Dimension of invariant measures for affine iterated function systems

Iterated function systems, Ruelle operators, and invariant projective measures

We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in...

Quantization Dimension for Conformal Iterated Function Systems

Quantization Dimension for Infinite Conformal Iterated Function Systems

The quantization dimension function for an F -conformal measure mF generated by an infinite conformal iterated function system satisfying the strong open set condition and by a summable Hölder family of functions is expressed by a simple formula involving the temperature function of the system. The temperature function is commonly used to perform a multifractal analysis, in our context of the m...

A Characterization of Hyperbolic Affine Iterated Function Systems

The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on R. Atsushi Kameyama [Distances on topological self-similar sets, in Fractal Geometry and Applications: A Jubilee of Benǒit Mandelbrot. Proceedings of Symposia in Pure Mathematics, Volume 72, Part 1, 2004, pages 117–129] asked the following fundamental question: Given a topolo...

Rigorous Approximation of Stationary Measures for Iterated Function Systems

We study the problem of the rigorous computation of the stationary measure of an IFS described by a stochastic mixture of two or more dynamical systems which are either all uniformly expanding on the interval, either all contractive. In the expanding case, the associated transfer operators satisfy a LasotaYorke inequality, and we compute rigorously the approximations in the L1 norm. The rigorou...