Geometric and combinatorial properties of self-similar multifractal measures
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Bibliographic record
Abstract
Abstract 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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
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| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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