✍ Scribed by Barnsley M.F., Demko S.
No coin nor oath required. For personal study only.
Iterated function systems (i.f.ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i.f.ss and occur as the supports of probability measures associated with functional equations. The existence of certain 'p-balanced' measures for i.f.ss is established, and these measures are uniquely characterized for hyperbolic i.f.ss. The Hausdorff-Besicovitch dimension for some attractors of hyperbolic i.f.ss is estimated with the aid of p-balanced measures. What appears to be the broadest framework for the exactly computable moment theory of p-balanced measures - that of linear i.f.ss and of probabilistic mixtures of iterated Riemann surfaces - is presented. This extensively generalizes earlier work on orthogonal polynomials on Julia sets. An example is given of fractal reconstruction with the use of linear i.f.ss and moment theory.
📜 SIMILAR VOLUMES
## Abstract Let {__S~i~__} be an iterated function system (IFS) on ℝ^__d__^ with attractor __K__. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, 𝓁}. We define the projection entropy function __h__~π~ on the space of invariant measures on Σ associated with the coding map π : Σ →