No 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.

No abstract

Let Ibea Banach space, B x its closed unit ball. We study several topological properties of B x with its weak topology. In particular, we consider spaces X such that (B x , weak) is a Polish topological space. If X has RNP and X* is separable, then B x is Polish; if B x is Polish, then X is somewhat reflexive. We also consider spaces X such that every closed subset of (B x , weak) is a Baire space. This is equivalent to property (PC), studied by Bourgain and Rosenthal.

No abstract