论文标题
具有异常局部维度特性的自相似度量
Self-similar measures with unusual local dimension properties
论文作者
论文摘要
令$μ$为满足有限类型条件的自相似度量。众所周知，这种度量的可实现的局部维度集是一个不相交的间隔的结合，其中某些间隔可能是退化点。尽管如此，是否可以实现这种可实现的本地维度的完全复杂性，但尚未显示出来。在本文中，我们提供了两种不同的结构。第一个是$μ$的度量，其中所有可实现的局部维度的集合是间隔联盟的结合和任意数量的分离点。第二个是一个度量$μ$，其中所有可实现的局部维度集是任意数量间隔间隔的结合。作为这些构造的应用，我们研究了这些措施的多纹状光谱$f_μ（α）$（α）$（α）$（α）$（α）$（q）$。我们给出了一个$μ$的示例，其中$f_μ（α）$不是凹的，其中$τ_μ（q）$具有两个非差异性。
Let $μ$ be a self-similar measure satisfying the finite type condition. It is known that the set of attainable local dimensions for such a measure is a union of disjoint intervals, where some intervals may be degenerate points. Despite this, it has not been shown if this full complexity of attainable local dimensions is achievable. In this paper we give two different constructions. The first is a measure $μ$ where the set of all attainable local dimensions is the union of an interval union and an arbitrary number of disjoint points. The second is a measure $μ$ where the set of all attainable local dimensions is the union of an arbitrary number of disjoint intervals. As an application to these construction, we study the multi-fractal spectrum $f_μ(α)$ and the $L^q$-spectrum $τ_μ(q)$ of these measures. We given an example of a $μ$ where $f_μ(α)$ is not concave, and where $τ_μ(q)$ has two points of non-differentiability.
