# Christopher Deninger: Dynamical systems for Arithmetic Schemes

**Time: **
Wed 2021-09-15 13.15 - 14.15

**Location: **
Zoom, meeting ID: 694 6016 6420

**Lecturer: **
Christopher Deninger (Münster)

**Abstract:** We attach infinite dimensional dynamical systems to arithmetic schemes *X* and in particular to systems of Diophantine equations over the integers and discuss some basic properties of these systems: in particular the relation between the periodic orbits and their lengths on the one hand and the closed points of *X* and their norms on the other. As a special case we obtain a dynamical system whose (compact packets of) periodic orbits correspond to the prime numbers *p* with the lengths being log *p*. We construct these dynamical systems as complex "points" of certain new ringed spaces involving sheaves of rational Witt vectors. In the zero dimensional case this is related to work of Kucharczyk and Scholze. If *X* is the spectrum of the *p*-adic integers and one takes *p*-adic "points" then a canonical subsystem is closely related to the Fargues–Fontaine curve in *p*-adic Hodge theory.

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