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Mariusz Urbański

Publications and source records attributed to Mariusz Urbański.

At least 37 records · Page 2Linked to original sources
arXiv2018

Quantization dimension for infinite conformal iterated function systems

The quantization dimension function for an $F$-conformal measure $m_F$ generated by an infinite conformal iterated function system satisfying the strong open set condition and by a summable Hölder family of functions is expressed by a simple formula involving the temperature function of the system. The temperature function is commonly used to perform the multifractal analysis, in our context of the measure $m_F$. The result in this paper extends a similar result of Lindsay and Mauldin established for finite conformal iterated function systems [Nonlinearity 15 (2002)].

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arXiv2018

Critically Finite Random Maps of an Interval

We consider random multimodal $C^3$ maps with negative Schwarzian derivative, defined on a finite union of closed intervals in $[0,1]$, onto the interval $[0,1]$ with the base space $Ω$ and a base invertible ergodic map $θ:Ω\toΩ$ preserving a probability measure $m$ on $Ω$. We denote the corresponding skew product map by $T$ and call it a critically finite random map of an interval. We prove that there exists a subset $AA(T)$ of $[0,1]$ with the following properties: (1) For each $t\in AA(T)$ a $t$-conformal random measure $ν_t$ exists. We denote by $λ_{t,ν_t,ω}$ the corresponding generalized eigenvalues of the corresponding dual operators $\mathcal{L}_{t,ω}^*$, $ω\inΩ$. (2) Given $t\ge 0$ any two $t$-conformal random measures are equivalent. (3) The expected topological pressure of the parameter $t$: $$\mathcal{E}P(t):=\int_Ω\logλ_{t,ν,ω}dm(ω) $$ is independent of the choice of a $t$-conformal random measure $ν$. (4) The function $$ AA(T)\ni t\longmapsto \mathcal{E}P(t)\in\mathbb R $$ is monotone decreasing and Lipschitz continuous. (5) With $b_T$ being defined as the supremum of such parameters $t\in AA(T)$ that $\mathcal{E}P(t)\ge 0$, it holds that $$ \mathcal{E}P(b_T)=0 \ \ \ {\rm and} \ \ \ [0,b_T]\subset \text{Int}(AA(T)). $$ (6) $\text{HD}(\mathcal{J}_ω(T))=b_T$ for $m$-a.e $ω\inΩ$, where $\mathcal{J}_ω(T)$, $ω\inΩ$, form the random closed set generated by the skew product map $T$. (7) $b_T=1$ if and only if $\bigcup_{Δ\in \mathcal{G}}Δ=[0,1]$, and then $\mathcal{J}_ω(T)=[0,1]$ for all $ω\inΩ$.

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arXiv2018

On the dimension spectrum of infinite subsystems of continued fractions

In this paper we study the dimension spectrum of continued fractions with coefficients restricted to infinite subsets of natural numbers. We prove that if $E$ is any arithmetic progression, the set of primes, or the set of squares $\{n^2\}_{n \in \mathbb{N}}$, then the continued fractions whose digits lie in $E$ have full dimension spectrum, which we denote by $DS(\mathcal{CF}_E)$. Moreover we prove that if $E$ is an infinite set of consecutive powers then the dimension spectrum $DS(\mathcal{CF}_E)$ always contains a non trivial interval. We also show that there exists some $E \subset \mathbb{N}$ and two non-trivial intervals $I_1, I_2$, such that $DS(\mathcal{CF}_E) \cap I_1=I_1$ and $DS(\mathcal{CF}_E) \cap I_2$ is a Cantor set. On the way we employ the computational approach of Falk and Nussbaum in order to obtain rigorous effective estimates for the Hausdorff dimension of continued fractions whose entries are restricted to infinite sets.

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arXiv2018

Random non-hyperbolic exponential maps

We consider random iteration of exponential entire functions, i.e. of the form ${\mathbb C}\ni z\mapsto f_λ(z):=λe^z\in\mathbb C$, $λ\in{\mathbb C}\setminus \{0\}$. Assuming that $λ$ is in a bounded closed interval $[A,B]$ with $A>1/e$, we deal with random iteration of the maps $f_λ$ governed by an invertible measurable map $θ:Ω\toΩ$ preserving a probability ergodic measure $m$ on $Ω$, where $Ω$ is a measurable space. The link from $Ω$ to exponential maps is then given by an arbitrary measurable function $η:Ω\longmapsto [A,B]$. We in fact work on the cylinder space $Q:={\mathbb C}/\sim$, where $\sim$ is the natural equivalence relation: $z\sim w$ if and only if $w-z$ is an integral multiple of $2πi$. We prove that then for every $t>1$ there exists a unique random conformal measure $ν^{(t)}$ for the random conformal dynamical system on $Q$. We further prove that this measure is supported on the, appropriately defined, radial Julia set. Next, we show that there exists a unique random probability invariant measure $μ^{(t)}$ absolutely continuous with respect to $μ^{(t)}$. In fact $μ^{(t)}$ is equivalent with $ν^{(t)}$. Then we turn to geometry. We define an expected topological pressure $\mathcal E P(t)\in{\mathbb R}$ and show that its only zero $h$ coincides with the Hausdorff dimension of $m$--almost every fiber radial Julia set $J_r(ω)\subset Q$, $ω\inΩ$. We show that $h\in (1,2)$ and that the omega--limit set of Lebesgue almost every point in $Q$ is contained in the real line $\mathbb R$. Finally, we entirely transfer our results to the original random dynamical system on $\mathbb C$. As our preliminary result, we show that all fiber Julia sets coincide with the entire complex plane $\mathbb C$.

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arXiv2018

The dimension spectrum of graph directed Markov systems

In this paper we study the dimension spectrum of general conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We perform a comprehensive study of the dimension spectrum addressing questions regarding its size and topological structure. As a corollary we obtain that the dimension spectrum of infinite conformal iterated function systems is compact and perfect. On the way we revisit the role of the parameter $θ$ in graph directed Markov systems and we show that new phenomena arise. We also establish topological pressure estimates for subsystems in the abstract setting of symbolic dynamics with countable alphabets. These estimates play a crucial role in our proofs regarding the dimension spectrum, and they allow us to study Hausdorff dimension asymptotics for subsystems. Finally we narrow our focus to the dimension spectrum of conformal iterated function systems and we prove, among other things, that the iterated function system resulting from the complex continued fractions algorithm has full dimension spectrum. We thus give a positive answer to the Texan conjecture for complex continued fractions.

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arXiv2017

A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation

We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular $m\times n$ matrices are both equal to $mn \big(1-\frac1{m+n}\big)$, thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis (preprint 2014) as well as answering a question of Bugeaud, Cheung, and Chevallier (preprint 2016). We introduce the notion of a $template$, which generalizes the notion of a $rigid$ $system$ (Roy, 2015) to the setting of matrix approximation. Our main theorem takes the following form: for any class of templates $\mathcal F$ closed under finite perturbations, the Hausdorff and packing dimensions of the set of matrices whose successive minima functions are members of $\mathcal F$ (up to finite perturbation) can be written as the suprema over $\mathcal F$ of certain natural functions on the space of templates. Besides implying KKLM's conjecture, this theorem has many other applications including computing the Hausdorff and packing dimensions of the set of points witnessing a conjecture of Starkov (2000), and of the set of points witnessing a conjecture of Schmidt (1983).

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arXiv2017

Badly approximable points on self-affine sponges and the lower Assouad dimension

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. In particular, for self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford-McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

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arXiv2017

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

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 ('98) resolving Sprindžuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss ('04), 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 Part I, we introduce and develop a systematic account of two classes of measures, which we call $quasi$-$decaying$ and $weakly$ $quasi$-$decaying$. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework (which has received much attention in recent years), thus proving a conjecture of KLW. We also prove the "inherited exponent of irrationality" version of this theorem, describing the relationship between the Diophantine properties of certain subspaces of the space of matrices and measures supported on these subspaces. In subsequent papers, we exhibit numerous examples of quasi-decaying measures, in support of the thesis that "almost any measure from dynamics and/or fractal geometry is quasi-decaying". In addition to the examples described above, we also prove (for example) that Gibbs measures (including conformal measures) of infinite iterated function systems are quasi-decaying, even if the systems in question do not satisfy the open set condition. We also discuss examples of non-extremal measures coming from dynamics, illustrating where the theory must halt.

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arXiv2016

Dimension rigidity in conformal structures

Let $Λ$ be the limit set of a conformal dynamical system, i.e. a Kleinian group acting on either finite- or infinite-dimensional real Hilbert space, a conformal iterated function system, or a rational function. We give an easily expressible sufficient condition, requiring that the limit set is not too much bigger than the radial limit set, for the following dichotomy: $Λ$ is either a real-analytic manifold or a fractal in the sense of Mandelbrot (i.e. its Hausdorff dimension is strictly greater than its topological dimension). Our primary focus is on the infinite-dimensional case. An important component of the strategy of our proof comes from the rectifiability techniques of Mayer and Urbański ('03), who obtained a dimension rigidity result for conformal iterated function systems (including those with infinite alphabets). In order to handle the infinite dimensional case, both for Kleinian groups and for iterated function systems, we introduce the notion of pseudorectifiability, a variant of rectifiability, and develop a theory around this notion similar to the theory of rectifiable sets. Our approach also extends existing results in the finite-dimensional case, where it unifies the realms of Kleinian groups, conformal iterated function systems, and rational functions. For Kleinian groups, we improve on the rigidity result of Kapovich ('09) by substantially weakening its hypothesis of geometrical finiteness. Moreover, our proof, based on rectifiability, is entirely different than that of Kapovich, which depends on homological algebra. Another advantage of our approach is that it allows us to use the "demension" of Štan$'$ko ('69) as a substitute for topological dimension. For example, we prove that any dynamically defined version of Antoine's necklace must have Hausdorff dimension strictly greater than 1 (i.e. the demension of Antoine's necklace).

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arXiv2016

Badly approximable vectors and fractals defined by conformal dynamical systems

We prove that if $J$ is the limit set of an irreducible conformal iterated function system (with either finite or countably infinite alphabet), then the badly approximable vectors form a set of full Hausdorff dimension in $J$. The same is true if $J$ is the radial Julia set of an irreducible meromorphic function (either rational or transcendental). The method of proof is to find subsets of $J$ that support absolutely friendly and Ahlfors regular measures of large dimension. In the appendix to this paper, we answer a question of Broderick, Kleinbock, Reich, Weiss, and the second-named author ('12) by showing that every hyperplane diffuse set supports an absolutely decaying measure.

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arXiv2016

Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings

Our monograph presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Our work unifies and extends a long list of results by many authors. We make it a point to avoid any assumption of properness/compactness, keeping in mind the motivating example of $\mathbb H^\infty$, the infinite-dimensional rank-one symmetric space of noncompact type over the reals. The monograph provides a number of examples of groups acting on $\mathbb H^\infty$ which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. Such examples often demonstrate the optimality of our theorems. We introduce a modification of the Poincaré exponent, an invariant of a group which gives more information than the usual Poincaré exponent, which we then use to vastly generalize the Bishop--Jones theorem relating the Hausdorff dimension of the radial limit set to the Poincaré exponent of the underlying semigroup. We give some examples based on our results which illustrate the connection between Hausdorff dimension and various notions of discreteness which show up in non-proper settings. We construct Patterson--Sullivan measures for groups of divergence type without any compactness assumption. This is carried out by first constructing such measures on the Samuel--Smirnov compactification of the bordification of the underlying hyperbolic space, and then showing that the measures are supported on the bordification. We study quasiconformal measures of geometrically finite groups in terms of (a) doubling and (b) exact dimensionality. Our analysis characterizes exact dimensionality in terms of Diophantine approximation on the boundary. We demonstrate that some Patterson--Sullivan measures are neither doubling nor exact dimensional, and some are exact dimensional but not doubling, but all doubling measures are exact dimensional.

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arXiv2016

Conformal graph directed Markov systems on Carnot groups

We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen's parameter. We illustrate our results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.

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arXiv2016

Diophantine approximation in Banach spaces

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.

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arXiv2015

Tukia's isomorphism theorem in CAT(-1) spaces

We prove a generalization of Tukia's ('85) isomorphism theorem which states that isomorphisms between geometrically finite groups extend equivariantly to the boundary. Tukia worked in the setting of real hyperbolic spaces of finite dimension, and his theorem cannot be generalized as stated to the setting of CAT($-1$) spaces. We exhibit examples of type-preserving isomorphisms of geometrically finite subgroups of finite-dimensional rank one symmetric spaces of noncompact type (ROSSONCTs) whose boundary extensions are not quasisymmetric. A sufficient condition for a type-preserving isomorphism to extend to a quasisymmetric equivariant homeomorphism between limit sets is that one of the groups in question is a lattice, and that the underlying base fields are the same, or if they are not the same then the base field of the space on which the lattice acts has the larger dimension. This in turn leads to a generalization of a rigidity theorem of Xie ('08) to the setting of finite-dimensional ROSSONCTs.

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arXiv2015

Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces

In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic '76 paper to more recent results of Hersonsky and Paulin ('02, '04, '07). Concrete examples of situations we consider which have not been considered before include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones ('97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson--Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.

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arXiv2014

Non-autonomous conformal iterated function systems and Moran-set constructions

We study non-autonomous conformal iterated function systems, with finite or countably infinite alphabet alike. These differ from the usual (autonomous) iterated function systems in that the contractions applied at each step in time are allowed to vary. (In the case where all maps are affine similarities, the resulting system is also called a "Moran set construction".) We shall show that, given a suitable restriction on the growth of the number of contractions used at each step, the Hausdorff dimension of the limit set of such a system is determined by an equation known as Bowen's formula. We also give examples that show the optimality of our results. In addition, we prove Bowen's formula for a class of infinite-alphabet-systems and deal with Hausdorff and packing measures for finite systems, as well as continuity of topological pressure and Hausdorff dimension for both finite and infinite systems. In particular, we strengthen the existing continuity results for infinite autonomous systems. As a simple application of our results, we show that, for a transcendental meromorphic function, the Hausdorff dimension of the set of transitive points (i.e., those points whose orbits are dense in the Julia set) is bounded from below by the hyperbolic dimension (in the sense of Shishikura).

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arXiv2013

Homogeneous kernerls and self similar sets

We consider singular integrals associated to homogeneous kernels on self similar sets. Using ideas from ergodic theory we prove, among other things, that in Euclidean spaces the principal values of singular integrals associated to real analytic, homogeneous kernels fail to exist a.e. on self similar sets satisfying some separation conditions. Furthermore in general metric groups, using similar techniques, we generalize a criterion of $L^2$-unboundedness for singular integrals on self similar sets.

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