Multifractal Decompositions of Digraph Recursive Fractals

Abstract

We prove that the multifractal decomposition behaves as expected for a family of sets K known as digraph recursive fractals, using measures μ of Markov type. For each value of a parameter α between a minimum αmin and maximum αmax, we define ‘multifractal components’ K(α) of K, and show that they are fractals in the sense of Taylor. The dimension f(α) of K(α) is computed from the data of the problem. The typical concave ‘multifractal f(α)’ dimension spectrum curve results. Under appropriate disjointness conditions, the multifractal components K(α) are given byK(α)={xɛK:limɛ↓()logμ(Bɛ(x))logdiamBɛ(x)=α} that is, K(α) consists of those points where μ has pointwise dimension α.

Book Review: Integral, probability, and fractal measures

“…As with self-similar sets, the IFSP approach to measures has been extended to socalled graph-directed constructions, to families of random contractive similarities and to families of non-linear contractions; see [1], [9], [17], [53], [55] for further references.…”

The spectrum of Poincaré recurrence

“…A Moran fractal is a Cantor-type set defined by iteration; it has a tractable structure and has been used extensively for estimating entropies and dimensions (see e.g. [1,11,17,18,21,22]). In our context, we will make use of a special dynamically defined (by the Bowen metric) Moran fractal (Definition 4.1).…”

Multifractal of the Appolonian tiling

“…V(i, j) E [l; d + l]*, 3n > 0 [{A(P, Y)}"]i,j > O-this represents the number of cylinders of (e,, starting at i and finishing at j). The largest zero of a polynomial is an analytic function of the coefficients in the region where it is a simple zero [19,Prop. 3:1], and this assures (by the Perron-Frobenius theory of nonnegative matrices) the existence and the uniqueness of the real y1 for which A(& y) has spectral radius equal to 1: we then get y1 = -F(P); fo r example, when /3 = 0, we have y1 = -F(O) = dr.…”