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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., Comment: 25 pages, 5 figures

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 K\'aroly 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. B\'ar\'any, K. Simon, B. Solomyak and A. \'Spiewak: 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.

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., Comment: 22 pages, 3 figures

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper Jordan, Pollicott and Simon (Commun. Math. Phys., 2007). For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure.

Quantization dimension has been computed for many invariant measures of dynamically defined fractals having well separated cylinders, that is, in the cases when the so-called Open Set Condition (OSC) holds. To attack the same problem in case of heavy overlaps between the cylinders, we consider a family of self-similar measures, for which the underlying Iterated Function System satisfies the so-called Weak Separation Property (WSP) but does not satisfy the OSC since complete overlaps occur in between the cylinders. The work in this paper also shows that the quantization dimension determined for the set of overlap self-similar construction satisfying the WSP has a relationship with the temperature function of the thermodynamic formalism.

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 $t_i\in\mathbb{Q}$., Comment: 54 pages

We consider one-parameter families of smooth uniformly contractive iterated function systems $\{f^\lambda_j\}$ on the real line. Given a family of parameter dependent measures $\{\mu_{\lambda}\}$ on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps $\Pi^\lambda$. The main novelty of our work is that the measures $\mu_\lambda$ 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 $(\Pi^\lambda)_*\mu_\lambda$, where we had to develop a new approach in place of earlier attempt which contains an error. Our main result states that if $\mu_\lambda$ are Gibbs measures for a family of H\"older continuous potentials $\phi^\lambda$, with H\"older continuous dependence on $\lambda$ and $\{\Pi^\lambda\}$ satisfy the transversality condition, then the projected measure $(\Pi^\lambda)_*\mu_\lambda$ is absolutely continuous for Lebesgue a.e.\ $\lambda$, 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).

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 ${\Bbb R}^d$., Comment: Minor changes. To appear in Ergodic Theory Dynam. Systems

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.

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, K\"aenm\"aki 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., Comment: 27 pages including Appendix; v2: minor updates, accepted version in RSE Proc A

This is the second part in a series of two papers. Here, we give an overview on the dimension theory of some dynamically defined function graphs, like Takagi and Weierstrass function, and we study the dimension of Markovian fractal interpolation functions and generalised Takagi functions generated by non-Markovian dynamics.

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.

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 Komj\'athy and Simon, Song-Havlin-Makse model, spherically symmetric trees, and supercritical Galton-Watson trees., Comment: 26 pages, 6 figures

We construct a family of planar self-affine carpets with overlaps using lower triangular matrices in a way that generalizes the original Gatzouras--Lalley carpets defined by diagonal matrices. Assuming the rectangular open set condition, Bara\'nski proved for this construction that for typical parameters, which can be explicitly checked, the inequalities between the Hausdorff, box and affinity dimension of the attractor are strict. We generalize this result to overlapping constructions, where we allow complete columns to be shifted along the horizontal axis or allow parallelograms to overlap within a column in a transversal way. Our main result is to show sufficient conditions under which these overlaps do not cause the drop of the dimension of the attractor. Several examples are provided to illustrate the results, including a self-affine smiley, a family of self-affine continuous curves, examples with overlaps and an application of our results to some three-dimensional systems., Comment: 12 figures; v2: improved presentation, updated references, added a three-dimensional example and an Appendix. Results unchanged

One of the most well known random fractals is the so-called Fractal percolation set. This is defined as follows: we divide the unique cube in $\mathbb{R}^d$ into $M^d$ congruent sub-cubes. For each of these cubes a certain retention probability is assigned with which we retain the cube. The interior of the discarded cubes contain no points from the Fractal percolation set. In the retained ones we repeat the process ad infinitum. The set that remains after infinitely many steps is the Fractal percolation set. The homogeneous case is when all of these probabilities are the same. Recently, there have been considerable developments in regards with the projection and slicing properties in the homogeneous case. In the first part of this note we give an account of some of these recent results and then we discuss the difficulties and provide some new partial results in the non-homogeneous case., Comment: 16 pages

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.

Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior $\left(A+\Gamma\right)^{\circ}$, when $\Gamma$ is a piecewise $\mathcal{C}^2$ curve and $A\subset \mathbb{R}^2.$ To begin, we give an example of a very large (full-measure, dense, $G_\delta$) set $A$ such that $\left(A+S^1\right)^{\circ}=\emptyset$, where $S^1$ denotes the unit circle. This suggests that merely the size of $A$ does not guarantee that $(A+S^1)^{\circ }\ne\emptyset$. If, however, we assume that $A$ is a kind of generalized product of two reasonably large sets, then $\left(A+\Gamma\right)^{\circ}\ne\emptyset$ whenever $\Gamma$ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of $C:=C_{\gamma}\times C_{\gamma}$, $\gamma \geq \frac{1}{3}$, pinned at any point of $C$ has non-empty interior, where $C_{\gamma}$ (see (1.1)) is the middle $1-2\gamma$ Cantor set (including the usual middle-third Cantor set, $C_{1/3}$). Our proof for the middle-third Cantor set requires a separate method. We also prove that $C+S^1$ has non-empty interior.

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 $A+\Gamma:=\left\{a+v:a\in A, v\in \Gamma \right\}$ when $A\subset \mathbb{R}^2$ and $\Gamma$ is a piecewise $\mathcal{C}^2$ curve. Assuming $\Gamma$ has non-vanishing curvature, we verify that (a) if $\dim_{\rm H} A \leq 1$, then $\dim_{\rm H} (A+\Gamma)=\dim_{\rm H} A +1$, where $\dim_{\rm H}$ denotes the Hausdorff dimension; (b) if $\dim_{\rm H} A>1$, then $Leb_2(A+\Gamma)>0$, where $Leb_2$ denotes the $2$-dimensional Lebesgue measure; (c) if $\dim_{\rm H} A=1$ and $H^1(A) < \infty$, then $Leb_2(A+\Gamma)=0$ 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 $t\in \mathbb{R}^2$.

Multifractal characteristics of the Internet traffic have been discovered and discussed in several research papers so far. However, the origin of this phenomenon is still not fully understood. It has been proven that the congestion control mechanism of the Internet transport protocol, i.e., the mechanism of TCP Reno can generate multifractal traffic properties. Nonetheless, TCP Reno does not exist in today's network any longer, surprisingly traffic multifractality has still been observed. In this paper we give the theoretical proof that TCP CUBIC, which is the default TCP version in the Linux world, can generate multifractal traffic. We give the multifractal spectrum of TCP CUBIC traffic and compare it with the multifractal spectrum of TCP Reno traffic. Moreover, we present the multifractal spectrum for a more general model, where TCP CUBIC and TCP Reno are special cases. Our results also show that TCP CUBIC produces less bursty traffic than TCP Reno., Comment: 42 pages, 9 figures

In this note we present some one-parameter families of homogeneous self-similar measures on the line such that - the similarity dimension is greater than $1$ for all parameters and - the singularity of some of the self-similar measures from this family is not caused by exact overlaps between the cylinders. We can obtain such a family as the angle-$\alpha$ projections of the natural measure of the Sierpi\'nski carpet. We present more general one-parameter families of self-similar measures $\nu_\alpha$, such that the set of parameters $\alpha$ for which $\nu_\alpha$ is singular is a dense $G_\delta$ set but this "exceptional" set of parameters of singularity has zero Hausdorff dimension.

As a continuation of a recent work of the same authors (arXiv:1504.07138), in this note we study the dimension theory of diagonally homogeneous triangular planar self-affine IFS., Comment: We weakened the assumptions of one our theorem and modified the formalization. The proofs became simpler. We also added new example

In this paper we consider fractal percolation random Cantor sets $E$ on the plane constructed with non-homogeneous probabilities. We focus on the case when the probabilities are large enough to guarantee that the almost sure dimension of $E$ is greater than $1$. Under this assumption in the case of homogeneous (equal) probabilities it was proved by Rams and the first author that the orthogonal projection of $E$ contains an interval simultaneously in all directions. Moreover, Peres and Rams proved the stronger result that the orthogonal projection of the natural measure on $E$ to every line is absolutely continuous with H\"older-continuous density. We point out that in the case of non-homogeneous probabilities neither of the two previous assertions remain valid. However, we also prove that in the non-homogeneous case every line whose tangent is neither a rational nor a Liouville number is non-exceptional. That is, almost surely for all of these directions the projection of $E$ contains some interval and the projection of the natural measure has H\"older-continuous density., Comment: This paper contains a crucial error in the proof of Lemma 4.3. which affects the main result Theorem 2.1. as well. We could not fix it yet

In this paper we consider diagonally affine, planar IFS $\Phi=\left\{S_i(x,y)=(\alpha_ix+t_{i,1},\beta_iy+t_{i,2})\right\}_{i=1}^m$. Combining the techniques of Hochman and Feng, Hu we compute the Hausdorff dimension of the self-affine attractor and measures and we give an upper bound for the dimension of the exceptional set of parameters.

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., Comment: 19 pages, 5 figures

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. {itemize}, Comment: Survey submitted for AFRT2012 conference

In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset \mathbb{R}^2 $ which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than 1 then the orthogonal projection to \textbf{every} line, the radial projection with \textbf{every} center, and distance set from \textbf{every} point contain intervals., Comment: to appear in ETDS

\emph{Fractal percolation} or \emph{Mandelbrot percolation} is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by $E\subset \mathbb{R}^2$ a typical realization of the fractal percolation on the plane, {itemize} If $\dim_{\rm H}E<1$ then for \textbf{all}lines $\ell$ the orthogonal projection $E_\ell$ of $E$ to $\ell$ has the same Hausdorff dimension as $E$, If $\dim_{\rm H}E>1$ then for any smooth real valued function $f$ which is strictly increasing in both coordinates, the image $f(E)$ contains an interval. {itemize} The second statement is quite interesting considering the fact that $E$ is almost surely a Cantor set (a {\it random dust}) for a large part of the parameter domain, see \cite{Chayes1988}. Finally, we solve a related problem about the existence of an interval in the algebraic sum of $d\geq 2$ one-dimensional fractal percolations., Comment: Corrected version

We investigate the dimension of intersections of the Sierpi\'nski gasket with lines. Our first main result describes a countable, dense set of angles that are exceptional for Marstrand's theorem. We then provide a multifractal analysis for the set of points in the projection for which the associated slice has a prescribed dimension., Comment: 19 pages, 4 figures

Albert-Laszlo Barabasi introduced a model which exhibits the bursty nature of the arrival times of events in systems determined by decisions of some humans. In Barabasi's model tasks are selected to execution according to some rules which depends on the priorities of the tasks. In this paper we generalize the selection rule of the A.-L. Barabasi priority queuing model. We show that the bursty nature of human behavior can be explained by a model where tasks are selected proportional to their priorities. In addition, we extend some of Vazquez's heuristic arguments to analytic proofs., Comment: 3 figures

Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabasi, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal $\Lambda $. With rigorous mathematical results we verify that our model captures some of the most important features of many real networks: the scale free and the high clustering properties. We also prove that the diameter is the logarithm of the size of the system. Using our (deterministic) fractal $\Lambda $ we generate random graph sequence sharing similar properties., Comment: 4 figures

In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson., Comment: Published in at http://dx.doi.org/10.1214/10-AOP558 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two independent copies. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a slightly more general construction of random Cantor sets. We also present a complete solution for the deterministic case., Comment: This replacement corrects an important omission in the proof of Theorem 1(a)

We prove that a purely unrectifiable self-similar set of finite 1-dimensional Hausdorff measure in the plane, satisfying the Open Set Condition, has radial projection of zero length from every point., Comment: 12 pages, 2 figures

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y x\}_{i=1}^m$, where $d_i\in \R$ and $\lambda_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of i.i.d. errors $y_1,y_2,...$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $\chi = E[\log(\lambda Y)]$ be the Lyapunov exponent. Assuming that $\chi < 0$, we obtain a family of conditional measures $\nu_y$ on the line, parametrized by $y = (y_1,y_2,...)$, the sequence of errors. Our main result is that if $h > |\chi|$, then $\nu_y$ is absolutely continuous with respect to the Lebesgue measure for a.e. $y$. We also prove that if $h < |\chi|$, then the measure $\nu_y$ is singular and has dimension $h/|\chi|$ for a.e. $y$. These results are applied to a randomly perturbed IFS suggested by Y. Sinai, and to a class of random sets considered by R. Arratia, motivated by probabilistic number theory., Comment: 22 pages