the measures of, suitably defined, large class of geometric annuli. We successfully do it. This problem is, in the existing literature, differently referred to by different authors; we call it the Thick Thin Annuli Property. Having this property established, we prove that for non--conformal systems the aforementioned distributions converge to the exponential one along sets of radii whose relative Lebesgue measure converges fast to one. But this is not all. In the context of conformal iterated function systems, we establish the Full Thin Annuli Property, which gives the same estimates for all radii. ln this way, we solve a long standing problem. As a result, we prove that the convergence to the exponential law holds along all radii for essentially all conformal iterated function systems and, with the help of the techniques of first return maps, for all aforementioned conformal dynamical systems.

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