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Multifractal Decompositions of Digraph Recursive Fractals
Author(s) -
Edgar G. A.,
Mauldin R. Daniel
Publication year - 1992
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-65.3.604
Subject(s) - multifractal system , mathematics , digraph , dimension (graph theory) , combinatorics , pointwise , fractal , fractal dimension , markov chain , discrete mathematics , hausdorff dimension , mathematical analysis , statistics
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.