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 α min and maximum α max , 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 ) = { 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.