论文标题
自相似性多重措施的几何和组合特性
Geometric and Combinatorial Properties of Self-similar Multifractal Measures
论文作者
论文摘要
对于任何自相似度量的$ $ $ $ $ $ $ $ \ mathbb {r} $，我们表明$μ$的分布受到仅根据IF的有限或可计数图管理的非负矩阵的产品控制的。这概括了从等额有限型案例中冯的净间隔构造。当该度量满足弱分离条件时，我们证明该有向图具有独特的吸引子。这使我们能够验证多重形式主义的限制，以限制$ \ mathbb {r} $的某些紧凑型子集，由有向图确定。当该措施相对于开放间隔满足广义有限类型条件时，有针对的图是有限的，我们证明，如果多型形式主义在\ Mathbb {r} $中以某个$ q \ after失败，则必须在吸引子中没有顶点的周期。作为直接应用，我们为一个不可数的IFS家族验证了具有精确重叠且没有对数可相称的收缩率的完整多重形式主义。
For any self-similar measure $μ$ in $\mathbb{R}$, we show that the distribution of $μ$ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the 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 $μ$ 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.
