Dynamical systems for arithmetic schemes

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Abstract: Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vectors to attach a new ringed space

W_{m a t h r m r a t} (X)

to every scheme

X

. We also define

R

-valued points

W_{m a t h r m r a t} (X) (R)

of

W_{m a t h r m r a t} (X)

for every commutative ring

R

. For normal schemes

X

of finite type over spec

m a t h b b Z

, using

W_{m a t h r m r a t} (X) (m a t h b b C)

we construct infinite dimensional

m a t h b b 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_{m a t h r m r a t} (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.

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