一类自相似测度的修正的多重分形形式
A Modified Multifractal Formalism for a Class of Self-Similar Measures
摘要：
在$\mathbb{R}^n$中的Borel测度$μ$的多重分形谱被定义为\[ f_μ(α) = \dim_H {x:\lim_{r\to 0} \frac{\log μ(B(x,r))}{\log r}=α}. \]对于开集条件下的自相似测度，这个函数和相关函数的行为是很好理解的;这种情况是非常规则的，并且由所谓的“多重分形形式主义”控制。最近有很多兴趣了解有多少理论延续到重叠的情况下;然而，在这种情况下知道的要少得多，而已知的情况清楚地表明，更复杂的现象是可能的。在这里，我们进行了一个完整的研究的多重分形结构的一类自相似措施的重叠，其中包括3倍卷积的康托措施。除其他外，我们证明了多重分形形式主义失败的许多这些措施，但它持有时，采取适当的限制。
英文摘要：
The multifractal spectrum of a Borel measure $μ$ in $\mathbb{R}^n$ is defined as \[ f_μ(α) = \dim_H {x:\lim_{r\to 0} \frac{\log μ(B(x,r))}{\log r}=α}. \] For self-similar measures under the open set condition the behavior of this and related functions is well-understood; the situation turns out to be very regular and is governed by the so-called ''multifractal formalism''. Recently there has been a lot of interest in understanding how much of the theory carries over to the overlapping case; however, much less is known in this case and what is known makes it clear that more complicated phenomena are possible. Here we carry out a complete study of the multifractal structure for a class of self-similar measures with overlap which includes the 3-fold convolution of the Cantor measure. Among other things, we prove that the multifractal formalism fails for many of these measures, but it holds when taking a suitable restriction.
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