Multifractal Decompositions of Digraph Recursive Fractals

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 by K ( α ) = { x ɛ K : lim ɛ ↓ ( ) log μ ( B ɛ ( x ) ) log diam B ɛ ( x ) = α } that is, K(α) consists of those points where μ has pointwise dimension α.