Dimensions and Measures in Infinite Iterated Function Systems

Infinite Iterated Function

We examine iterated function systems consisting of a countably innnite number of contracting mappings (IIFS). We state results analogous to the well-known case of nitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. Comparing the descriptive power of IFS and IIFS as mechanisms deen...

The Doubling Property of Conformal Measures of Infinite Iterated Function Systems

The Hutchinson–barnsley Theory for Infinite Iterated Function Systems

We show that some results of the Hutchinson–Barnsley theory for finite iterated function systems can be carried over to the infinite case. Namely, if {Fi : i ∈ N} is a family of Matkowski’s contractions on a complete metric space (X, d) such that (Fix0)i∈N is bounded for some x0 ∈ X, then there exists a non-empty bounded and separable set K which is invariant with respect to this family, that i...

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...

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...