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Mariusz Urbanski

Publications and source records attributed to Mariusz Urbanski.

At least 19 recordsLinked to original sources
arXiv2022

Geometry of measures in random systems with complete connections

We study new relations between countable iterated function systems (IFS) with overlaps, Smale endomorphisms and random systems with complete connections. We prove that stationary measures for countable conformal IFS with overlaps and placedependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps S, and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the geometry of the system.

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arXiv2021

Smale endomorphisms over graph-directed Markov systems

We study Smale skew product endomorphisms (introduced in [27]) now over countable graph directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply for instance to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We give also applications in ergodic number theory, for example to the continued fraction expansion, and the backward fractions expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions $\mathcal T_β$ of $β$-maps $T_β$, for arbitrary $β> 1$.

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arXiv2020

Thermodynamic formalism for coarse expanding dynamical systems

We consider a class of dynamical systems, which we call weakly coarse expanding, which is a generalization to the postcritically infinite case of expanding Thurston maps as discussed by Bonk-Meyer and is closely related to coarse expanding conformal systems as defined by Haissinsky-Pilgrim. We prove existence and uniqueness of equilibrium states for a wide class of potentials, as well as statistical laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and a large deviation principle. Further, if the system is defined on the 2-sphere, we prove all such results even in presence of periodic (repelling) branch points.

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arXiv2020

Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Elliptic Functions

The ultimate goal of our book is to present a unified approach to the dynamics, ergodic theory, and geometry of elliptic functions from $\C$ to $\oc$. We consider elliptic functions as a most regular class of transcendental meromorphic functions. Poles form an essential feature of such functions but the set of critical values is finite and an elliptic function is "the same" on its of its fundamental regions. In a sense this is the class of transcendental meromorphic functions which resembles rational functions most. On the other hand, the differences are huge. We will touch on them in the course of this introduction. In order to comprehensively cover the dynamics and geometry of elliptic functions we make large preparations. This is done in the first two parts of the book: Part 1, "Ergodic Theory and Measures" and Part 2,"Geometry and Conformal Measures". We intend our book to be as self contained as possible and we use essentially all major results of Part~1 and Part~2 in Part~3 and Part~4 dealing with elliptic functions. This book can be thus treated as a fairly comprehensive account of dynamics, ergodic theory, and fractal geometry of elliptic functions but also as a reference book (with proofs) for many results of geometric measure theory, finite and infinite abstract ergodic theory, Young's towers, measure--theoretic Kolmogorov--Sinai entropy, thermodynamic formalism, geometric function theory (in particular Koebe's Distortion Theorems and Riemann--Hurwitz Formulas), various kinds of conformal measures, conformal graph Directed Markov systems and iterated function systems, classical general theory of elliptic functions, and topological dynamics of transcendental meromorphic functions.

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arXiv2017

Asymptotic Counting in Conformal Dynamical Systems

In this article we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered hy a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These have direct applications to a variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. A fairly complete collection of asymptotic counting results for them is presented. Our new approach is founded on spectral properties of complexified Ruelle--Perron--Frobenius operators and Tauberian theorems as used in classical problems of prime number theory.

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arXiv2017

Real Analyticity for random dynamics of transcendental functions

Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron-Frobenius operator are assumed to converge. We also provide a Bowen's formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application states real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions.

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arXiv2017

Random Dynamics of Transcendental Functions

This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order and whose derivative satisfies some growth condition at infinity. This class contains most of the classical families of transcendental functions and goes much beyond. Based on uniform versions of Nevanlinna's value distribution theory we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with exponential decay of correlations and a central limit theorem are shown. This part relies on our construction of new positive invariant cones that are adapted to the setting of unbounded phase spaces. This setting rules out the use of Hilbert's metric along with the usual contraction principle. However these cones allow us to apply a contraction argument stemming from Bowen's initial approach.

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arXiv2016

Non-Escaping Sets in Conformal Dynamical Systems and Singular Perturbations of Perron-Frobenius Operators

The study of escape rates for a ball in a dynamical systems has been much studied. Understanding the asymptotic behavior of the escape rate as the radius of the ball tends to zero is an especially subtle problem. In the case of hyperbolic conformal systems this has been addressed by various authors. In this paper we consider a far more general realm of conformal maps where the analysis is correspondingly more complicated. We prove the existence of escape rates and calculate them in the context of countable alphabets, either finite or infinite, uniformly contracting conformal graph directed Markov systems with their special case of conformal countable alphabet iterated function systems. This goal is achieved by developing the appropriate theory of singular perturbations of Perron-Frobenius (transfer) operators associated with countable alphabet subshifts of finite type and Hölder continuous summable potentials. This is the key ingredient for further results about other conformal systems. These include topological Collet-Eckmann multimodal interval maps and rational maps of the Riemann sphere (an equivalent formulation is to be uniformly hyperbolic on periodic points), and also a large class of transcendental meromorphic functions.

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arXiv2015

Overlap functions for measures in conformal iterated function systems

We study conformal iterated function systems (IFS) $\mathcal S = \{ϕ_i\}_{i \in I}$ with arbitrary overlaps, and measures $μ$ on limit sets $Λ$, which are projections of equilibrium measures $\hat μ$ with respect to a certain lift map $Φ$ on $Σ_I^+ \times Λ$. No type of Open Set Condition is assumed. We introduce a notion of overlap function and overlap number for such a measure $\hat μ$ with respect to $\mathcal S$; and, in particular a notion of (topological) overlap number $o(\mathcal S)$. These notions take in consideration the $n$-chains between points in the limit set. We prove that $o(\mathcal S, \hat μ)$ is related to a conditional entropy of $\hat μ$ with respect to the lift $Φ$. Various types of projections to $Λ$ of invariant measures are studied. We obtain upper estimates for the Hausdorff dimension $HD(μ)$ of $μ$ on $Λ$, by using pressure functions and $o(\mathcal S, \hat μ)$. In particular, this applies to projections of Bernoulli measures on $Σ_I^+$. Next, we apply the results to Bernoulli convolutions $ν_λ$ for $λ\in (\frac 12, 1)$, which correspond to self-similar measures determined by composing, with equal probabilities, the contractions of an IFS with overlaps $\mathcal S_λ$. We prove that for all $λ\in (\frac 12, 1)$, there exists a relation between $HD(ν_λ)$ and the overlap number $o(\mathcal S_λ)$. The number $o(\mathcal S_λ)$ is approximated with integrals on $Σ_2^+$ with respect to the uniform Bernoulli measure $ν_{(\frac 12, \frac 12)}$. We also estimate $o(\mathcal S_λ)$ for certain values of $λ$.

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arXiv2015

Countable Alphabet Random Subhifts of finite type with weakly positive transfer operator

We deal with countable alphabet locally compact random subshifts of finite type (the latter merely meaning that the symbol space is generated by an incidence matrix) under the absence of Big Images Property and under the absence of uniform positivity of the transfer operator. We first establish the existence of random conformal measures along with good bounds for the iterates of the Perron-Frobenius operator. Then, using the technique of positive cones and proving a version of Bowen's type contraction (see \cite{Bow75}), we also establish a fairly complete thermodynamical formalism. This means that we prove the existence and uniqueness of fiberwise invariant measures (giving rise to a global invariant measure) equivalent to the fiberwise conformal measures. Furthermore, we establish the existence of a spectral gap for the transfer operators, which in the random context precisely means the exponential rate of convergence of the normalized iterated transfer operator. This latter property in a relatively straightforward way entails the exponential decay of correlations and the Central Limit Theorem.

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arXiv2015

Random countable iterated function systems with overlaps and applications

We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy, the dimensional exactness of the projections of invariant measures from the shift space, and we give a formula for their dimension, in the context of random infinite conformal iterated function systems with overlaps. There exist many differences between our case and the finite deterministic case studied in [7], and we introduce new methods specific to the infinite and random case. We apply our results towards a problem related to a conjecture of Lyons about random continued fractions ([10]), and show that for Lebesgue-almost all parameters λ> 0, the invariant measure ν_λis exact dimensional. The finite IFS determining these continued fractions is not hyperbolic, but we can associate to it a random infinite IFS of contractions which have overlaps. We study then also other large classes of random countable iterated function systems with overlaps, namely: a) several types of random iterated function systems related to Kahane-Salem sets; and b) randomized infinite IFS in the plane which have uniformly bounded number of disc overlaps. For all the above classes, we find lower and upper estimates for the pointwise (and Hausdorff, packing) dimensions of the invariant measures.

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arXiv2014

Shrinking targets for non-autonomous dynamical systems corresponding to Cantor series expansions

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the non-autonomous dynamical system on the interval $[0,1)$, viewed as $\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilizing our theorem. In particular, we provide a Diophantine approximation interpretation of our scheme.

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arXiv2014

Diophantine properties of measures invariant with respect to the Gauss map

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. Finally, we answer in the negative a question posed by Kleinbock, Lindenstrauss, and Weiss, by constructing a family of measures on the real line which are Ahlfors regular and yet do not satisfy a 0-1 law for approximability.

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arXiv2013

Transversality Family of Expanding Rational Semigroups

This paper deals with both complex dynamical systems and conformal iterated function systems. We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a $d$-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 2. We also show that a family of small perturbations of Sierpiński gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each complex number $a$ with $|a|\neq 0,1$, the family of small perturbations of the semigroup generated by ${z^{2}, az^2} $ satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.

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arXiv2012

Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We estimate the Bowen parameters (zeros of the pressure functions) and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $F$ and the Lyapunov exponent of $F$ with respect to the maximal entropy measure for $F$. Moreover, we show that the equality holds if and only if the generators $f_{j}$ are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a linear fractional transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than two.

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arXiv2012

Regularity and irregularity of fiber dimension of non-autonomous dynamical systems

This note concerns non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. We provide a necessary and sufficient condition for holomorphic stability which leads to Hölder continuity of dimensions of hyperbolic non-autonomous Julia sets with respect to the $l^\infty$-topology on the parameter space. On the other hand we show that, for some particular family, the Hausdorff and packing dimension functions are not differentiable at any point and that these dimensions are not equal on an open dense set of the parameter space still with respect to the $l^\infty$-topology.

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arXiv2011

Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.

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