论文标题
$ \ mathbb {t}^2 $上的一些平滑流量的Prime Orbits
Prime orbits for some smooth flows on $\mathbb{T}^2$
论文作者
论文摘要
我们考虑一类平滑的混合流量$ t^{α，γ} $ on $ \ mathbb {t}^2 $，带有一个退化的固定点$ x_0 \ in \ mathbb {t}^2 $ of Power类型$γ\ in（-1,0）$。我们证明，对于$g_δ$密度集$α\ in \ mathbb {t} $，$ t^{α，γ} $的素数定理沿着完整的上层密度子序列。特别是，在\ Mathbb {t}^2 \ setMinus \ {x_0 \} $中，每一个$ x \ in prime orbit $ \ mathbb {t}^2 $。 我们还表明，在$ \ mathbb {t}^2 $上，有一类平滑的弱混合流，该流量为prime number定理所持的。实际上，我们表明存在一组密集的平滑函数（在统一拓扑中），该功能量定理定量（带有错误项$ \ log^{ - a} n $）。
We consider a class of smooth mixing flows $T^{α,γ}$ on $\mathbb{T}^2$ with one degenerated fixed point $x_0\in \mathbb{T}^2$ of power type $γ\in (-1,0)$. We prove that for a $G_δ$ dense set of $α\in \mathbb{T}$, a prime number theorem for $T^{α,γ}$ holds along a full upper density subsequence. In particular it follows that for every $x\in \mathbb{T}^2\setminus\{x_0\}$, the prime orbit $\mathbb{T}^2$. We also show that there exists a class of smooth weakly mixing flows on $\mathbb{T}^2$ for which a prime number theorem holds. In fact we show that there exists a dense set of smooth functions (in the uniform topology) for which prime number theorem holds quantitatively (with an error term $\log^{-A}N$).
