We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + λ_i Y x\}_{i=1}^m$, where $d_i\in \R$ and $λ_i>0$ 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 $χ= E[\log(λY)]$ be the Lyapunov exponent. Assuming that $χ< 0$, we obtain a family of conditional measures $ν_y$ on the line, parametrized by $y = (y_1,y_2,...)$, the sequence of errors. Our main result is that if $h > |χ|$, then $ν_y$ is absolutely continuous with respect to the Lebesgue measure for a.e. $y$. We also prove that if $h < |χ|$, then the measure $ν_y$ is singular and has dimension $h/|χ|$ 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.
In this paper, we investigate parameter families of iterated function systems and continuity. Specifically, if we have a set of iterated function systems that depend continuously on a parameter, which properties of the invariant sets will vary continuously? We show here that the invariant sets will vary continuously as a function of the parameter. Furthermore, we give some sufficient conditions under which the Hausdorff dimension will depend continuously on the parameter. Lastly, we give a counter-example where the dimension is not a continuous function of the parameter.
We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy, the dimensional exactness of the projections of invariant measures from the shift space, and we give a formula for their dimension, in the context of random infinite conformal iterated function systems with overlaps. There exist many differences between our case and the finite deterministic case studied in [7], and we introduce new methods specific to the i…
We consider iterated function systems on the interval with random perturbation. Let $Y_ε$ be uniformly distributed in $[1- ε, 1 + ε]$ and let $f_i \in C^{1+α}$ be contractions with fixpoints $a_i$. We consider the iterated function system $\{Y_εf_i + a_i (1 - Y_ε) \}_{i=1}^n$, were each of the maps are chosen with probability $p_i$. It is shown that the invariant density is in $L^2$ and the $L^2$-norm does not grow faster than $1/\sqrtε$, as $ε$ vanishes.
We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems with overlaps $\mathcal S$. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for iterated function systems with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If $ν= π_*μ$ is a self-conformal measure, then $HD(ν) < \frac{h(μ)}{|χ(μ)|}$ i…
In this paper we extend previous work on IFSs without overlap. Our method involves systems of operators generalizing the more familiar Cuntz relations from operator algebra theory, and from subband filter operators in signal processing.
We review aspects of an important paper by Robert Strichartz concerning reverse iterated function systems (i.f.s.) and fractal blowups. We compare the invariant sets of reverse i.f.s. with those of more standard i.f.s. and with those of inverse i.f.s. We describe Strichartz' fractal blowups and explain how they may be used to construct tilings of $\mathbb{R}^{n}$ even in the case where the i.f.s. is overlapping. We introduce and establish the notion of "tops" of blowups. Our motives are not pure: we seek to show that a simple i.f.s. and an idea of Strichartz, can be used to create complicated …
In this paper we consider Iterated Function Systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exp…
In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine's theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of $\mathbb{Q}^d$ contained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary …
We study conformal iterated function systems (IFS) $\mathcal S = \{ϕ_i\}_{i \in I}$ with arbitrary overlaps, and measures $μ$ on limit sets $Λ$, which are projections of equilibrium measures $\hat μ$ with respect to a certain lift map $Φ$ on $Σ_I^+ \times Λ$. No type of Open Set Condition is assumed. We introduce a notion of overlap function and overlap number for such a measure $\hat μ$ with respect to $\mathcal S$; and, in particular a notion of (topological) overlap number $o(\mathcal S)$. These notions take in consideration the $n$-chains between points in the limit set. We prove that $o(\…
In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation. This result shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems our results apply to include those arising from Bernoulli convol…
In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of $1$-variable and $\infty$-variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap. Under the additional assumption of the uniform strong separati…
Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small scales. Iterated function systems satisfying these properties were shown to exist by the author and by Bárány and Käenmäki. In this paper we prove a general theorem on the existence of such iterated function systems within a parameterised family. This theorem shows that if a parameterised family contains two independent subfamilies, and the set of parameters…
This paper discusses, certain algebraic, analytic, and topological results on partial iterated function systems($IFS_p$'s). Also, the article proves the Collage theorem for partial iterated function systems. Further, it provides a method to address the points in the attractor of a partial iterated function system and obtain results related to the address of points in the attractor. The completeness of the partial metric space of contractions with a fixed contractivity factor is proved, under suitable conditions. Also, it demonstrates the continuity of the map that associates each contraction i…
Let $\bm p_0,...,\bm p_{m-1}$ be points in ${\mathbb R}^d$, and let $\{f_j\}_{j=0}^{m-1}$ be a one-parameter family of similitudes of ${\mathbb R}^d$: $$ f_j(\bm x) = λ\bm x + (1-λ)\bm p_j, j=0,...,m-1, $$ where $λ\in(0,1)$ is our parameter. Then, as is well known, there exists a unique self-similar attractor $S_λ$ satisfying $S_λ=\bigcup_{j=0}^{m-1} f_j(S_λ)$. Each $\bm x\in S_λ$ has at least one address $(i_1,i_2,...)\in\prod_1^\infty\{0,1,...,m-1\}$, i.e., $\lim_n f_{i_1}f_{i_2}... f_{i_n}({\bf 0})=\bm x$. We show that for $λ$ sufficiently close to 1, each $\bm x\in S_λ\setminus\{\bm p_0,..…
Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then the underlying iterated function system contains an exact overlap. In recent years significant progress has been made towards these conjectures. Hochman proved that if the Hausdorff dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then there are cylinders which are super-exponentially close at all small scales. Several years later, Shmerkin proved an analogous statement for the $L…
The affinity dimension is a number associated to an iterated function system of affine maps, which is fundamental in the study of the fractal dimensions of self-affine sets. De-Jun Feng and the author recently solved a folklore open problem, by proving that the affinity dimension is a continuous function of the defining maps. The proof also yields the continuity of a topological pressure arising in the study of random matrix products. I survey the definition, motivation and main properties of the affinity dimension and the associated SVF topological pressure, and give a proof of their continui…
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.
We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping intervals. Numerical techniques are described to test this conjecture, and a partial rigorous result in this direction is proven.
We study new relations between countable iterated function systems (IFS) with overlaps, Smale endomorphisms and random systems with complete connections. We prove that stationary measures for countable conformal IFS with overlaps and placedependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps S, and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the ge…
We investigate whether the Hutchinson operator associated with the iterated function system (IFS) is continuous. It clarifies several partial results scattered across recent literature. While the main example for IFS with strict attractor was provided by the family of contractions (the so-called hyperbolic system), the accent was put on ensuring that various contractivity-like conditions are preserved when the Hutchinson operator is induced, unless very recently it was discovered that strict attractors are quite often present for a large class of noncontractive maps, namely projective maps. Th…
Let $E$ be the self-similar set generated by the {\it iterated function system} {\[ f_0(x)=\frac{x}β,\quad f_1(x)=\frac{x+1}β, \quad f_{β+1}=\frac{x+β+1}β \]}with $β\ge 3$. {Then} $E$ is a self-similar set with complete {overlaps}, i.e., $f_{0}\circ f_{β+1}=f_{1}\circ f_1$, but $E$ is not totally self-similar. We investigate all its generating iterated function systems, give the spectrum of $E$, and determine the Hausdorff dimension and Hausdorff measure of $E$ and of the sets which contain all points in $E$ having finite or infinite different triadic codings.
We investigate the use of iterated function system (IFS) models for data analysis. An IFS is a discrete dynamical system in which each time step corresponds to the application of one of a finite collection of maps. The maps, which represent distinct dynamical regimes, may act in some pre-determined sequence or may be applied in random order. An algorithm is developed to detect the sequence of regime switches under the assumption of continuity. This method is tested on a simple IFS and applied to an experimental computer performance data set. This methodology has a wide range of potential uses:…
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 a…
We study skew product lifts and overlap numbers for equilibrium measures μ_ψof Hölder continuous potentials ψon such lifts. We find computable formulas and estimates for the overlap numbers in several concrete significant cases of systems with overlaps. In particular we obtain iterated systems which are asymptotically irrational-to-1 and absolutely continuous on their limit sets. Then we look into the general structure of the Rokhlin conditional measures of μ_ψwith respect to different fiber partitions associated to the lift Φ, and find relations between them. Moreover we prove an estimate on …
For a self-similar set in $\mathbb{R}^d$ that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least $1$. In this paper, it is shown that the Assouad dimension of such a set is the sum of the dimension of the vector space spanned by the set of $\textit{overlapping directions}$ and the Assouad dimension of the orthogonal projection of the set self-similar set onto the orthogonal complement of that vector space. This result is applied to give sufficient conditions on th…
The pressure function $P(A, s)$ plays a fundamental role in the calculation of the dimension of "typical" self-affine sets, where $A=(A_1,\ldots, A_k)$ is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on $A$. As a consequence, we show that the dimension of "typical" self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated …
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 a…
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…
We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the "overlap set" $\Ok$ is finite, and which are "invertible" on the attractor $A$, the sense that there is a continuous surjection $q: A\to A$ whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that $q$ is not a local homeomorphism precisely at $\Ok$. We suppose also that there is a rational function $p$ with the Julia set $J$ such that $(A,q)$ and $(J,p)$ are conjugate. We prove that if $A$ has bounded turning and $p$ has no parabol…
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 semig…
We study continuity and discontinuity properties of some popular measure-dimension mappings under some topologies on the space of probability measures in this work. We give examples to show that no continuity can be guaranteed under general weak, setwise or TV topology on the space of measures for any of these measure-dimension mappings. However, in some particular circumstances or by assuming some restrictions on the measures, we do have some (semi-)continuity results. We then apply our continuity results to concerning measures appearing in two kinds of infinite iterated function systems, nam…
We show how to turn the question of the absolute continuity of Bernoulli convolutions into one of counting the growth of the number of overlaps in the system. When the contraction parameter is a hyperbolic algebraic integer, we turn this question of absolute continuity into a question involving the ergodic theory of cocycles over domain exchange transformations.
We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the deterministic case. The overlapping case is more complicated; we introduce the notion of finite type for random homogeneous iterated function systems and give a formula for the local dimensions of finite type, regular, random homogeneous self-similar measures in terms of Lyapunov exponents of certain transition matrices. We show that almost all points with respect t…
We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at small scales there are super-exponentially close cylinders. This is a step towards the folklore conjecture that such a drop in dimension is explained only by exact overlaps, and confirms the conjecture in cases where the contraction parameters are algebraic. It also gives an affirmative answer to a conjecture of Furstenberg, showing that the projections of the "1-dimensional Sierpinski gasket" in irrational directio…
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 […
