无共振的Cantor测度的卷积
Convolutions of Cantor measures without resonance
摘要：
用$μ_a$表示随机和$(1-a) \sum_{j=0}^\infty ω_j a^j$的分布，其中$P(ω_j=0)=P(ω_j=1)=1/2$和所有选择都是独立的。对于$0<a<1/2$，度量$μ_a$在$C_a$上得到支持，中心康托集是通过从闭联合区间开始，删除长度为$(1-2a)$的开放中心区间，并在每个剩余区间上归纳迭代此过程而获得的。研究卷积$μ_a * (μ_b \circ S_λ^{-1})$，其中$S_λ(x)=λx$是一个重标度映射.证明了若$\log b/\log a$是无理数，且$λ\neq 0$是无理数，则有\[ D(μ_a *(μ_b\circ S_λ^{-1})) = \min(\dim_H(C_a)+\dim_H(C_b),1), \]，其中$D$表示测度的相关维数、Hausdorff维数或packing维数.我们还表明，也许令人惊讶的是，对于不可数的许多值的$λ$的卷积$μ_{1/4} *(μ_{1/3}\circ S_λ^{-1})$是一个奇异的措施，虽然$\dim_H(C_{1/4})+\dim_H(C_{1/3})>1$和$\log (1/3) /\log (1/4)$是不合理的。
英文摘要：
Denote by $μ_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty ω_j a^j$, where $P(ω_j=0)=P(ω_j=1)=1/2$ and all the choices are independent. For $0<a<1/2$, the measure $μ_a$ is supported on $C_a$, the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length $(1-2a)$, and iterating this process inductively on each of the remaining intervals. We investigate the convolutions $μ_a * (μ_b \circ S_λ^{-1})$, where $S_λ(x)=λx$ is a rescaling map. We prove that if the ratio $\log b/\log a$ is irrational and $λ\neq 0$, then \[ D(μ_a *(μ_b\circ S_λ^{-1})) = \min(\dim_H(C_a)+\dim_H(C_b),1), \] where $D$ denotes any of correlation, Hausdorff or packing dimension of a measure. We also show that, perhaps surprisingly, for uncountably many values of $λ$ the convolution $μ_{1/4} *(μ_{1/3}\circ S_λ^{-1})$ is a singular measure, although $\dim_H(C_{1/4})+\dim_H(C_{1/3})>1$ and $\log (1/3) /\log (1/4)$ is irrational.
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