A class of continua that are not attractors of any IFS

This paper presents a sufficient condition for a continuum in ? ⁿ to be embeddable in ? ⁿ in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ?² that is not an attractor of any weak iterated function system is also given.     

Approximation of measures by Markov processes and homogeneous affine iterated function systems

It is shown that under certain conditions, attractive invariant measures for iterated function systems (indeed for Markov processes on locally compact spaces) depend continuously on parameters of the system. We discuss a special class of iterated function systems, the homogeneous affine ones, for which an inverse problem is easily solved in principle by Fourier transform methods. We show that a probability measureμ onR ⁿ can be approximated by invariant measures for finite iterated function systems of this class if \(\hat \mu (t)/\hat \mu (a^T t)\) is positive definite for some nonzero contractive linear mapa:R ⁿ →R ⁿ . Moments and cumulants are also discussed.     

IFS attractors and Cantor sets

We build a metric space which is homeomorphic to a Cantor set but cannot be realized as the attractor of an iterated function system. We give also an example of a Cantor set K in R³ such that every homeomorphism f of R³ which preserves K coincides with the identity on K.     

Singularity and L2-dimension of self-similar measures

We study self-similar measures defined by non-uniformly contractive iterated function systems of similitudes with overlaps. In the case the contraction ratios of the similitudes are exponentially commensurable, we describe a method to compute the L²-dimension of the associated self-similar measures. Our result allows us to determine the singularity of some of such measures.     

Lipscomb’s L(A) Space Fractalized in l p (A)

In this paper, by using some of our new results concerning the shift space for an infinite IFS (see A. Mihail and R. Miculescu, The shift space for an infinite iterated function system, Math. Rep. Bucur. 11 (2009), 21?C32), we show that, for an infinite set A, the embedded version of the Lipscomb space L(A) in l ^p (A), ${p \in [1,\infty)}$ , with the metric induced from l ^p (A), denoted by ${\omega_p^A}$ , is the attractor of an infinite iterated function system comprising affine transformations of l ^p (A). In this way we provide a generalization of the positive answer that we gave to an open problem of J.C. Perry (see Lipscomb??s universal space is the attractor of an infinite iterated function system, Proc. Amer. Math. Soc. 124 (1996), 2479?C2489) in one of our previous works (see R. Miculescu and A. Mihail, Lipscomb space ?? ^A is the attractor of an infinite IFS containing affine transformations of l ²(A), Proc. Amer. Math. Soc. 136 (2008), 587?C592). Moreover, as a byproduct, we provide a generalization of Corollary 15 from Perry??s paper by proving that ${\omega_p^A}$ is a closed subset of l ^p (A).     

Invariant Measure Associated with a Generalized Countable Iterated Function System

Generalized countable iterated function systems (GCIFS) are an extension of countable iterated function systems by considering contractions from X × X into X instead of contractions on the compact metric space X into itself. For a GCIFS endowed with a system of probabilities we associate an invariant and normalized Borel measure whose support is just the attractor of the respective GCIFS, extending the classical Hutchinson’s construction.     

V-variable fractals: Fractals with partial self similarity

We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V-variable fractal sets or measures. These V-variable fractals can also be obtained from the points on the attractor of a single deterministic iterated function system. Existence, uniqueness and approximation results are established under average contractive assumptions. We also obtain extensions of some basic results concerning iterated function systems.     

Bessel sequences of exponentials on fractal measures

Jorgensen and Pedersen have proven that a certain fractal measure ν has no infinite set of complex exponentials which form an orthonormal set in L²(ν). We prove that any fractal measure μ obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a Riesz basic sequence, or more generally a Bessel sequence, in L²(μ) such that the frequencies have positive Beurling dimension.     

Diophantine approximation and self-conformal measures

It is proved that the Hausdorff measure on the limit set of a finite conformal iterated function system is strongly extremal, meaning that almost all points with respect to this measure are not multiplicatively very well approximable. This proves Conjecture 10.6 from (on fractal measures and Diophantine approximation, preprint, 2003). The strong extremality of all (S,P)-invariant measures is established, where S is a finite conformal iterated function system and P is a probability vector. Both above results are consequences of the much more general Theorem 1.5 concerning Gibbs states of Hölder families of functions.     

Iterated maximal functions in variable exponent Lebesgue spaces

In this paper we study the iterated Hardy?CLittlewood maximal operator in variable exponent Lebesgue spaces with exponent allowed to reach the value 1. We use modulars where the L ^p(·)-modular is perturbed by a logarithmic-type function, and the results hold also in the more general context of such Musielak?COrlicz spaces.     

Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.     

Boundedness for iterated commutators on the mixed norm spaces

This paper studies the iterated commutators on mixed norm spaces L²(?) characterizing the conjugate holomorphic symbols for which the corresponding iterated commutators are bounded by using the Bergman geometry, properties of holomorphic functions and related analysis.     

On some algorithmic problems regarding the hairpin completion

We consider a few algorithmic problems regarding the hairpin completion. The first problem we consider is the membership problem of the hairpin and iterated hairpin completion of a language. We propose an O(nf(n)) and O(n²f(n)) time recognition algorithm for the hairpin completion and iterated hairpin completion, respectively, of a language recognizable in O(f(n)) time. We show that the n factor is not needed in the case of hairpin completion of regular and context-free languages. The n² factor is not needed in the case of iterated hairpin completion of context-free languages, but it is reduced to n in the case of iterated hairpin completion of regular languages. We then define the hairpin completion distance between two words and present a cubic time algorithm for computing this distance. A linear time algorithm for deciding whether or not the hairpin completion distance with respect to a given word is connected is also discussed. Finally, we give a short list of open problems which appear attractive to us.     

Hausdorff dimension of self-similar sets with overlaps

We provide a simple formula to compute the Hausdorff dimension of the attractor of an overlapping iterated function system of contractive similarities satisfying a certain collection of assumptions. This formula is obtained by associating a non-overlapping infinite iterated function system to an iterated function system satisfying our assumptions and using the results of Moran to compute the Hausdorff dimension of the attractor of this infinite iterated function system, thus showing that the Hausforff dimension of the attractor of this infinite iterated function system agrees with that of the attractor of the original iterated function system. Our methods are applicable to some iterated function systems that do not satisfy the finite type condition recently introduced by Ngai and Wang.      

Ends of iterated function systems

Guided by classical concepts, we define the notion of ends of an iterated function system and prove that the number of ends is an upper bound for the number of nondegenerate components of its attractor. The remaining isolated points are then linked to idempotent maps. A commutative diagram illustrates the natural relationships between the infinite walks in a semigroup and components of an attractor in more detail. We show in particular that, if an iterated function system is one-ended, the associated attractor is connected, and ask whether every connected attractor (fractal) conversely admits a one-ended system.     

Infinite ϕ-periodic graphs

Given a graph-valued function ?, a graph G is called ? -periodic if there is some integer p ≥ 1 such that G is isomorphic to ?^p(G). In this paper it is shown how to construct ?-periodic graphs, given essentially an embedding of a graph H as a subgraph of some iterated ?-graph ?^p (H), provided ? obeys certain axioms. Then the results are applied to some graph-valued functions known in the literature.     

Euclidean self-similar sets generated by geometrically independent sets

For every integer n>0, we consider all iterated function systems generated by n+1 Euclidean similarities acting on Rn whose fixed points form the set of vertices of an n-simplex, and characterize the nature of attractors of such iterated function systems in terms of contractivity factors of their generators.     

On the asymptotic behaviour of the empirical random field of the brownian motion

Let ξ_t, t ? 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ??∫^t₀?(ξ_s) ds is investigated, where ? belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm.     

Chaos game representation of the Dst index and prediction of geomagnetic storm events

This paper proposes a two-dimensional chaos game representation (CGR) for the D_st index. The CGR provides an effective method to characterize the multifractality of the D_st time series. The probability measure of this representation is then modeled as a recurrent iterated function system in fractal theory, which leads to an algorithm for prediction of a storm event. We present an analysis and modeling of the D_st time series over the period 1963–2003. The numerical results obtained indicate that the method is useful in predicting storm events one day ahead.