We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partially ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only restriction that a metric space considered has the Heine–Borel property. As a by–product, we also obtain so...

We prove singularity of some distributions of random continued fractions that correspond to iterated function systems with overlap and a parabolic point. These arose while studying the conductance of GaltonWatson trees. §

Let (X, d) be a compact metric space, and let an iterated function system (IFS) be given on X, i.e., a finite set of continuous maps σi: X → X, i = 0, 1, · · · , N − 1. The maps σi transform the measures μ on X into new measures μ i . If the diameter of σi1 ◦ · · · ◦ σik (X) tends to zero as k → ∞, and if pi > 0 satisfies ∑ i pi = 1, then it is known that there is a unique Borel probability mea...

The term fractal was first introduced by Mandelbrot to denote sets with highly irregular structures. Mandelbrot and others have then used fractals to model various natural phenomena (for example, [19, 2]). A mathematical development of fractals via the theory of (non-random) self-similar sets and measures was given by Hutchinson [12]. A family of contractive maps {Sj}^ on E m is called an itera...

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 R d , 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 R d , (the support of µ) and the mea...

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τi, i = 1, . . . , N , in some Hausdorff space Y . There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X ⊂ Y . Typically, some kind of contractivity...

This paper provides a method to predict magnetic storm events based on the time series of the Dst index over the period 1981–2002. This method is based on the multiple scaling of the measure representation of the Dst time series. This measure is modeled as a recurrent iterated function system, which leads to a method to predict storm patterns included in its attractor. Numerical results are pro...

We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine tiles. Among the interesting aspects are that even if the affine contraction underlying the iterated function system is not conjugated to a similarity we obtain an upperand and lower-bound for its Hausdorff dimension. In fact, we obtain the exact value for the dimension if the ...

Recall that a Borel probability measure μ on IR is called extremal if μ-almost every number in IR is not very well approximable. In this paper, we prove extremality (and implying it the exponentially fast decay property (efd)) of conformal measures induced by 1-dimensional finite parabolic iterated function systems. We also investigate the doubling property of these measures and we estimate fro...

In the paper we address the problem of computation in recurrent neural networks (RNN). In the rst part we provide a formal analysis of the dynamical behavior of a RNN with a single self{recurrent unit in the hidden layer, show how such a RNN may be designed to perform an (unrestricted) counting task and describe a generalization of the counter network that performs binary stack operations. In t...