arXiv subjects

Natalia Jurga

Publications and source records attributed to Natalia Jurga.

At least 19 recordsLinked to original sources

arXiv2026

Dynamical blanket times

We introduce dynamical blanket times, which quantify how quickly the empirical distribution along a typical orbit approximates the invariant measure. These can be viewed as a measure-theoretic analogue of the previously introduced \emph{dynamical cover time}, which measures how quickly an orbit becomes dense in the space. Motivated by analogous comparability questions for random walks on graphs, we investigate how much longer it takes for a dynamical system to ``blanket'' than to ``cover''. For finite-branch, uniformly expanding interval maps, we obtain upper bounds on the expected blanket time in terms of the spatial scale and the precision of approximation. In the special case where the invariant measure is absolutely continuous with respect to Lebesgue, this yields comparability between the expected blanket and cover times, uniformly across all sufficiently small scales. Our approach combines two main ingredients. First, we establish large deviation estimates for hitting times which are uniform over both the target location and the spatial scale. Second, using methods from multifractal analysis, we construct a finite discretisation of the invariant measure which reduces the problem to a suitably controlled discrete model.

math.DS↗

arXiv2024

Cover times in dynamical systems

Given a one-dimensional dynamical system we study its cover time, which quantifies the rate at which orbits become dense in the state space. Using transfer operator tools for dynamical systems with holes and inducing techniques, for a wide class of uniformly hyperbolic and non-uniformly hyperbolic systems we obtain an asymptotic formula for the expected cover time in terms of the decay rate of the measure of the ball of minimum measure. Applications include the Gauss map and Manneville-Pomeau maps.

math.DS↗

arXiv2024

Self-projective sets

Self-projective sets are natural fractal sets which describe the action of a semigroup of matrices on projective space. In recent years there has been growing interest in studying the dimension theory of self-projective sets, as well as progress in the understanding of closely related objects such as Furstenberg measures. The aim of this survey is twofold: first to motivate the study of these objects from several different perspectives and second to make the study of these objects more accessible for readers with expertise in iterated function systems.

math.DS↗

arXiv2023

Hausdorff dimension of the Rauzy gasket

The Rauzy gasket is the attractor of a parabolic, nonconformal iterated function system on the projective plane which describes an exceptional parameter set in various important topological and dynamical problems. Since 2009 there have been several attempts to calculate the Hausdorff dimension of the Rauzy gasket, which is a challenging problem due to the combination of the parabolicity and nonconformality in the geometry. In this paper we settle this question by proving that the Hausdorff dimension of the Rauzy gasket equals the (projective) affinity dimension. The key technical result underpinning this is a partial generalisation of work of Hochman and Solomyak to the $\mathrm{SL}_3(\mathbb{R})$ setting, where we establish the exact dimension of stationary (Furstenberg) measures supported on the Rauzy gasket. The dimension results for both stationary measures and attractors are established in broader generality and extend recent work on projective iterated function systems to higher dimensions.

math.DS↗

arXiv2022

Non-existence of the box dimension for dynamically invariant sets

One of the key challenges in the dimension theory of smooth dynamical systems is in establishing whether or not the Hausdorff, lower and upper box dimensions coincide for invariant sets. For sets invariant under conformal dynamics, these three dimensions always coincide. On the other hand, considerable attention has been given to examples of sets invariant under non-conformal dynamics whose Hausdorff and box dimensions do not coincide. These constructions exploit the fact that the Hausdorff and box dimensions quantify size in fundamentally different ways, the former in terms of covers by sets of varying diameters and the latter in terms of covers by sets of fixed diameters. In this article we construct the first example of a dynamically invariant set with distinct lower and upper box dimensions. Heuristically, this describes that if size is quantified in terms of covers by sets of equal diameters, a dynamically invariant set can appear bigger when viewed at certain resolutions than at others.

math.DS↗

arXiv2022

Parabolic carpets

We introduce and study a family of non-conformal and non-uniformly contracting iterated function systems. We refer to the attractors of such systems as parabolic carpets. Roughly speaking they may be thought of as nonlinear analogues of self-affine carpets which are allowed to have parabolic fixed points. We compute the $L^q$-spectrum of a class of weak Gibbs measures supported on parabolic carpets as well as the box dimensions of the carpet itself.

math.DS↗

arXiv2021

On the convergence rate of the chaos game

This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting iterated function system of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford-McMullen carpets we completely characterise the family of probability vectors which minimise the Minkowski dimension of Bernoulli measures. Interestingly, these vectors have not appeared in any other aspect of Bedford-McMullen carpets before.

math.DS↗

arXiv2020

Ledrappier-Young formulae for a family of nonlinear attractors

We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier-Young formula.

math.DS↗

arXiv2020

Box dimensions of $(\times m, \times n)$-invariant sets

We study the box dimensions of sets invariant under the toral endomorphism $(x, y) \mapsto (m x \text{ mod } 1, \, n y \text{ mod } 1)$ for integers $n>m \geq 2$. The basic examples of such sets are Bedford-McMullen carpets and, more generally, invariant sets are modelled by subshifts on the associated symbolic space. When this subshift is topologically mixing and sofic the situation is well-understood by results of Kenyon and Peres. Moreover, other work of Kenyon and Peres shows that the Hausdorff dimension is generally given by a variational principle. Therefore, our work is focused on the box dimensions in the case where the underlying shift is not topologically mixing and sofic. We establish straightforward upper and lower bounds for the box dimensions in terms of entropy which hold for all subshifts and show that the upper bound is the correct value for coded subshifts whose entropy can be realised by words which can be freely concatenated, which includes many well-known families such as $β$-shifts, (generalised) $S$-gap shifts, and transitive sofic shifts. We also provide examples of transitive coded subshifts where the general upper bound fails and the box dimension is actually given by the general lower bound. In the non-transitive sofic setting, we provide a formula for the box dimensions which is often intermediate between the general lower and upper bounds.

math.DS↗

arXiv2020

How long is the Chaos Game?

In the 1988 textbook "Fractals Everywhere" M. Barnsley introduced an algorithm for generating fractals through a random procedure which he called the "chaos game". Using ideas from the classical theory of covering times of Markov chains we prove an asymptotic formula for the expected time taken by this procedure to generate a $δ$-dense subset of a given self-similar fractal satisfying the open set condition.

math.DS↗

arXiv2020

The Hausdorff dimension of self-projective sets

Given a finite set $\mathcal{A} \subseteq \mathrm{SL}(2,\mathbb{R})$ we study the dimension of the attractor $K_\mathcal{A}$ of the iterated function system induced by the projective action of $\mathcal{A}$. In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of $K_\mathcal{A}$ is given by the minimum of 1 and the critical exponent, under the assumption that $\mathcal{A}$ satisfies certain discreteness conditions and a Diophantine property. Our approach combines techniques from the theories of iterated function systems and Möbius semigroups, and allows us to discuss the continuity of the Hausdorff dimension, as well as the dimension of the support of the Furstenberg measure.

math.DS↗

arXiv2020

Dimension spectrum of infinite self-affine iterated function systems

Given an infinite iterated function system (IFS) $\mathcal{F}$, we define its dimension spectrum $D(\mathcal{F})$ to be the set of real numbers which can be realised as the dimension of some subsystem of $\mathcal{F}$. In the case where $\mathcal{F}$ is a conformal IFS, the properties of the dimension spectrum have been studied by several authors. In this paper we investigate for the first time the properties of the dimension spectrum when $\mathcal{F}$ is a non-conformal IFS. In particular, unlike dimension spectra of conformal IFS which are always compact and perfect (by a result of Chousionis, Leykekhman and Urbański, Selecta 2019), we construct examples to show that $D(\mathcal{F})$ need not be compact and may contain isolated points.

math.DS↗

arXiv2019

The box dimensions of exceptional self-affine sets in $\mathbb{R}^3$

We study the box dimensions of self-affine sets in $\mathbb{R}^3$ which are generated by a finite collection of generalised permutation matrices. We obtain bounds for the dimensions which hold with very minimal assumptions and give rise to sharp results in many cases. There are many issues in extending the well-established planar theory to $\mathbb{R}^3$ including that the principal planar projections are (affine distortions of) self-affine sets with overlaps (rather than self-similar sets) and that the natural modified singular value function fails to be sub-multiplicative in general. We introduce several new techniques to deal with these issues and hopefully provide some insight into the challenges in extending the theory further.

math.DS↗

arXiv2019

Effective estimates on the top Lyapunov exponent for random matrix products

We study the top Lyapunov exponents of random products of positive $2 \times 2$ matrices and obtain an efficient algorithm for its computation. As in the earlier work of Pollicott, the algorithm is based on the Fredholm theory of determinants of trace-class linear operators. In this article we obtain a simpler expression for the approximations which only require calculation of the eigenvalues of finite matrix products and not the eigenvectors. Moreover, we obtain effective bounds on the error term in terms of two explicit constants: a constant which describes how far the set of matrices are from all being column stochastic, and a constant which measures the minimal amount of projective contraction of the positive quadrant under the action of the matrices.

math.DS↗

arXiv2019

Approximating integrals with respect to stationary probability measures of iterated function systems

We study fast approximation of integrals with respect to stationary probability measures associated to iterated functions systems on the unit interval. We provide an algorithm for approximating the integrals under certain conditions on the iterated function system and on the function that is being integrated. We apply this technique to estimate Hausdorff moments, Wasserstein distances and Lyapunov exponents of stationary probability measures.

math.DS↗

arXiv2019

Analyticity of the affinity dimension for planar iterated function systems with matrices which preserve a cone

The sub-additive pressure function $P(s)$ for an affine iterated function system (IFS) and the affinity dimension, defined as the unique solution $s_0$ to $P(s_0)=1$, were introduced by K. Falconer in his seminal 1988 paper on self-affine fractals. The affinity dimension prescribes a value for the Hausdorff dimension of a self-affine set which is known to be correct in generic cases and in an increasing range of explicit cases. It was shown by Feng and Shmerkin in 2014 that the affinity dimension depends continuously on the IFS. In this article we prove that when the linear parts of the affinities which define the IFS are $2 \times 2$ matrices which strictly preserve a common cone, the sub-additive pressure is locally real analytic as a function of the matrix coefficients of the linear parts of the affinities. In this setting we also show that the sub-additive pressure is piecewise real analytic in $s$, implying that the affinity dimension is locally analytic in the matrix coefficients. Combining this with a recent result of Bárány, Hochman and Rapaport we obtain results concerning the analyticity of the Hausdorff dimension for certain families of planar self-affine sets.

math.DS↗

arXiv2018

Dimensions of equilibrium measures on a class of planar self-affine sets

We study equilibrium measures (Käenmäki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure. In particular, we do this by showing that the Käenmäki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.

math.DS↗

arXiv2018

A new proof of the dimension gap for the Gauss map

Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map $T(x)= \frac{1}{x} \mod 1$ satisfy a `dimension gap' meaning that for some $c>0$, $\sup_{\mathbf{p}} \dim μ_{\mathbf{p}} <1-c$, where $μ_{\mathbf{p}}$ denotes the (pushforward) Bernoulli measure for the countable probability vector $\mathbf{p}$. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.

math.DS↗