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.

In this paper we describe the spectral properties of semigroups expanding maps acting on Polish spaces, considering both sequences transfer operators along infinite compositions dynamics and integrated operators. We prove that there exists a limiting behaviour for such operators, these semigroup actions admit equilibrium states with exponential decay correlations several limit theorems. The ref...

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...

We provide a formal proof of equivalence between the class of fractals created by Recursive Turtle Programs (RTP) and Iterated Affine Transformations (IAT). We begin by reviewing RTP ( a geometric interpretation of non-bracketed L-systems with a single production rule) and IAT (Iterated Function Systems restricted to affine transformations). Next, we provide a simple extension to RTP that gener...

We consider attractors for certain types of random dynamical systems. These are skew-product systems whose base transformations preserve an ergodic invariant measure. We discuss definitions of invariant sets, attractors and invariant measures for deterministic and random dynamical systems. Under assumptions that include, for example, iterated function systems, but that exclude stochastic differ...

Visualization provides powerful tools for the investigation of dynamical systems. The application of various visualization techniques to complex and chaotic dynamical systems is discussed. The interactive specification and modification of strange attractors allow an easier understanding of the underlying dynamics. Graphical time series analysis visualizes time series and phase space reconstruct...

in this paper, we study l1 -solutions of the following two-direction poly-scale refinement equationwe prove that the vector space of all l1 -solutions of the above equation is at most one-dimensional andconsists of compactly supported functions of constant sign. we also show that any- l1 solution of the aboveequation is either positive or negative on its support under a special assumption. with...