Dimension bounds for invariant measures of bi-Lipschitz 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

The Directional Dimension of Subanalytic Sets Is Invariant under Bi-lipschitz Homeomorphisms

Let A ⊂ R be a set-germ at 0 ∈ R such that 0 ∈ A. We say that r ∈ S is a direction of A at 0 ∈ R if there is a sequence of points {xi} ⊂ A\{0} tending to 0 ∈ R such that xi ‖xi‖ → r as i → ∞. Let D(A) denote the set of all directions of A at 0 ∈ R. Let A, B ⊂ R be subanalytic set-germs at 0 ∈ R such that 0 ∈ A ∩ B. We study the problem of whether the dimension of the common direction set, dim(D...

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...

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...