自相似多重分形测度的几何与组合性质
Geometric and Combinatorial Properties of Self-similar Multifractal Measures
AI总结：
本文通过构造有向图控制自相似测度的分布，推广净区间方法，在弱分离及广义有限类型条件下验证多重分形形式体系，并应用于具有精确重叠的不可数IFS族。
摘要：
对于$\mathbb{R}$中的任意自相似测度$\mu$，我们证明$\mu$的分布由非负矩阵的乘积控制，这些乘积受一个仅依赖于迭代函数系统（IFS）的有限或可数图支配。这推广了Feng在等压缩有限类型情形下的净区间构造。当测度满足弱分离条件时，我们证明该有向图具有唯一吸引子。这使得我们能够验证$\mu$在由该有向图确定的$\mathbb{R}$的某些紧子集上的限制满足多重分形形式体系。当测度关于某个开区间满足广义有限类型条件时，该有向图是有限的，并且我们证明：若多重分形形式体系在某个$q\in\mathbb{R}$处失效，则必然存在一个不含吸引子中顶点的环。作为直接应用，我们验证了一个不可数族具有精确重叠且压缩比不满足对数可公度性的迭代函数系统的完全多重分形形式体系。
英文摘要：
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.
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