The structure of the set of local dimensions of a self-similar measure has been studied by numerous mathematicians, initially for measures that satisfy the open set condition and, more recently, for measures on $\mathbb{R}$ that are of finite type. In this paper, our focus is on finite type measures defined on the torus, the quotient space $\mathbb{R}\backslash \mathbb{Z}$. We give criteria which ensures that the set of local dimensions of the measure taken over points in special classes generates an interval. We construct a non-trivial example of a measure on the torus that admits an isolated point in its set of local dimensions. We prove that the set of local dimensions for a finite type measure that is the quotient of a self-similar measure satisfying the strict separation condition is an interval. We show that sufficiently many convolutions of Cantor-like measures on the torus never admit an isolated point in their set of local dimensions, in stark contrast to such measures on $\mathbb{R}$. Further, we give a family of Cantor-like measures on the torus where the set of local dimensions is a strict subset of the set of local dimensions, excluding the isolated point, of the corresponding measures on $\mathbb{R}$.

中文翻译：

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环面上有限类型测度的局部维数

许多数学家已经研究了自相似测度的局部维数集的结构，最初是针对满足开集条件的测度，最近是针对有限类型的 $\mathbb{R}$ 上的测度. 在本文中，我们的重点是定义在圆环上的有限类型度量，商空间 $\mathbb{R}\backslash \mathbb{Z}$。我们给出了确保对特殊类中的点采取的度量的局部维度集产生间隔的标准。我们在环面上构建了一个测量的非平凡示例，该示例允许其局部维度集中的孤立点。我们证明，作为满足严格分离条件的自相似测度的商的有限类型测度的局部维度集是一个区间。我们表明，在环面上有足够多的类似康托尔测度的卷积从不承认其局部维度集中的孤立点，这与 $\mathbb{R}$ 上的此类测度形成鲜明对比。此外，我们在环上给出了一系列类似康托尔的测度，其中局部维度集是 $\mathbb{R}$ 上相应测度的局部维度集的严格子集，不包括孤立点。