论文标题
分析偏斜产品的质数定理
Prime number theorem for analytic skew products
论文作者
论文摘要
我们在$ 2 $ -TORUS $ \ MATHBB {T}^2 $上建立了所有独特的Ergodic，Analytic Skew产品的质量定理。更准确地说，对于每个不合理的$α$和每$ 1 $ - 周期性的真实分析$ g：\ Mathbb {r} \ to \ to \ Mathbb {r} $零是指零，让$ T_ {α，G}：\ Mathbb { （x+α，y+g（x））$。我们证明，如果$ t_ {α，g} $是独特的ergodic，那么，对于每一个$（x，y）\ in \ mathbb {t}^2 $，序列$ \ {t_ {α，g}^p（x，g}^p（x，x，y）\} $都在$ \ mathbb p $ p $ p $ p $ prover上。这是一类天然，非代数和平滑动力学系统的第一个示例，该系统为此定理所保留。我们还表明，如果$ g $仅在$ \ mathbb {t}^2 $上连续存在，则不一定会保持这样的素数定理。
We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus $\mathbb{T}^2$. More precisely, for every irrational $α$ and every $1$-periodic real analytic $g:\mathbb{R}\to\mathbb{R}$ of zero mean, let $T_{α,g} : \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be defined by $(x,y) \mapsto (x+α,y+g(x))$. We prove that if $T_{α, g}$ is uniquely ergodic then, for every $(x,y) \in \mathbb{T}^2$, the sequence $\{T_{α, g}^p(x,y)\}$ is equidistributed on $\mathbb{T}^2$ as $p$ traverses prime numbers. This is the first example of a class of natural, non-algebraic and smooth dynamical systems for which a prime number theorem holds. We also show that such a prime number theorem does not necessarily hold if $g$ is only continuous on $\mathbb{T}^2$.
