Dynamical systems for arithmetic schemes

Abstract

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vector rings to attach a new ringed space $W_{\mathrm{rat}}\left(X\right)$ to every scheme $X$. We also define $R$-valued points $W_{\mathrm{rat}}\left(X\right)\left(R\right)$ of $W_{\mathrm{rat}}\left(X\right)$ for every commutative ring $R$. For normal schemes $X$ of finite type over $\mathrm{spec}Z$, using $W_{\mathrm{rat}}\left(X\right)\left(ℂ\right)$ we construct infinite dimensional $R$-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 $W_{\mathrm{rat}}\left(X\right)$ for $X$ the spectrum of a $p$-adic local number ring are related to the points of the Fargues–Fontaine curve.