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Balázs Bárány

Publications and source records attributed to Balázs Bárány.

At least 37 records · Page 2Linked to original sources
arXiv2023

Lagrange-like spectrum of perfect additive complements

Two infinite sets $A$ and $B$ of non-negative integers are called \emph{perfect additive complements of non-negative integers}, if every non-negative integer can be uniquely expressed as the sum of elements from $A$ and $B$. In this paper, we define a Lagrange-like spectrum of the perfect additive complements ($\mathfrak{L} $ for short). As a main result, we obtain the smallest accumulation point of the set $\mathfrak{L} $ and prove that the set $\mathfrak{L} $ is closed. Other related results and problems are also contained.

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arXiv2023

Scaling limits of self-conformal measures

We show that any self-conformal measure $μ$ on $\mathbb{R}$ is uniformly scaling and generates an ergodic fractal distribution. This generalizes existing results by removing the need for any separation condition. We also obtain applications to the prevalence of normal numbers in self-conformal sets, the resonance between self-conformal measures on the line, and projections of self-affine measures on carpets.

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arXiv2023

On the dimension of planar self-affine sets with non-invertible maps

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition, the dimension equals to the affinity dimension for a typical choice of the linear parts of the non-invertible mappings, furthermore, we show that the dimension is strictly smaller than the affinity dimension for certain choices of parameters.

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arXiv2022

Recurrence rates for shifts of finite type

Let $Σ_{A}$ be a topologically mixing shift of finite type, let $σ:Σ_{A}\toΣ_{A}$ be the usual left-shift, and let $μ$ be the Gibbs measure for a Hölder continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system $(Σ_{A},σ)$ that hold $μ$-almost surely. In particular, given a function $ψ:\mathbb{N}\to \mathbb{N}$ we are interested in the following set $$R_ψ=\{{\texttt i}\in Σ_{A}:i_{n+1}\ldots i_{n+ψ(n)+1}=i_1\ldots i_{ψ(n)}\textrm{ for infinitely many }n\in\mathbb{N}\}.$$ We provide sufficient conditions for $μ(R_ψ)=1$ and sufficient conditions for $μ(R_ψ)=0$. As a corollary of these results, we discover a new critical threshold where the measure of $R_ψ$ transitions from zero to one. This threshold was previously unknown even in the special case of a non-uniform Bernoulli measure defined on the full shift. The proofs of our results combine ideas from Probability Theory and Thermodynamic Formalism. In our final section we apply our results to the study of dynamics on self-similar sets.

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arXiv2022

Spectrum of weighted Birkhoff average

Let $\{s_n\}_{n\in\N}$ be a decreasing nonsummable sequence of positive reals. In this paper, we investigate the weighted Birkhoff average $\frac{1}{S_n}\sum_{k=0}^{n-1}s_kϕ(T^kx)$ on aperiodic irreducible subshift of finite type $Σ_{\bf A}$ where $ϕ: Σ_{\bf A}\mapsto \R$ is a continuous potential. Firstly, we show the entropy spectrum of the weighed Birkhoff averages remains the same as that of the Birkhoff averages. Then we calculate the packing spectrum of the weighed Birkhoff averages. It turns out that we can have two cases, either the packing dimension of every level set equals to its Hausdorff dimension or for every nonempty level set it is equal to the packing dimension of the whole space.

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arXiv2022

Typical absolute continuity for classes of dynamically defined measures

We consider one-parameter families of smooth uniformly contractive iterated function systems $\{f^λ_j\}$ on the real line. Given a family of parameter dependent measures $\{μ_λ\}$ on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps $Π^λ$. The main novelty of our work is that the measures $μ_λ$ depend on the parameter, whereas up till now it has been usually assumed that the measure on the symbolic space is fixed and the parameter dependence comes only from the natural projection. This is especially the case in the question of absolute continuity of the projected measure $(Π^λ)_*μ_λ$, where we had to develop a new approach in place of earlier attempt which contains an error. Our main result states that if $μ_λ$ are Gibbs measures for a family of Hölder continuous potentials $ϕ^λ$, with Hölder continuous dependence on $λ$ and $\{Π^λ\}$ satisfy the transversality condition, then the projected measure $(Π^λ)_*μ_λ$ is absolutely continuous for Lebesgue a.e.\ $λ$, such that the ratio of entropy over the Lyapunov exponent is strictly greater than $1$. We deduce it from a more general almost sure lower bound on the Sobolev dimension for families of measures with regular enough dependence on the parameter. Under less restrictive assumptions, we also obtain an almost sure formula for the Hausdorff dimension. As applications of our results, we study stationary measures for iterated function systems with place-dependent probabilities (place-dependent Bernoulli convolutions and the Blackwell measure for binary channel) and equilibrium measures for hyperbolic IFS with overlaps (in particular: natural measures for non-homogeneous self-similar IFS and certain systems corresponding to random continued fractions).

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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.

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arXiv2021

Dynamically defined subsets of generic self-affine sets

In dynamical systems, shrinking target sets and pointwise recurrent sets are two important classes of dynamically defined subsets. In this article we introduce a mild condition on the linear parts of the affine mappings that allow us to bound the Hausdorff dimension of cylindrical shrinking target and recurrence sets. For generic self-affine sets in the sense of Falconer, that is by randomising the translation part of the affine maps, we prove that these bounds are sharp. These mild assumptions mean that our results significantly extend and complement the existing literature for recurrence on self-affine sets.

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arXiv2021

On the multifractal spectrum of weighted Birkhoff averages

In this paper, we study the topological spectrum of weighted Birkhoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.

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arXiv2021

On the Hausdorff measure of shrinking target sets on self-conformal sets

In this article, we study the Hausdorff measure of shrinking target sets on self-conformal sets. The Hausdorff dimension of the sets we are interested in here was established by Hill and Velani in 1995. However, until recently, little more was known about the Hausdorff measure of these particular sets. In this paper we provide a complete characterisation of the Hausdorff measure of these sets, obtaining a dichotomy for the Hausdorff measure which is determined by the convergence or divergence of a sum depending on the radii of our "shrinking targets". Our main result complements earlier work of Levesley, Salp, and Velani~(2007), and recent work of Baker (2019).

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arXiv2020

Super-exponential condensation without exact overlaps

We exhibit self-similar sets on the line which are not exponentially separated and do not generate any exact overlaps. Our result shows that the exponential separation, introduced by Hochman in his groundbreaking theorem on the dimension of self-similar sets, is too weak to describe the full theory.

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arXiv2020

Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than $1$ then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to $1$. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit [BLMS, 2020]. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

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arXiv2020

Assouad dimension of planar self-affine sets

We calculate the Assouad dimension of a planar self-affine set $X$ satisfying the strong separation condition and the projection condition and show that $X$ is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set $X$ adheres to very strong tangential regularity by showing that any two points of $X$, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets.

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arXiv2020

Dimension of the repeller for a piecewise expanding affine map

In this paper, we study the dimension theory of a class of piecewise affine systems in euclidean spaces suggested by Michael Barnsley, with some applications to the fractal image compression. It is a more general version of the class considered in the work of Keane, Simon and Solomyak [The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition. {\it Fund. Math.}, {\bf 180}(3):279-292, 2003] and can be considered as the continuation of the works [On the dimension of self-affine sets and measures with overlaps. {\it Proc. Amer. Math. Soc.}, {\bf 144}(10):4427-4440, 2016], [On the dimension of triangular self-affine sets. {\it Erg. Th. \& Dynam. Sys.}, to appear.] by the authors. We also present some applications of our results for the generalized Takagi functions and fractal interpolation functions.

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arXiv2020

On the dimension of self-similar measures with complicated overlaps

In this paper, we investigate the Hausdorff dimension of the invariant measures of the iterated function system (IFS) $\{αx, βx, γx+(1-γ)\}$. We provide an "almost every" type result by a direct application of the results of Feng and Hu; and Kamalutdinov and Tetenov.

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arXiv2019

Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles

In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of self-affine sets, and the theory of random matrix products, it has often been found useful to assume positivity of the matrix entries in order to simplify or make feasible certain types of calculation. It is natural to ask how positivity may be relaxed or generalised in a way which enables similar calculations to be made in more general contexts. On the one hand one may generalise by considering almost additive or asymptotically additive potentials which mimic the properties enjoyed by the logarithm of the norm of a positive matrix cocycle; on the other hand one may consider matrix cocycles which are dominated, a condition which includes positive matrix cocycles but is more general. In this article we explore the relationship between almost additivity and domination for planar cocycles. We show in particular that a locally constant linear cocycle in the plane is almost additive if and only if it is either conjugate to a cocycle of isometries, or satisfies a property slightly weaker than domination which is introduced in this paper. Applications to matrix thermodynamic formalism are presented.

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arXiv2019

Dimension Theory of some non-Markovian repellers Part I: A gentle introduction

Michael Barnsley introduced a family of fractals sets which are repellers of piecewise affine systems. The study of these fractals was motivated by certain problems that arose in fractal image compression but the results we obtained can be applied for the computation of the Hausdorff dimension of the graph of some functions, like generalized Takagi functions and fractal interpolation functions. In this paper we introduce this class of fractals and present the tools in the one-dimensional dynamics and nonconformal fractal theory that are needed to investigate them. This is the first part in a series of two papers. In the continuation there will be more proofs and we apply the tools introduced here to study some fractal function graphs.

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