We consider iterated function systems on the interval with random perturbation. Let Yε be uniformly distributed in [1 − ε, 1 + ε] and let fi ∈ C be contractions with fixpoints ai. We consider the iterated function system {Yεfi + ai(1− Yε)}i=1, were each of the maps are chosen with probability pi. It is shown that the invariant density is in L and the L-norm does not grow faster than 1/ √ ε, as ...

In earlier work [5], aimed at developing an approach to signal processing that can be applied as well to nonlinear systems as linear ones, we produced mathematical models of digital communications channels that took the form of iterated function systems (IFS). For finite-dimensional systems these models have observability properties indicating they could be used for signal processing applicatio...

The Ruelle operator theorem has been studied extensively both in dynamical systems and iterated function systems. In this paper we study the Ruelle operator theorem for non-expansive systems. Our theorems give some sufficient conditions for the Ruelle operator theorem to be held for a non-expansive system.

We consider calculation of the dimensions of self-affine fractals and multifractals that are the attractors of iterated function systems specified in terms of upper triangular matrices. Using methods from linear algebra we obtain explicit formulae for the dimensions that are valid in many cases.

Given a bounded open subset Ω of Rd (d 1) and a positive finite Borel measure μ supported on Ω with μ(Ω) > 0, we study a Laplace-type operator μ that extends the classical Laplacian. We show that the properties of this operator depend on the multifractal structure of the measure, especially on its lower L∞-dimension dim∞(μ). We give a sufficient condition for which the Sobolev space H 1 0 (Ω) i...

The concept of Self-Organized Criticality (SOC) was proposed in an attempt to explain the widespread appearance of power-law in nature. It describes a mechanism in which a system reaches spontaneously a state where the characteristic events (avalanches) are distributed according to a power law. We present a dynamical systems approach to Self-Organized Criticality where the dynamics is described...