确定性序列对CPE正规性的破坏
Destruction of CPE-normality along deterministic sequences
AI总结：
本文证明确定性集合（除表面结构外）会破坏任何非独立同分布且具有完全正熵的移位不变测度的正规性，推广了算术级数和Gauss测度的已知结果，并举例说明在CPE类之外，正规性保持与确定性、熵和不相交性可任意组合共存。
摘要：
设$\mu$为有限或可数字母表$\Lambda$上$\Lambda^{\mathbb N}$中的一个移位不变测度。我们称无限子集$S=\{s_1,s_2,\dots\}\subset\mathbb N$（其中$s_1<s_2<\dots$）“保持（破坏）$\mu$-正规性”，如果对于任何对$\mu$通用的$x=(x_1,x_2,\dots)\in\Lambda^{\mathbb N}$，序列$x|_S=(x_{s_1},x_{s_2},\dots)$是（不是）对$\mu$通用的。由Kamae和Weiss可知，若$\mu$是i.i.d.的，则任何具有正下密度的确定性集合都保持$\mu$-正规性。我们证明，确定性集合（除了我们称为“表面的”那种非常原始结构的集合之外）会破坏任何非i.i.d.且具有完全正熵（CPE）的测度$\mu$的$\mu$-正规性。这推广了Heersink和Vandehey关于算术级数和Gauss测度（与连分数变换相关）的结果。我们给出几个例子表明，在CPE测度类之外，$\mu$-正规性的保持可以与三个参数的几乎任意组合共存：$S$的确定性（或其缺失）、$\mu$的熵（零或正），以及$\mu$与从$S$导出的测度之间的不相交性（或其缺失）。
英文摘要：
Let $μ$ be a shift-invariant measure on $Λ^{\mathbb N}$, where $Λ$ is a finite or countable alphabet. We say that an infinite subset $S=\{s_1,s_2,\dots\}\subset\mathbb N$ (where $s_1<s_2<\dots$) "preserves (destroys) $μ$-normality" if, for any $x=(x_1,x_2,\dots)\inΛ^{\mathbb N}$ generic for $μ$, the sequence $x|_S=(x_{s_1},x_{s_2},\dots)$ is (is not) generic for $μ$. It is known from Kamae and Weiss that if $μ$ is i.i.d. then any deterministic set of positive lower density preserves $μ$-normality. We show that deterministic sets, except ones with a very primitive structure that we call "superficial", destroy $μ$-normality for any non-i.i.d. measure $μ$ with completely positive entropy (CPE). This generalizes Heersink and Vandehey's result for arithmetic progressions and the Gauss measure (associated to the continued fraction transformation). We give several examples showing that, outside the class of measures with CPE, $μ$-normality preservation can coexist with nearly any combination of three parameters: determinism of $S$ (or its lack), entropy of $μ$ (zero or positive), and disjointness (or its lack) between $μ$ and the measures derived from $S$.
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