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 $\bm x\in S_λ\setminus\{\bm p_0,...,\bm p_{m-1}\}$ has $2^{\aleph_0}$ different addresses. If $λ$ is not too close to 1, then we can still have an overlap, but there exist $\bm x$'s which have a unique address. However, we prove that almost every $\bm x\in S_λ$ has $2^{\aleph_0}$ addresses, provided $S_λ$ contains no holes and at least one proper overlap. We apply these results to the case of expansions with deleted digits.
Furthermore, we give sharp sufficient conditions for the Open Set Condition to fail and for the attractor to have no holes.
These results are generalisations of the corresponding one-dimensional results, however most proofs are different.
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 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 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 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…
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 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 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 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 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 meas…
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(\…
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…
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…
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…
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.
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…
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 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…
Let $\{S_i\}_{i=1}^{l}$ be an iterated function system(IFS) on $\mathbb{R}^d$ with attractor K. Let $π$ be the canonical projection. In this paper we define a new concept called "projection pressure" $P_π(ϕ)$ for $ϕ\in C(\mathbb{R}^d)$ under certain affine IFS, and show the variational principle about the projection pressure. Furthermore we check that the unique zero root of "projection pressure" still satisfies Bowen's equation when each $S_i$ is the similar map with the same compression ratio. Using the root of Bowen's equation, we can get the Hausdorff dimension of the attractor $K$.
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…
We prove that certain families of homogenous affine iterated function systems in $\mathbb{R}^d$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters. These examples demonstrate that in the theorems of Falconer and Jordan-Pollicott-Simon on the almost sure dimensions of self-affine sets and measures, the set of exceptional translation parameters can be a dense set. The proof combines results from the literature on self-affine tilings of $\mathbb{R}^d$ with an adaptation of a classic argument of Erdős on the s…
Given $λ\in(0, 1)$, let $E_λ$ be the self-similar set generated by the iterated function system $\{x/3,(x+λ)/3,(x+2)/3\}$. Then $E_λ$ is a self-similar set with overlaps. We obtain the necessary and sufficient condition for $E_λ$ to be totally self-similar, which is a concept first introduced by Broomhead, Montaldi, and Sidorov in 2004. When $E_λ$ is totally self-similar, all its generating IFSs are investigated, and the size of the set of points having finite triadic codings is determined. Besides, we give some properties of the spectrum of $E_λ$ and show that the spectrum of $E_λ$ vanishes i…
This paper is an exposition, with some new applications, of our results on the growth of entropy of convolutions. We explain the main result on $\mathbb{R}$, and derive, via a linearization argument, an analogous result for the action of the affine group on $\mathbb{R}$. We also develop versions of the results for entropy dimension and Hausdorff dimension. The method is applied to two problems on the border of fractal geometry and additive combinatorics. First, we consider attractors $X$ of compact families $Φ$ of similarities of $\mathbb{R}$. We conjecture that if $Φ$ is uncountable and $X$ i…
Let $(Σ_A, σ)$ be a subshift of finite type and let $M(x)$ be a continuous function on $Σ_A$ taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where $M(x)$ takes finite values $M_1, ..., M_m$. The pressure function reduces to $P(q):=\lim_{n\to \infty}\frac{1}{n} \log \sum_{J \in \sum_{A, n}} \|M_J\|^q$. The expression is important when we consider the multifractal formalism for certain ite…
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…
