叶空间上的数论与动力系统

具有$p-$ adic断面的层合空间的标度群流和Lefschetz迹公式

Scaling group flow and Lefschetz trace formula for laminated spaces with $p-$adic transversal

在他的方法来解析数论C。Deninger建议，对于Riemann zeta函数$\hatζ(s)$（resp.在有限域$\mathbb{F}_q$上的光滑投影曲线$Y$的ζ函数$ζ_Y(s)$，$q=p^f$）），人们可能会联想到一个叶状的黎曼层空间$(S_{\mathbb{Q}}, \mathcal{F}, g, ϕ^t)$（resp. $(S_Y, \mathcal{F}, g, ϕ^t)$）赋予流$ϕ^t$的作用，其原始紧轨道应对应于$\mathbb{Q}$的素数（分别为）。$Y$）。这种叶状空间和流动$ϕ^t$的存在仍然是未知的，除非$Y$是椭圆曲线（见Deninger）。受后一种情形的启发，我们引入了一类叶层空间$(S=\frac{\mathcal{L}\times \R^{+*}}{q^\Z}, \mathcal{F}, g, ϕ^t)$，其中$\mathcal{L} $是局部$D\times \Z_p^m$，$D$是$\mathbb{C}.$的开圆盘。假设$ \mathcal{L}$上的叶调和形式是局部横截常数，证明了流$ϕ^t$作用于叶向Hodge上同调$H^j_τ$的Lefschetz迹公式（$(S,\mathcal{F})$），它非常类似于$\mathbb{F}_q$上（一般）光滑曲线的zeta函数的显式公式。我们还证明了$ϕ^t$对$H^1_τ$作用的无穷小生…

An adic dynamical system related to the Delannoy numbers

We introduce an adic (Bratteli-Vershik) dynamical system based on a diagram whose path counts from the root are the Delannoy numbers. We identify the ergodic invariant measures, prove total ergodicity for each of them, and initiate the study of the dimension group and other dynamical properties.

一类多项式动力系统的度增长与非线性伪随机数生成器

On the Degree Growth in Some Polynomial Dynamical Systems and Nonlinear Pseudorandom Number Generators

本文研究了一类由多元多项式迭代生成的动力系统，并估计了这些迭代的度增长。我们使用这些估计约束指数和沿这些动力系统的轨道，并表明，他们承认更强的估计比在一般情况下，因此可以用于伪随机数生成。

A Dynamical Proof of the Prime Number Theorem

We present a new, elementary, dynamical proof of the prime number theorem.

数域上动力系统的S-积分预周期点

S-integral preperiodic points for dynamical systems over number fields

给定数域$K$上定义的有理映射$ϕ: {\mathbb P}^1\to {\mathbb P}^1$，证明了$P$在每一个地方满足一定局部条件的$S$关于非预周期点$P$为整数的$ϕ$ -预周期点的有限性结果.这证明了S的一个猜想的一个特例。是的

Number theoretical properties of Romik's dynamical system

We study a dynamical system that was originally defined by Romik in 2008 using an old theorem of Berggren concerning Pythagorean triples. Romik's system is closely related to the Farey map on the unit interval which generates an additive continued fraction algorithm. We explore some number theoretical properties of the Romik system. In particular, we prove an analogue of Lagrange's theorem in the case of the Romik system on the unit quarter circle, which states that a point possesses an eventually periodic digit expansion if and only if the point is defined over a real quadratic extension fiel…

动力系统的同余恒等式

Congruence Identities Arising From Dynamical Systems

通过对某些区间映射的所有周期的周期点的个数进行计数，得到了无穷多个新的数论同余恒等式。

素数定理的动力学推广以及加法与乘法半群作用的不交性

Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions

本研究建立两个蕴含多项乘性数论经典结果的遍历定理，将Sarnak的Möbius不交性猜想重构于新动力学框架，提出聚焦加乘半群作用不交性的猜想扩展并证明其若干特例。

The Dynamical Manin-Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps

We prove the Dynamical Bogomolov Conjecture for endomorphisms of P^1\times P^1 defined over a number field. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we prove the Dynamical Manin-Mumford Conjecture for endomorhisms of P^1\times P^1 defined over the complex numbers.

关于具有有界动力学典范高度点的数量

On the number of points with bounded dynamical canonical height

本文研究具有有界动力学典范高度的点的数量，利用 Hsia 的动力学高度 zeta 函数，给出特殊情形下的显式公式，并推导出点数的渐近行为。

A dynamical construction of small totally $p$-adic algebraic numbers

We give a dynamical construction of an infinite sequence of distinct totally $p$-adic algebraic numbers whose Weil heights tend to the limit $\frac{\log p}{p-1}$, thus giving a new proof of a result of Bombieri-Zannier. The proof is essentially equivalent to the explicit calculation of the Arakelov-Zhang pairing of the maps $σ(x)=x^2$ and $ϕ_p(x)=\frac{1}{p}(x^p-x)$.

Dynamical systems and uniform distribution of sequences

We give a survey on classical and recent applications of dynamical systems to number theoretic problems. In particular, we focus on normal numbers, also including computational aspects. The main result is a sufficient condition for establishing multidimensional van der Corput sets. This condition is applied to various examples.

动力系统的广义数论自旋链联系及期望值

Generalized Number Theoretic Spin Chain-Connections to Dynamical Systems and Expectation Values

通过引入一个新的参数x>=0，我们推广了数论自旋链，一个基于Farey分数的一维统计模型。这允许我们在链的长度上写递归关系。这些关系与Lewis三项方程密切相关，后者在Selberg函数的研究中非常有用。然后，我们利用这些关系和自旋取向变换。我们发现一个简单的连接与传递算子的模型的动力系统中的不稳定性。此外，我们能够计算某些自旋期望值明确的自由能或相关长度。这些期望值中的一些似乎与相变的机制直接相关。