Local dimensions of self-similar measures satisfying the finite neighbour condition. (English) Zbl 1514.28007

The authors investigate sets of local dimensions for self-similar measures in \(\mathbb{R}\) satisfying the so-called finite neighbour condition, which is formally stronger than the weak separation condition (WSC) but satisfied in all known examples. Under a mild technical assumption, we establish that the set of attainable local dimensions is a finite union of (possibly singleton) compact intervals. The number of intervals is bounded above by the number of non-trivial maximal strongly connected components of a finite directed graph construction depending only on the governing iterated function system. Additionally, an explanation of how these results allow computations of the sets of local dimensions in many explicit cases is given. These results contextualize and generalize a vast amount of prior work on sets of local dimensions for self-similar measures satisfying the WSC.


28A80 Fractals
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