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Geometric and combinatorial properties of self-similar multifractal measures. (English) Zbl 07682668

Summary: For any self-similar measure \(\mu\) in \(\mathbb{R}\), we show that the distribution of \(\mu\) is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the iterated function system of similarities (IFS). This generalizes the net interval construction of Feng from the equicontractive finite-type case. When the measure satisfies the weak separation condition, we prove that this directed graph has a unique attractor. This allows us to verify the multifractal formalism for restrictions of \(\mu\) to certain compact subsets of \(\mathbb{R}\), determined by the directed graph. When the measure satisfies the generalized finite-type condition with respect to an open interval, the directed graph is finite and we prove that if the multifractal formalism fails at some \(q\in\mathbb{R}\), there must be a cycle with no vertices in the attractor. As a direct application, we verify the complete multifractal formalism for an uncountable family of IFSs with exact overlaps and without logarithmically commensurable contraction ratios.

MSC:

28A80 Fractals
37C45 Dimension theory of smooth dynamical systems
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