Dynamical systems for arithmetic schemes

TL;DR

This paper introduces a new sheafified rational Witt vector construction for schemes, leading to infinite-dimensional dynamical systems that connect algebraic geometry with topological dynamics and $p$-adic geometry.

Contribution

It develops a novel intrinsic approach to associate dynamical systems to schemes via sheafified rational Witt vectors, extending previous extrinsic methods.

Findings

Constructed infinite-dimensional $ ext{R}$-dynamical systems from schemes.

Linked periodic orbits of these systems to closed points of schemes.

Connected $p$-adic points to the Fargues-Fontaine curve.

Abstract

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vectors to attach a new ringed space $W_{\mathrm{rat}} (X)$ to every scheme $X$. We also define $R$-valued points $W_{\mathrm{rat}} (X) (R)$ of $W_{\mathrm{rat}} (X)$ for every commutative ring $R$. For normal schemes $X$ of finite type over spec $\mathbb{Z}$, using $W_{\mathrm{rat}} (X) (\mathbb{C})$ we construct infinite dimensional $\mathbb{R}$-dynamical systems whose periodic orbits are related to the closed points of $X$. Various aspects of these topological dynamical systems are studied. We also explain how certain $p$-adic points of $W_{\mathrm{rat}} (X)$ for $X$ the spectrum of a $p$-adic local number ring are related to the points of the Fargues-Fontaine curve. The new intrinsic construction of the dynamical systems generalizes and clarifies the original extrinsic construction in v.1 and v.2. Many further results have been added.