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Local dimensions of measures of finite type on the torus. (English) Zbl 1447.28004

The authors introduce a concept of a finite type measure on the torus \(\mathbb{T} \cong \mathbb{R}/\mathbb{Z}\) that are quotients of equicontractive, self-similar measures on \(\mathbb{R}\), and develop a general method for computing the local dimensions of such measures. Using this method, a simple formula for the local dimensions at the periodic points is obtained. Moreover, it is proved that if a self-similar measure arises from an iterated function system that satisfies the strong separation condition, then the set of local dimensions of the quotient measure is not only an interval but coincides with the set of local dimensions of the original measure. A first non-trivial example of a quotient measure on \(\mathbb{T}\) whose set of local dimensions admits an isolated point is also provided.

MSC:

28A78 Hausdorff and packing measures
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
28A80 Fractals
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