论文标题
沿确定性序列的CPE正态的破坏
Destruction of CPE-normality along deterministic sequences
论文作者
论文摘要
令$μ$为$λ^{\ mathbb n} $上的转换不变的度量，其中$λ$是有限或可计数的字母。我们说一个无限子集$ s = \ {s_1，s_2，\ dots \} \ subset \ subset \ mathbb n $（其中$ s_1 <s_1 <s_1 <s_2 <\ dots $）“保留（destroys）$μ$ - normality”如果任何$ x =（x_1，x_2，x_2，y \ in） $μ$，序列$ x | _s =（x_ {s_1}，x_ {s_2}，\ dots）$是（不是）$μ$的通用。从卡玛（Kamae）和魏斯（Weiss）知道，如果$μ$是I.I.D。然后，任何确定性的正密度阳性集合都保留了$μ$ normality。我们显示的是确定性集，除了具有非常原始的结构的设置，我们称为“表面”，破坏了任何非i.i.d的$μ$ normality。用完全正熵（CPE）测量$μ$。这概括了Heersink和Vandehey的算术渐进率和高斯度量（与持续的分数转换有关）的结果。我们举了几个例子，表明，除了CPE的一系列措施外，$μ$ - 规范保存可以与几乎任何三个参数的组合共存：$ s $（或缺乏）的确定论，$μ$（零或阳性）的熵（零或正面），以及$μ$之间的差异（或缺乏）和$μ$之间的度量。
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 $S$.
