arXiv subjects

Károly Simon

Publications and source records attributed to Károly Simon.

At least 19 recordsLinked to original sources

arXiv2026

Hausdorff dimension of planar box-like self-affine sets with rotations

We study the Hausdorff dimension of proximal planar self-affine sets whose defining maps have diagonal or anti-diagonal linear parts, with both types present. Such systems are irreducible but not strongly irreducible. Under a projective exponential separation condition, we show that the Hausdorff dimension equals the minimum of $2$ and the affinity dimension, thereby extending the dimension formula of Hochman--Rapaport to all proximal irreducible planar systems. When the affinity dimension is at most an explicit threshold determined by the Lyapunov exponents, the formula holds assuming only that there are no exact overlaps. Above the threshold, it holds under the strong open set condition. These dimension formulae are proved by approximation with finite diagonal subsystems. Assuming projective exponential separation, we also determine the pointwise dimensions of the coordinate projections of the equilibrium state. Below the threshold, the projections are not exact-dimensional: the pointwise dimension of each exists almost everywhere and takes exactly two values, each on a set of positive measure. At or above the threshold, both projections are exact-dimensional with dimension one.

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arXiv2026

Open dynamics with a moving hole: inhomogeneous IFS and exponential separation condition

Let $\mathcal F=\{f_i(x)=r_i x+t_i\}_{i=0}^{b-1}$ be a self-similar iterated function system (IFS) on $\mathbb R$ with $b\ge 3$, and let $K_{\mathcal F}$ be the associated self-similar set. Given $m\in\mathbb N$, for any sequence $ω=ω^0ω^1\ldots$ with each $ω^i\in \{0,1,\ldots,b-1\}^m$ a block of length $m$, let \[ K_{\mathcal F}^ω:=π_{\mathcal F}\left(\{(d_i)\in \{0,1,\ldots,b-1\}^{\mathbb N}: d_{k+1}d_{k+2}\ldots d_{k+m}\neω^k\quad\forall k\ge 0\}\right), \] where $π_{\mathcal F}$ is the natural coding map from $\{0,1,\ldots,b-1\}^{\mathbb N}$ to $K_{\mathcal F}$. The set $K_{\mathcal F}^ω$ is called the survivor set of an open dynamical system with a moving hole related to $ω$. When $\mathcal F$ is a homogeneous IFS satisfying the open set condition (OSC), the dimensions of $K_{\mathcal F}^ω$ were determined by the first author, Sun and Wang (Adv. Math., 2026). In this paper we generalize these results to inhomogeneous IFSs. When $\mathcal F$ satisfies the OSC, we give explicit formulae for the dimensions of $K_{\mathcal F}^ω$. In particular, we show that the packing dimension and upper box-counting dimension of $K_{\mathcal F}^ω$ always coincide, while the Hausdorff dimension and lower box-counting dimension of $K_{\mathcal F}^ω$ are not necessarily equal, which is different from the homogeneous case. Furthermore, when $\mathcal F$ satisfies the exponential separation condition and the similarity dimension $s_{\mathcal F}\le 1$, we are able to determine the Hausdorff dimension of $K_{\mathcal F}^ω$. Finally, we apply our results to badly approximable numbers and non-recurrence points in the setting of inhomogeneous IFS satisfying the OSC.

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arXiv2026

Random Survivor Sets with Moving Holes in Bedford-McMullen Carpets

Let $2\le m < n$ be integers, and let $K(\mathcal{A})$ be the Bedford--McMullen carpet generated by $\{(x,y)\mapsto((x+i)/m,(y+j)/n):(i,j)\in\mathcal{A}\}$, where $\mathcal{A}$ is a subset of $\{0,1,\ldots,m-1\}\times\{0,1,\ldots,n-1\}$. Write $\mathcal{B}$ for the set of occupied columns, and assume that $\#\mathcal{B}\ge2$ and each column contains at least two symbols of $\mathcal{A}$. At each vertex of the column tree $\mathcal{B}^*$, we independently choose one forbidden symbol from $\mathcal{A}$ according to a prescribed distribution. Let $\mathbb{P}$ denote the resulting product measure on $\mathcal{A}^{\mathcal{B}^*}$. This gives a random survivor set $K_ω(\mathcal{A})$ with $ω\in\mathcal{A}^{\mathcal{B}^*}$. We show that, for $\mathbb{P}$-a.e. $ω$, \begin{equation*} \dim_H K_ω(\mathcal{A})=\frac{1}{\log m}\log\left(\int \sum_{b\in\mathcal{B}}\ell_b(ω)^{\frac{\log m}{\log n}} d\mathbb{P}(ω)\right), \end{equation*} where $\ell_b(ω)$ is the number of surviving symbols in column $b$ at the root.

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arXiv2026

Coding multiplicity in self-similar sets on the line

We study coding multiplicity for self-similar sets on the real line. When the similarity dimension exceeds one, almost every point with respect to the natural self-similar measure has infinitely many codings. We focus on the case when the similarity dimension is at most one. Our analysis proceeds through the recurrence properties and the size of the critical set, defined as the union of the pairwise intersections of the first-level pieces. Our main structural result shows that when the critical set is finite, unboundedness of coding multiplicity is equivalent to periodic recurrence to the critical set. Moreover, every commensurable system with a finite critical set has uniformly bounded coding multiplicity. This conclusion is sharp in the incommensurable setting. We construct a one-parameter family of four-map systems such that for Lebesgue almost every parameter, the critical set is a singleton and the open set condition fails, while the attractor contains a point with infinitely many codings. More generally, for transversal families, the set of parameters admitting an infinite coding fibre is dense among those for which the strong separation condition fails. We also obtain a quantitative lower bound for the Hausdorff dimension of this parameter set. Such parameters can form a meagre set. We continue the study of the geometric size of branching, obtaining dimension formulae for typical coding fibres and almost-sure upper bounds on the dimension of the critical set in transversal families. In contrast, we construct an exponentially separated system with no dimension drop whose critical set has the same Hausdorff dimension as the attractor and whose coding map has a fibre of continuum cardinality. Although we work in the one-dimensional setting, several results and arguments extend to higher dimensions.

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arXiv2026

Dimension properties of a family of random self-similar iterated function systems

In this note we present a dimension formula for certain randomized self-similar sets. The self-similar sets we are considering are $d$-dimensional generalizations of rational projections of inhomogeneous sponges: they have overlaps, but the overlapping structure is in some sense well-organized. The randomization happens according to a labelled Galton-Watson tree corresponding to the cylinders of the IFS, for example as in the case of the Mandelbrot percolation set. The dimension formula is given in terms of a pressure function corresponding to the expectation matrices, which describes the overlapping structure and the randomness of the system.

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arXiv2025

Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously

We prove a universal projection theorem, giving conditions on a parametrized family of maps $Π_λ: X \to \mathbb{R}^d$ and a collection M of measures on X under which for almost every $λ$ equality $\dim_H Π_λμ= \min\{d, \dim_H μ\}$ holds for all measures $μ\in M$ simultaneously (i.e. on a full measure set of $λ$'s independent of $μ$). We require family $Π_λ$ to satisfy a transversality condition and collection M to satisfy a new condition called relative dimension separability. Under the same assumptions, we also prove that if the Assouad dimension of X is smaller than d, then for almost every $λ$, projection $Π_λ$ is nearly bi-Lipschitz (i.e. with pointwise $α$-Hölder inverse for every $α\in (0,1)$) at $μ$-a.e. x, for all $μ\in M$ simultaneously. Our setting encompasses families of orthogonal projections, natural projections corresponding to conformal iterated function systems, and non-autonomous or random IFS. As applications, we provide novel results on the multifractal analysis, giving formula for the Hausdorff dimension of a level set of the local dimension for a typical (w.r.t the translation parameter) self-similar measure on the line, valid for the full range spectrum (including the decreasing part of the spectrum; previous results were covering only the increasing part). Among another applications, we prove that given a parametrized contracting conformal IFS satisfying the transversality condition, for almost every parameter the dimension formula holds for all ergodic shift-invariant measures simultaneously. We also prove that the dimension part of the Marstrand's projection theorem holds simultaneously for the collection of all ergodic measures on a strongly separated self-conformal set and for the collection of all Gibbs measures on a self-conformal set (without any separation).

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arXiv2024

Interior points and Lebesgue measure of overlapping Mandelbrot percolation sets

We consider a special one-parameter family of d-dimensional random, homogeneous self-similar iterated function systems (IFSs) satisfying the finite type condition. The object of our study is the positivity of Lebesgue measure and the existence of interior points in these random sets and in particular the existence of an interesting parameter interval where the attractor has positive Lebesgue measure, but empty interior almost surely conditioned on the attractor not being empty. We give a sharp bound on the critical probability for the case of positivity Lebesgue measure using the theory of multitype branching processes in random environments and in some special cases on the critical probability for the existence of interior points. Using a recent result of Tom Rush, we provide a family of such random sets where there exists a parameter interval for which the corresponding attractor has a positive Lebesgue measure, but empty interior almost surely conditioned on the attractor not being empty.

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arXiv2024

Typical dimension and absolute continuity for classes of dynamically defined measures, Part II : exposition and extensions

This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of ergodic measures for parametrized families of smooth iterated function systems (IFS) on the line. Under some assumptions, most crucially, a transversality condition, we obtain formulas for the Hausdorff dimension of the measure and absolute continuity for almost every parameter in the appropriate parameter region. The main novelty of [1] and the present paper is that not only the IFS, but also the ergodic measure in the symbolic space, whose push-forward we consider, depends on the parameter. This includes many interesting families of measures, in particular, invariant measures for IFS's with place-dependent probabilities and natural (equilibrium) measures for smooth IFS's. One of the goals of this paper is to present an exposition of [1] in a more reader-friendly way, emphasizing the ideas and proof strategies, but omitting the more technical parts. This exposition/survey is based in part on the series of lectures by Károly Simon at the Summer School "Dynamics and Fractals" in 2023 at the Banach Center, Warsaw. The main new feature, compared to [1], is that we consider multi-parameter families; in other words, the set of parameters is allowed to be multi-dimensional. This broadens the scope of applications. A new application considered here is to a class of Furstenberg-like measures. [1] B. Bárány, K. Simon, B. Solomyak and A. Śpiewak: Typical absolute continuity for classes of dynamically defined measures. Advances in Mathematics, Volume 399, 2022, 108258, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2022.108258.

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arXiv2024

Multitype branching processes in random environments with not strictly positive expectation matrices

It is well known that under some conditions the almost sure survival probability of a multitype branching processes in random environment is positive if the Lyapunov exponent corresponding to the expectation matrices is positive, and zero if the Lyapunov exponent is negative. The goal of this note is to establish similar results when certain positivity conditions on the expectation matrices are not met. One application of such a result is to classify the positivity of Lebesgue measure of certain overlapping random self-similar sets in the line.

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arXiv2022

Dimension and measure of sums of planar sets and curves

Considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the measure and dimension of $A+Γ:=\left\{a+v:a\in A, v\in Γ\right\}$ when $A\subset \mathbb{R}^2$ and $Γ$ is a piecewise $\mathcal{C}^2$ curve. Assuming $Γ$ has non-vanishing curvature, we verify that (a) if $\dim_{\rm H} A \leq 1$, then $\dim_{\rm H} (A+Γ)=\dim_{\rm H} A +1$, where $\dim_{\rm H}$ denotes the Hausdorff dimension; (b) if $\dim_{\rm H} A>1$, then $Leb_2(A+Γ)>0$, where $Leb_2$ denotes the $2$-dimensional Lebesgue measure; (c) if $\dim_{\rm H} A=1$ and $H^1(A) < \infty$, then $Leb_2(A+Γ)=0$ if and only if $A$ is an irregular (purely unrectifiable) $1$-set. Here, $H^1$ denotes the $1$-dimensional Hausdorff measure. Items (a) and (b) follow from previous works of Wolff and Oberlin using Fourier analysis. In this article, we develop an approach using nonlinear projection theory which gives new proofs of (a) and (b) and the first proof of (c). Item (c) has a number of consequences: if a circle is thrown randomly on the plane, it will almost surely not intersect the four corner Cantor set. Moreover, the pinned distance set of an irregular $1$-set has $1$-dimensional Lebesgue measure equal to zero at almost every pin $t\in \mathbb{R}^2$.

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arXiv2022

Projections of the random Menger sponge

Using a similar random process to the one which yields the fractal percolation sets, starting from the deterministic Menger sponge we get the random Menger sponge. We examine its orthogonal projections from the point of Hausdorff dimension, Lebesgue measure and existence of interior points. We obtain these results as special cases of our theorems stated for random self-similar IFSs. These are obatained by a random process similar to the fractal percolation, applied for the cylinder sets of a deterministic self-similar IFS, as in arXiv:1212.1345. In this paper the associated deterministic IFS on the line is of the special form $\mathcal{S}=\left\{\frac{1}{L}x+t_i \right\} _{i=1}^{m}$, where $L\in\mathbb{N}$, $L\geq 2$ and $t_i\in\mathbb{Q}$.

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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

Dimension estimates for $C^1$ iterated function systems and repellers. Part II

This is the second part of our study of the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on ${\Bbb R}^d$. In the first part we proved that the upper box-counting dimension of the attractor of any $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parametrized families of $C^1$ IFSs, and show that these upper bounds give actually the dimensions for "typical" $C^1$ IFSs under this transversality condition. Moreover we verify the GTC for some parametrized families of $C^1$ IFSs on ${\Bbb R}^d$.

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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

Dimension estimates for $C^1$ iterated function systems and repellers. Part I

This is the first article in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for $C^1$ expanding maps on Riemannian manifolds.

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arXiv2020

Projections of Mandelbrot percolation in higher dimensions

We consider fractal percolation (or Mandelbrot percolation) which is one of the most well studied example of random Cantor sets. Rams and the first author studied the projections (orthogonal, radial and co-radial) of fractal percolation sets on the plane. We extend their results to higher dimension.

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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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arXiv2019

Transfinite fractal dimension of trees and hierarchical scale-free graphs

In this paper, we introduce a new concept: the transfinite fractal dimension of graph sequences motivated by the notion of fractality of complex networks proposed by Song et al. We show that the definition of fractality cannot be applied to networks with `tree-like' structure and exponential growth rate of neighborhoods. However, we show that the definition of fractal dimension could be modified in a way that takes into account the exponential growth, and with the modified definition, the fractal dimension becomes a proper parameter of graph sequences. We find that this parameter is related to the growth rate of trees. We also generalize the concept of box dimension further and introduce the transfinite Cesaro fractal dimension. Using rigorous proofs we determine the optimal box-covering and transfinite fractal dimension of various models: the hierarchical graph sequence model introduced by Komjáthy and Simon, Song-Havlin-Makse model, spherically symmetric trees, and supercritical Galton-Watson trees.

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