Disintegration results for fractal measures and applications to Diophantine approximation

We define a perturbed iterated function system (pIFS) in $\R^d$ as, loosely speaking, a sequence of iterated function systems (IFSs) whose constituent transformations converge towards some limiting IFS. We define the attractor of such a system in a similar style to that of an IFS, and prove that such a set exists uniquely. We define a partially perturbed IFS (ppIFS) to be a perturbed IFS with a constant tail. In a setup with similitudes and the strong separation condition we show that a pIFS attractor can be approximated by a sequence of ppIFS attractors in such a way that the Hausdorff measure is preserved in the limit. We use this result to calculate the exact Hausdorff measure of the pIFS attractor from that of the limiting IFS.

As is well known, when a self-similar set satisfies the strong separation condition, all self-similar measures are doubling. In this paper, we further prove that all Markov measures are doubling on a self-similar set with the strong separation condition. Subsequently, we focus on self-similar measures and Markov measures on Sierpinski carpets. Without the strong separation condition, Sierpinski carpets can be divided into different types. In each case, we fully characterize doubling self-similar measures and doubling Markov measures on a Sierpinski carpet.

Diophantine approximation and self-conformal measures

For a dust-like self-similar set (generated by an IFS of similarities with the strong separation condition), Elekes, Keleti and Máthé found an invariant, called algebraic dependence number, by considering its generating IFSs and isometry invariant self-similar measures. We find an intrinsic quantitative characterisation of this number: it is the dimension over \mathbb{Q} of the vector space generated by the logarithms of all the common ratios of infinite geometric sequences in the gap length set, minus 1. Using this, we present a lower bound on the minimal cardinality of generating IFSs (with or without separation conditions) in terms of the gap lengths of a dust-like self-similar set. We also establish an analogous result for dust-like graph-directed attractors on complete metric spaces, and present a new proof of the logarithmic commensurability theorem for IFSs with the strong separation condition. These are new applications of the ratio analysis method and the gap sequence.

Abstract. The natural projection of a parameter lower (upper) distri-bution set for a self-similar measure on a self-similar set satisfying theopen set condition is the cylindrical lower or upper local dimension setfor the Legendre self-similarmeasure which is derived from the self-similarmeasure and the self-similar set. 1. IntroductionRecently, we [1] investigated the relation between spectral classes of a self-similar Cantor set in a set theoretical sense. More recently, using the parameterdistribution, we find the parallel results for the self-similar set (attractor of theIFS consisting of n(≥ 2) similitudes satisfying the OSC (open set condition))instead of the self-similar Cantor set (attractor of the IFS consisting of 2 simil-itudes satisfying the SSC (strong separation condition)), which leads to a gen-eralization of [1]. In this paper, we define the Legendre self-similar measureson the self-similar set which is derived from the self-similar measure and theself-similar set. Using the Legendre self-similar measures on the self-similar set,we give full relationship between the natural projection of a parameter lower(upper) distribution set for a self-similar measure on a self-similar set and thecylindrical lower or upper local dimension set for the Legendre self-similar mea-sures.2. PreliminariesLet N and R be the set of positive integers and the set of real numbersrespectively. An attractor K in the d-dimensional Euclidean space R

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 which 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 that our results apply to include those arising from Bernoulli convolutions, the{0,1,3}\{0,1,3\}problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of some well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the{0,1,3}\{0,1,3\}problem has positive Lebesgue measure.For eacht∈[0,1]t\in [0,1]we letΦt\Phi _tbe the iterated function system given byΦt≔{ϕ1(x)=x2,ϕ2(x)=x+12,ϕ3(x)=x+t2,ϕ4(x)=x+1+t2}.\begin{equation*} \Phi _{t}≔\Big \{\phi _1(x)=\frac {x}{2},\phi _2(x)=\frac {x+1}{2},\phi _3(x)=\frac {x+t}{2},\phi _{4}(x)=\frac {x+1+t}{2}\Big \}. \end{equation*}We prove that eitherΦt\Phi _tcontains an exact overlap, or we observe Khintchine like behaviour. Our analysis shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures.Last of all, we introduce a property of an iterated function system that we call being consistently separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous. We include several explicit examples of consistently separated iterated function systems.

Fractal vector measures and vector calculus on planar fractal domains

We deal with the question of continuity of numerical values of Hausdorff measures in parametrised families of linear (similarity) and conformal dynamical systems by developing the pioneering work of Lars Olsen and the work [SUZ]. We prove Hölder continuity of the function ascribing to a parameter the numerical value of the Hausdorff measure of either the corresponding limit set or the corresponding Julia set. We consider three cases. Firstly, we consider the case of parametrised families of conformal iterated function systems in $\mathbb{R}$k with k ⩾ 3. Secondly, we consider all linear iterated function systems consisting of similarities in $\mathbb{R}$k with k ⩾ 1. In either of these two cases, the strong separation condition is assumed. In the latter case the Hölder exponent obtained is equal to 1/2. Thirdly, we prove such Hölder continuity for analytic families of conformal expanding repellers in the complex plane $\mathbb{C}$. Furthermore, we prove the Hausdorff measure function to be piecewise real–analytic for families of naturally parametrised linear IFSs in $\mathbb{R}$ satisfying the strong separation condition. On the other hand, we also give an example of a family of linear IFSs in $\mathbb{R}$ for which this function is not even differentiable at some parameters.

In this paper we study the absolute continuity of self-similar measures defined by iterated function systems (IFS) whose contraction ratios are not uniform. We introduce a transversality condition for a multi-parameter family of IFS and study the absolute continuity of the corresponding self-similar measures. Our study is a natural extension of the study of Bernoulli convolutions by Solomyak, Peres, et al.

The vector-valued measure generated by the self-conformal iterated function system (IFS) was introduced by the author (2004 J. Math. Anal. Appl. 299 341–56). Its variation is scalar measure. This paper proves that the Lq-spectrum of the variation measure is differentiable and the corresponding multifractal formalism holds, if the relevant IFS satisfies the open set condition. This result can be applied to the study of self-conformal (scalar) measures generated by certain IFS with overlaps.

For some fractal measures it is a very difficult problem in general to prove the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In fact there are examples of extremely sparse sets that are not even Bessel spectra. In this article, we investigate this problem for general fractal measures induced by iterated function systems (IFS). We prove some existence results of spectra associated with Hadamard pairs. We also obtain some characterizations of Bessel spectrum in terms of finite matrices for affine IFS measures, and one sufficient condition of frame spectrum in the case that the affine IFS has no overlap.

Let μ be the attracting measure of a condensation system associated with a self-similar measure ν. We determine the upper and lower quantization dimension of μ under the strong separation condition.

In this work, we focus on the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure that determines the modified lower box dimension Moran fractal sets. The equivalence of these measures for a class of Moran is shown by having a strong separation condition. We give a sufficient condition for the equality of the Hewitt–Stromberg dimension, Hausdorff dimension, and packing dimensions. As an application, we obtain some relevant conclusions about the Hewitt–Stromberg measures and dimensions of the image measure of a τ-invariant ergodic Borel probability measures. Moreover, we give some statistical interpretation to dimensions and corresponding geometrical measures.

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 to this measure are described by a distinguished subset called the essential class, and that the dimension of the support can be computed almost surely from knowledge of this essential class. For a special subcase, that we call commuting, we prove that the set of attainable local dimensions is almost surely a closed interval. Particular examples of such random measures are analyzed in more detail.

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics.