Recurrent iterated function systems

Constructive Approximation - Recurrent iterated function systems generalize iterated function systems as introduced by Barnsley and Demko [BD] in that a Markov chain (typically with some zeros in...     

Reich-type iterated function systems

In this paper, we introduce the concept of Reich-type iterated function system and prove the existence and uniqueness of the attractor of such a system. Moreover, we study the properties of the canonical projection from the code space onto the attractor of such a system. We also present an iterated function system consisting of continuous Reich contractions having more than one attractor.     

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.     

Sensitivity of iterated function systems

The present work is concerned with the equicontinuity and sensitivity of iterated function systems (IFSs). Here, we consider more general case of IFSs, i.e. the IFSs generated by a family of relations. We generalize the concepts of transitivity, sensitivity and equicontinuity to these kinds of systems. This note investigates the relationships between these concepts. Then, several sufficient conditions for sensitivity of IFSs are presented. We introduce the notion of weak topologically exact for IFSs generated by a family of relations. It is proved that non-minimal weak topologically exact IFSs are sensitive. That yields to different examples of non-minimal sensitive systems which are not an M-system. Moreover, some interesting examples are given which provide some facts about the sensitive property of IFSs.     

Dual systems of algebraic iterated function systems

We study a class of graph-directed iterated function systems on R$\mathbb{R}$ with algebraic parameters, which we call algebraic GIFS. We construct a dual IFS of an algebraic GIFS, and study the relations between the two systems. We determine when a dual system satisfies the open set condition, which is fundamental. For feasible Pisot systems, we construct the left and right Rauzy–Thurston tilings, and study their multiplicities and decompositions. We also investigate their relation with codings space, domain-exchange transformation, and the Pisot spectrum conjecture. The dual IFS provides a unified and simple framework for Rauzy fractals, β-tilings and related studies, and allows us gain better understanding.     

Wavelets for iterated function systems

We construct a wavelet and a generalised Fourier basis with respect to some fractal measure given by a one-dimensional iterated function system. In this paper we will not assume that these systems are given by linear contractions generalising in this way some previous work of Dutkay, Jorgensen, and Pedersen to the non-linear setting. As a byproduct we are able to provide a Fourier basis also for such linear fractals like the Middle Third Cantor Set which have been left out by previous approaches.     

Infinite non-conformal iterated function systems

We consider a class of iterated function systems consisting of a countable infinity of non-conformal contractions, extending both the self-affine limit sets of Lalley and Gatzouras as well as the infinite iterated function systems of Mauldin and Urbański. Natural examples include the sets of points in the plane obtained by taking the binary expansion along the vertical and the continued fraction expansion along the horizontal and deleting certain pairs of digits. We prove that the Hausdorff dimension of the limit set is equal to the supremum of the dimensions of compactly supported ergodic measures, which are given by a Ledrappier and Young type formula. In addition we consider the multifractal analysis of Birkhoff averages for countable families of potentials. We obtain a conditional variational principle for the level sets.     

Dimension theory of iterated function systems

Let {S_i} be an iterated function system (IFS) on ?^d with attractor K. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, ??}. We define the projection entropy function h_π on the space of invariant measures on Σ associated with the coding map π : Σ → K and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (respectively, the direct product of finitely many conformal IFSs), without any separation condition, the projection of an ergodic measure under π is always exactly dimensional and its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (respectively, the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFSs, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. © 2008 Wiley Periodicals, Inc.     

On single-matrix graph-directed iterated function systems

We study graph-directed function systems where each contraction in the system has the form fe(x)=A−1(x+de), where A is an expanding matrix. We show that a certain discreteness implies the open set condition, and the latter implies the strong open set condition. Hausdorff measures and dimensions (w.r.t. a weak norm) of the invariant sets are investigated. The stationary Markov measures of the system are proved to be translation invariant.     

Graphs induced by iterated function systems

For an iterated function system (IFS) of similitudes, we define two graphs on the representing symbolic space. We show that if the self-similar set \(K\) has positive Lebesgue measure or the IFS satisfies the weak separation condition, then the graphs are hyperbolic; moreover the hyperbolic boundaries are homeomorphic to the self-similar sets.     

Correlation dimension for iterated function systems

The correlation dimension of an attractor is a fundamental dynamical invariant that can be computed from a time series. We show that the correlation dimension of the attractor of a class of iterated function systems in is typically uniquely determined by the contraction rates of the maps which make up the system. When the contraction rates are uniform in each direction, our results imply that for a corresponding class of deterministic systems the information dimension of the attractor is typically equal to its Lyapunov dimension, as conjected by Kaplan and Yorke.

     

Subdivision schemes for iterated function systems

We identify iterated function systems and regular Borel measures such that the matrix subdivision process relative to a finite family converges if and only if satisfies certain spectral properties.     

Tilings from graph directed iterated function systems

Geometriae Dedicata - A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system (GIFS), based on a combinatorial structure we call a...     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps. The authors are supported in part by an HKRGC grant.     

Overlapping iterated function systems on a segment

Overlapping iterated function systems generate families of injective mappings from the attractor onto shift-invariant subsets of the code space. In this paper we consider an example of such a family for the uniformly linear systems of iterated functions on the unit segment.     

Fourier frequencies in affine iterated function systems

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in Rd, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X,μ) where X is a compact subset of Rd (the support of μ), and the measure μ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L²(X,μ); and (2) explicit constructions of Fourier bases from the given data defining the IFS.     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of \mathbbR^d{{\mathbb{R}}^d} . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps.     

Constructing quasisymmetric functions via graph-directed iterated function systems

Using Tukia’s method for representing a quasisymmetric function as a quasisymmetric sieve, we generalize his modification to the Salem scheme and find a sufficient condition for the collection of functions that realize a structure parametrization of a graph-directed function system of a particular form (a one-dimensional multizipper) to consist of quasisymmetric functions. We give an asymptotically sharp estimate for the quasisymmetry coefficient of these functions in terms of the dilation coefficients of the mappings constituting a given multizipper, which yields a substantially more general method for constructing quasisymmetric functions than Tukia’s construction.