Dynamical systems for arithmetic schemes

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.


Introduction
In the present paper we construct infinite-dimensional continuous-time dynamical systems X 0 attached to integral normal schemes X 0 which are flat and of finite type over spec Z. The closed points x 0 of X 0 correspond bijectively to compact packets Γ x 0 of periodic orbits of length log Nx 0 . Here Nx 0 = |κ(x 0 )| is the norm of x 0 , the number of elements in the (finite) residue field κ(x 0 ) of x 0 . If p = char κ(x 0 ), then Γ x 0 is fibred over the compact group Aut(F × p )/Aut(F p ) =Ẑ × (p) /pẐ , with fibres the periodic orbits in Γ x 0 . Each periodic orbit of X 0 lies in exactly one packet Γ x 0 . There are no fixed points of the flow. The existence of periodic orbits rests on the existence of Frobenius elements for x 0 in the Galois group of the function field K 0 of X 0 . The space X 0 carries a generalized 1codimensional foliation such that the flow maps leaves to leaves. The existence of such systems X 0 for arithmetic schemes X 0 was predicted by earlier work, c.f. [Den00], [Den98], [Den01], [Den08]. Whether the particular dynamical systems constructed in this paper or suitable subsystems can be useful or require modification remains to be seen. Working with X 0 itself, the greatest difficulty is that X 0 is infinite dimensional instead of being 2 dim X 0 + 1-dimensional as suggested by arithmetic topology and the above references. There are many natural questions about X 0 but we adress only the most basic one, namely the connectedness of X 0 in a natural topology, c.f. Corollary 16. We prove that X 0 has at most finitely many connected components and if X 0 ⊗ Q Q(µ ∞ ) is connected, then X 0 is connected. The proof uses de Jong's theory of alterations if dim X 0 ≥ 2.
In [KS16] Kucharczyk and Scholze realize the absolute Galois groups of fields containing all roots of unity as covering groups on certain spaces. Part of the generic fibre of our space • X(C) is contained in one of their spaces.
I am deeply indepted to the CRCs 478 and 878 for their longterm support. Moreover I am very grateful to Umberto Zannier for the invitation to the Scuola Normale in Pisa where the construction of the dynamical systems was found. I also thank the HIM in Bonn for supporting a fruitful stay.
2 The N -and Q >0 -dynamical systems Let X 0 be an arithmetic scheme by which we mean an integral normal scheme which is flat and of finite type over spec Z. Let K 0 be the function field of X 0 .
Note that every open subscheme X ′ 0 of X 0 is again an arithmetic scheme with the same function field K 0 . Our constructions depend on the choice of a radicially closed field extension K of K 0 which is Galois over K 0 . Radicially closed means that every equation T N − a for a ∈ K decomposes into linear factors. Equivalently, K × is divisible and K contains all roots of unity. For example we can take an algebraic closure of K 0 or the maximal solvable extension of K 0 in an algebraic closure. Let G be the Galois group of K over K 0 and let X be the normalization of X 0 in K. The group G acts on X over X 0 . Let C be an algebraically closed field of characteristic zero. We are thinking of the complex numbers or of C p , the completion of an algebraic closure of Q p for a prime number p.
Proof. If X 0 = spec R 0 is affine we have X = spec R where R is the normalization of R 0 in K. Hence R 0 = R G and the claim is a special case of [KS16,Lemma 4.9]. The general case follows because by construction π is integral and in particular affine: We have where {X i 0 } is an open affine covering of X 0 . Moreover π −1 (X i 0 (C)) = X i (C) where X i → X i 0 is the normalization of X i 0 in K. Hence π −1 (X i 0 (C))/G = X i 0 (C) and therefore the map X(C)/G → X 0 (C) is bijective.
A point of X 0 (C) consists of a scheme theoretic point x 0 of X 0 together with an embedding of its residue field κ(x 0 ) into C. Such field embeddings exist only if char κ(x 0 ) = char C = 0. Hence X 0 (C) does not "see" the points x 0 in the fibres of X 0 over spec F p . Moreover there is no Frobenius endomorphism of X 0 (C). We propose generalizations of the notion of C-valued point of X 0 which adress these issues. There are several variants of the following construction. They depend on the choice of a suitable condition E on the characters P × below.

Definition 2. a) Let
• X(C) be the set of pairs P = (x, P × ) where x ∈ X and P × : κ(x) × → C × is a multiplicative homomorphism satisfying condition E.
b) The group G acts on • X(C) from the right by (x, P × ) σ = (σ −1 (x), P × • σ) for σ ∈ G. We set Here the strongest condition E for our purposes, E = E f requires ker P × to be finite. The weakest one, E = E max requires that (ker P × ) tors is finite and that im P × ⊂ µ(C) implies that κ(x) × is torsion. Other interesting conditions are E f d , resp. E f d 0 , that(ker P × ) tors is finite, and dim Q (ker P × ) ⊗ Q < ∞ resp. dim Q (ker P × | κ(x 0 ) × ) ⊗ Q < ∞. Whenever we do not specify the condition E, we assume the implications E f ⊂ E ⊂ E max and in particular that the groups (ker P × ) tors are finite. We have implications E f ⊂ E f d ⊂ E f d 0 ⊂ E max and the corresponding dynamical systems satisfy the corresponding inclusions.
We will identify the two points of view, writing P = (p, P × ) and p P = p occasionally. Then we have P σ = (p, P × ) σ where P σ = P • σ.
There are many considerations in arithmetic geometry leading to relaxing the condition of additivity but keeping multiplicativity: For example the Teichmüller lift in Witt vector theory which is only multiplicative, the work on F 1 -geometry for which we refer to the survey [Lor18], ideas in arithmetic topology [Mor12] and its dynamical enhancement [Den02], and recently the work of Kucharczyk and Scholze [KS16].
We could have asked for P × to be injective like in the classical case where P is also additive, but since we want an action by commuting Frobenius endomorphisms for all prime numbers p we must at least allow finite kernels ker P × .
We now define the Frobenius actions. Let N be the multiplicative monoid of positive integers. For ν ∈ N define a map F ν : For this to make sense we have to assume that condition E is such that P In the affine case where P = (p, P × ) can be viewed as a multiplicative map P : R → C we have F ν (P )(r) = P (r ν ) = P (r) ν for all r ∈ R .
Letting ν ∈ N act via F ν we obtain a right action of N on • X(C) which commutes with the G-action. Hence we obtain self maps F ν and an N -action on the quotient Note that we have natural inclusions X(C) ⊂ • X(C) and by Proposition 1 also X 0 (C) ⊂ • X 0 (C). More generally, for any field k with a multiplicative embedding i : corresponds to a point x ∈ X and an inclusion κ(x) where P × is the composition If k is algebraically closed, we have X(k)/G = X 0 (k) by [KS16,Lemma 4.9], and we obtain an inclusion i * : X 0 (k) ֒→ Lemma 3. Let k be a countable field and let χ µ : µ(k) → C × be a homomorphism. Then there is a homomorphism χ : k × → C × with χ| µ(k) = χ µ and ker χ = ker χ µ . In particular there is an injective homomorphism k × ֒→ C × .
Proof. We may assume that k is algebraically closed. The sequences are both split since µ(k) and µ(C) are divisible. Choose splittings and consider the induced isomorphisms Here k × /µ(k) and C × /µ(C) are uniquely divisible abelian groups and therefore Q-vector spaces. We have Hence there is a Q-linear injection from k × /µ(k) into C × /µ(C). Together with χ µ and the above decompositions we obtain a homomorphism χ : k × → C × prolonging χ µ with ker χ = ker χ µ .
For any point x ∈ X the residue field k = κ(x) is countable. Using Lemma 3 it follows that the map pr X : Hence the induced map of orbit spaces • X 0 (C) → X/G is surjective as well. The going-up theorem implies that the normalization map X → X 0 and hence the induced map X/G → X 0 are surjective as well. By composition we obtain a natural surjection pr X 0 : In conclusion we see that contrary to X 0 (C) the new set of points • X 0 (C) carries information about all points of X 0 . Enlarging • X(C) and • X 0 (C) we may extend the N -action to an action by the multiplicative group Q >0 of positive rational numbers. For ν ∈ N and x ∈ X, the maps ( ) ν : O X,x → O X,x and hence also ( ) ν : κ(x) × → κ(x) × are surjective since by our assumptions, K × is divisible and O X,x is integrally closed in K.
Thus the maps F ν : • X(C) → • X(C) and hence also F ν : We therefore have to extend the backward orbits of the N -operation. This is done by passing to the colimit over the set N ordered by divisibility, the transition maps being the F ν for ν ∈ N . Thus we seť The maps F ν extend to bijections F ν onX(C) andX 0 (C) and using them the points ofX(C) can be written in the form The actions of N on • X(C) and • X 0 (C) extend uniquely to actions by Q >0 onX(C) andX 0 (C). Thus q = ν/ν ′ with ν, ν ′ ∈ N acts by F q := F ν • F −1 ν ′ . Note that the projections pr X and pr X 0 above extend Q >0 -equivariantly to projections pr X :X(C) −→ X and pr X 0 :X 0 (C) −→ X 0 .
Here we let Q >0 act trivially on X and X 0 . The N -equivariant G-action of • X(C) extends uniquely to a Q >0 -equivariant G-action onX(C) and the natural Q >0 -equivariant projectionπ induces a Q >0 -equivariant bijection: The fibres of pr X 0 :X 0 (C) → X 0 are Q >0 -invariant. We will now analyze the structures of the Q >0 -sets C x 0 = pr −1 In this case κ(x 0 ) is a finite field and for every closed point x ∈ X over x 0 the residue field κ(x) is an algebraic closure of κ(x 0 ) since K and hence O X,x , being integrally closed in K contain all roots of unity. The reduction map O X,x → κ(x) induces an isomorphism Here p = char κ(x 0 ) and µ (p) (S) denotes the group of prime-to-p order roots of unity in a ring S. Consider the commutative diagram where pr = pr X and The fibre pr −1 0 (x 0 ) consists of the G-orbits of all pairs (x, P × ) where x is a closed point of X over x 0 and P × : κ(x) × → C × satisfies condition E. Since κ(x) × is torsion this means that ker P × = (ker P × ) tors is finite or equivalently finite cyclic since finite subgroups of L × for a field L are cyclic. Given x and using the isomorphism (2), any choice of isomorphisms µ (p) (K) gives an element of pr −1 0 (x 0 ). The structure of pr −1 0 (x 0 ) and of C x 0 as N -resp. Q >0sets is determined as follows. Fix a closed point x over x 0 and let G x be the stabilizer subgroup of x in G. It surjects onto Gal Chap.V, § 2, n o 3, Proposition 6]. Because of (3), the group of automorphisms of the abelian group κ(x) × is given byẐ × (p) whereẐ (p) = l =p Z l . We have a natural inclusion Here on the left, Nx 0 corresponds to the Frobenius automorphism y → y N x 0 in the Galois group. The monoidẐ × (p) × N acts on the set S of homomorphisms P × : κ(x) × → C × with finite cyclic kernel by pre-composition. Fix a multiplicative embedding i : κ(x) × ֒→ C × . It follows that we have aẐ × Two elements (a, ν) and (a ′ , ν ′ ) are in the same fibre of this map if and only if ν ′ /ν = p n for some n ∈ Z and a = p n a ′ . Any closed point y in X over x 0 is conjugate to our chosen point x by an element of G. This follows from (1) and [Mat89, Theorem 5 vi)]. We have canonical N -equivariant bijections Explicitely this gives the N -equivariant bijection: Composing with (6) we get the N -equivariant surjection: Two elements (a, ν) and (a ′ , ν ′ ) are in the same fibre if and only if ν ′ /ν = p n for some n ∈ Z and a = p n a ′ . The later condition depends only on the class of n mod deg x 0 . Passing to the Q >0 -extensions i.e. to colim N we obtain a Q >0 -equivariant bijection: Here, for a right Γ-modul A and a left Γ-module B we write A × Γ B for the quotient of A × B by the left Γ-action given by γ(a, b) = (aγ −1 , γb). The Γorbit of (a, b) is denoted by [a, b]. It follows that all points P 0 ∈ C x 0 have isotropy subgroup (Q >0 ) P 0 = Nx Z 0 . Namely, the equation [a, r]q = [a, r], i.e. [a, rq] = [a, r] means that for some n ∈ Z we have a = p n a inẐ × (p) /NxẐ 0 and rq = p n r. This is equivalent to p n ∈ NxẐ 0 and q = p n i.e. to q ∈ Nx Z 0 . We may also write (9) as a Q >0 -equivariant bijection: It shows that C x 0 fibres over the compact group with fibres the Q >0 -orbits in C x 0 . After this discussion of the Q >0 -sets C x 0 we mention one further item before proving the basic Theorem 4 below. The maps (8), (9) and the fibration map are non-canonical and depend on our choices of x and i. On the other hand, the projection More generally consider the map ρ : For ν ∈ N let ν x be the prime-to-char κ(x) part of ν. If char κ(x) = 0 then ν x = ν. We have Recall the projection pr X 0 :X 0 (C) → X 0 and for a prime p seť This is the set of points is the set of points with κ(x) of characteristic zero. Formulas (12) and (13) show that ρ induces Q >0 -equivariant maps and We obtain a natural Q >0 -equivariant surjective map The projection (11) is the restriction of ρ : Concerning the Q >0 -action onX 0 (C) we have the following main result: Theorem 4. The following decomposition holds, where x 0 runs over the closed Proof. Replacing P 0 ∈X 0 (C) with F ν (P 0 ) for some ν ∈ N does not change the isotropy subgroup. Hence it is sufficient to determine the points P 0 ∈ • X 0 (C) whose isotropy subgroup is non-trivial. The points P 0 in . This is equivalent to the existence of some σ ∈ G with σx = x and Here σ is the automorphism of κ(x) over κ(x 0 ) induced by σ where x 0 = π(x).

Condition (18) means that the subgroup
Here m x is the maximal ideal in O X,x .
The proof of the lemma will be given below. For q = 1, using the lemma, it follows that (ker P × ) ⊗ Q = κ(x) × ⊗ Q and hence that im P × is torsion. Since condition E implies E max , it follows that κ(x) × is torsion. By the structure theory of fields we conclude that κ(x) ⊂ F p for some p. But then the point x must be closed and The remaining assertions in the theorem have already been shown in our discussion of C x 0 . In particular, we have q ∈ Nx Z 0 . Thus, modulo the lemma, Theorem 4 is proved.
Hence we have ). This implies: In the commutative group ring Z σ of the cyclic group σ generated by σ we have The group ring Z σ acts on κ(x) × . Setting y = z ω we obtain:

This implies that
Thus we have ). This implies the Lemma and hence Theorem 4. ✷ Remark 6. For conditions E ⊂ E f d 0 the proof of Theorem 4 does not require Lemma 5. As before (18) is the residue field of a point on a scheme of finite type over Z it follows that κ(x 0 ) is a finite field and that x 0 is a closed point of X 0 . We now finish the proof as before.
The condition that (ker P × ) tors is finite seems to be fundamental, in particular for the later connectedness results. However by itself it is not enough to ensure that Theorem 4 holds. It does force the points P 0 ∈X 0 (C) over characteristic zero points of X 0 to have trivial stabilizer (Q >0 ) P 0 = 1. However over all positive characteristic points of X 0 it allows points P 0 with (Q >0 ) P 0 = 1 whereas we want this to happen only over the closed points of X 0 : Theorem 7. Define • X(C) etc as before under the only condition that all the groups (ker P × ) tors are finite. For P 0 ∈X 0 (C) let x 0 = pr 0 (P 0 ) ∈ X 0 . Then the following assertions hold: We may assume that in q = ν/ν ′ , ν = ν ′ the integers ν, ν ′ ∈ N are coprime. Let µ (νν ′ ) (K) be the group of roots of unity in K whose orders are prime to ν and ν ′ . Then we have Namely, if a prime number l divides ν i − ν ′ i , then it must be prime to ν and ν ′ because otherwise it would divide them both. Hence the right hand side is contained in µ (νν ′ ) (K). Now assume that ζ has order N where N is prime to ν and ν ′ .
and the reverse inclusion follows. Together with the lemma it follows that µ l ∞ (K) maps is finite. It follows that νν ′ > 1 can have only one prime divisor, namely Since ν = ν ′ are coprime it follows that ν = p n and ν ′ = 1 or ν = 1 and ν ′ = p n for some n ≥ 1. Thus we have q = ν/ν ′ ∈ p Z . We now assume that ν = p n , ν ′ = 1, the following argument being similar in the case ν = 1, ν ′ = p n . We know that the group image. The same is therefore true for its Pontrjagin dual map p n − a :Ẑ (p) → Z (p) . Here σ acts on µ (p) by ζ → ζ a for some a ∈Ẑ × (p) whereẐ (p) = l =p Z l . Writing a = (a l ) it follows that all maps p n − a l : Z l → Z l for l = p have finite image. This implies a l = p n for all l and hence a = p n inẐ × (p) . Since σ is trivial on κ(x 0 ) and hence on κ( Then the open affine subscheme spec R of X contains the points x ∈ X over x 0 . Choose such an x and let p be the corresponding prime ideal in R and let p 0 = p ∩ R 0 be the prime is surjective. We now show that it is an isomorphism. Its kernel consists of elements ζ = y Nx 0 σ(y) −1 ∈ µ(κ(p)) for some y ∈ κ(p) × . Choose some N ≥ 1 prime to p with ζ N = 1. For z = y N we then have σ(z) = z Nx 0 . Letting i be such that σ i (z) = z we see that z is an (N i x 0 − 1)-th root of unity in κ(p) × . Hence z and therefore also y lie in µ(κ(p)) = F × p . But then, by the choice of σ we have σ(y) = y Nx 0 and therefore ζ = 1. Using the isomorphism µ (p) (K) In this section we assume that C is the field of complex numbers.
Let R >0 be the group of positive real numbers under multiplication and consider the suspension It is the quotient ofX 0 (C) × R >0 by the left Q >0 -action given by The group R >0 acts on X 0 via the second factor: We may also view X 0 as a continuous time dynamical system by letting t ∈ R act by e t . We write φ t for this action i.e. φ t ([P 0 , u]) = [P 0 , ue t ]. Dynamical systems of this type can be studied with the methods of [Far04]. For a closed The Q >0 -bijection (10) induces an R >0 -bijection The next result is an immediate consequence of Theorem 1.
Theorem 8. The following decomposition holds, where x 0 runs over the closed For any point The theorem asserts that the closed points x 0 of X 0 correspond bijectively to the compact "packets" Γ x 0 of periodic orbits of length log Nx 0 in the R-dynamical system X 0 . Any periodic orbit γ in X 0 is contained in Γ x 0 for a uniquely determined closed point x 0 of X 0 . In our previous analogies between arithmetic schemes and foliated dynamical systems, we assumed that the closed points should corresponded bijectively to the periodic orbits. This point of view was suggested by analytic results for path-connected phase spaces. The correspondence in Theorem 8 is a little bit more involved.
We now introduce topologies on our spaces. We begin with the case of an affine arithmetic scheme X 0 = spec R 0 and write X = spec R. Viewing • X(C) as a set of multiplicative maps P : R → C we give • X(C) the topology of pointwise convergence. It is the subspace topology induced by the Tychonov topology of Lemma 9. For affine arithmetic schemes X 0 , the natural surjective map pr X : for all n, i.e. p n = P −1 n (0) ⊃ I. Since P n (r) → P (r) for all r ∈ R, it follows that P (r) = 0 for r ∈ I and hence p = P −1 (0) ⊃ I i.e. P ∈ pr −1 X (A). Hence pr −1 X (A) is closed and therefore pr X is continuous.
It is an open subscheme of X = spec R and there is a commutative diagram to (x, P × ) or in terms of multiplicative maps, P ′ is sent to P ′ | R . It follows that the inclusion is continuous.
, then P ∈ A ′ and we are done. If P ∈ • X ′ (C) then P = P ′ | R . For the multiplicative maps P ′ n , P ′ : R ′ → C we have P ′ n (r) → P ′ (r) for each r ∈ R as n → ∞. Let p ′ = P ′ −1 (0) ⊂ R ′ and p = P −1 (0) ⊂ R be the prime ideals belonging to P ′ resp. P . We have p = p ′ ∩ R. Since X ′ = spec R ′ ⊂ X = spec R is an open immersion, the local rings of X ′ in p ′ and X in p are canonically isomorphic. Thus the inclusion R ⊂ R ′ induces a commutative diagram: Here P, P ′ also denote the unique multiplicative extensions of P and P ′ to Since P n (s) → P (s) = 0 for n → ∞ we have P n (s) = 0 for n ≥ n 0 (s). Pointwise convergence P n → P on R gives: for n → ∞. Thus we have P ′ n → P ′ pointwise on R ′ and therefore P ′ ∈ A ⊂ A ′ since P n ∈ A and A is closed in Now let X 0 be any arithmetic scheme. For any affine open (and hence arithmetic) subscheme X ′ 0 of X 0 consider the inclusion • X(C)/G with the quotient topology. Using Lemma 9 one sees that pr X : • X(C) −→ X and hence also pr X 0 : Lemma 11. For any arithmetic scheme X 0 , the group G acts by homeomorphisms on • X(C) and the injective maps F ν : • X(C) ֒→ Proof. We first assume that X 0 = spec R 0 is affine. Consider the equations and Here χ is in the finite group Hom(µ ν (K), C × ). It follows that F ν ( •

X(C)) is both closed and open in
• X(C). Now let A be a closed subset of • X(C). If P n ∈ F ν (A) and P n → P in • X(C), it follows from (21) that P ∈ F ν ( • X(C)). Writing P n = F ν (Q n ) and P = F ν (Q) it follows that Q n (r) → Q(r) for any r ∈ R since any r has the form r = s ν for some s ∈ R. Since A is closed we see that Q ∈ A and hence P ∈ F ν (A). Thus F ν (A) is closed. The continuity of F ν being clear, it follows that F ν induces a homeomorphism onto its image F ν ( • X(C)). Since for any subset Z ⊂ , then by definition of the topology of Namely, any point (x, P × ) ∈ • X(C) in the intersection would satisfy P × | µ(κ(x)) = 1, contradicting the finiteness of (ker P × ) tors .
Recall that the maps F ν extend to bijections F ν :X(C) ∼ − →X(C). We givě X(C) = colim N • X(C) the inductive limit topology. It is the finest topology such that for all ν ∈ N the inclusions The map is continuous and we have seen that In conclusion, F ν is a homeomorphism for every ν ∈ N and therefore also for every ν ∈ Q >0 . Assertion c) is clear.
We giveX 0 (C) =X(C)/G the quotient topology. The projections ) : • X(C) −→ X is continuous for each ν ∈ N . Since pr X = pr X • F ν , this follows from the continuity of pr X : • X(C) → X which was noted before Lemma 11.
Proof. It suffices to show that pr −1 X (η) is dense inX(C). Since pr −1 X (η) anď X(C) are Q >0 -invariant and sinceX(C) is covered by the Q >0 -translates of the subspace Hence it suffices to show that pr −1 (η) is dense in • X(C) if X 0 = spec R 0 is affine. We will first do this in case the chosen radicially closed field K is an algebraic closure of K 0 . Let Hom E (K × , C × ) be the set of homomorphisms χ : K × → C × satisfying condition E. We have an identification We will also view χ as a multiplicative map χ : R → C by extending χ | R\0 by We have to show that every neighborhood of P contains a point from pr −1 (η). Since P (0) = 0 = χ(0), this means that given any finite subset T ⊂ R \ 0 and some ε > 0 there is an element χ ∈ Hom E (K × , C × ) such that |χ(r) − P (r)| < ε for all r ∈ T . (25) We will actually find some χ with finite kernel i.e. satisfying condition E f ⊂ E. We first define a character χ µ : µ(K) → C × , which will be the restriction of χ to µ(K). If the characteristic p of κ(p) is zero we have µ(K) ֒→ κ(p) × and we set χ µ (ζ) = P (ζ) = P × (ζ mod p) for ζ ∈ µ(K) .
If p is non-zero, we have a direct product decomposition and it suffices to define χ µ on each factor. We set noting the inclusion µ (p) (K) ֒→ κ(p) × . On µ p ∞ (K) the map P is identically 1 and we cannot set χ µ = P because then χ µ and hence χ would have a kernel with infinite torsion. Instead we choose some M ≥ 0 which is bigger than the p-valuations of the orders of all the roots of unity in the finite set T . Fix an isomorphism ι : In both cases p = 0 and p = 0 the so defined character χ µ : µ(K) → C × has a finite kernel and we have At a later stage we may have to increase the chosen M. The characters χ that we construct are of the following type. Since µ(K) is divisible the sequence of abelian groups splits. Let α : K × → µ(K) and β : K × /µ(K) → K × be corresponding splittings i.e. α(ζ) = ζ for ζ ∈ µ(K) and β(x) = xα(x) −1 where we set x = x mod µ(K). Note that α(β(x)) = 1. Set Z(1) = 2πiZ and Q(1) = 2πiQ inside of C and choose a complement C ′ to Q(1) as Q-vector spaces, i.e. C = Q(1) ⊕ C ′ . Then the homomorphism is injective. Our characters χ are defined by composition Here χ ′ : K × /µ(K) → C ′ is an injective linear map of Q-vector spaces to be constructed and the top horizontal map sends x to (α(x), x). Explicitely, we have: It is then immediate that χ | µ(K) = χ µ and ker χ = ker χ µ is finite .
In defining χ ′ , so that the resulting character χ is close to P on T , the elements t ∈ T ∩ p are the problematic ones since P (t) = 0. The naive approach which works for t ∈ T \ p, using linear algebra after tensoring with Q runs into the problem that 0 0 is not defined. Let K 1 be the finite extension of K 0 generated by T and let R 1 = R ∩ K 1 be the integral closure of R 0 in K 1 . Since Z is universally Japanese, R 1 is a finite R 0 -algebra and in particular of finite type over Z. As an integral Noetherian scheme, X 1 = spec R 1 has an alteration, cf. [dJ96]. This is a proper, dominant hence surjective morphism f : Z → X 1 from an integral regular scheme Z whose function field L is a finite extension of K 1 . Let z ∈ Z be a point with f (z) = p 1 := p ∩ R 1 . Then we have inclusions, where Since we supposed that K = K 0 was an algebraic closure of K 0 we may assume that L ⊂ K in this diagram. Let m z be the maximal ideal in the local ring O z . The map R 1,p 1 ⊂ O z is local, i.e. p 1 R 1,p 1 = m z ∩ R 1,p 1 and hence p 1 = m z ∩ R 1 . By the Auslander-Buchsbaum theorem, the regular, Noetherian local ring O z is a unique factorization domain. We therefore have a decomposition of multiplicative monoids Here π 1 , π 2 , . . . are a complete set of pairwise non-associated prime elements of O z . If T ∩ p = T ∩ p 1 ⊂ m z is non-empty we can write its elements s 1 , . . . , s m in the form Here N ≥ 1, ν ij ≥ 0 and for each i there is some 1 ≤ j ≤ N with ν ij ≥ 1.
We will come back to this expression later. If T \ p = T \ p 1 is non-empty its elements t 1 , . . . , t n lie in R × 1,p 1 ⊂ O × z . Note that P restricts to a homomorphism R × 1,p 1 → C × since on R × 1,p 1 it is the composition Let Γ be the subgroup of R × 1,p 1 /µ(K 1 ) ⊂ K × /µ(K) generated by t 1 , . . . , t n . Since Γ is finitely generated and torsion-free, it is free of some rank l ≥ 1. Let γ 1 , . . . , γ l be a Z-basis of Γ. Then we have for some k ij ∈ Z and 1 ≤ i ≤ n .
We do not know if ξ i ∈ R × 1,p 1 , but since ξ i differs from an element of R × 1,p 1 only by a root of unity, we know that P (ξ j ) = 0 for all 1 ≤ j ≤ l. We need χ µ and P to agree on all ζ i for 1 ≤ i ≤ n. This will be the case if the integer M ≥ 0 in (26) is bigger than the p-valuations of the orders of all the roots of unity ζ i for 1 ≤ i ≤ n. Choosing some M that satisfies this condition as well, χ µ is finally defined. We have Thus, if χ ′ is defined in such a way, that for the corresponding χ the values P (ξ j ) and χ(ξ j ) are very close to each other, it will follow that P (t i ) and Note here that α(ξ j ) = α(β(γ j )) = 1. Choose z j ∈ C with exp z j = P (ξ j ) and choose Q-linearly independent complex numbers w 1 , . . . , w l ∈ C ′ such that each w j is very close to z j for each j. Note that C ′ is dense in C since it has to contain two R-linearly independent vectors. We obtain an injective Q-linear map χ ′ : Γ ⊗ Q ֒→ C ′ by setting χ ′ (γ j ) = w j for 1 ≤ j ≤ l. Then P (ξ j ) and χ(ξ j ) will be close to each other as desired. We extend χ ′ from Γ⊗Q ⊂ (R × 1,p 1 /µ(K 1 ))⊗Q ⊂ (O × z /µ(L))⊗Q to an injective Q-linear map The isomorphism (29) gives a decomposition of Q-vector spaces The values of χ ′ on the first summand on the right are already determined. In the formula resulting from (30), the values χ ′ (u i ) are therefore given. We extend χ ′ to a Q-linear injection in such a way that the values Re χ ′ (π 1 ), . . . , Re χ ′ (π N ) are close to −∞. Then the real parts of will be close to −∞ as well. It is decisive here that all ν ij ≥ 0 and for each i there is an index j with ν ij ≥ 1. In this way we can make as small as we wish and hence χ(s i ) as close to P (s i ) = 0 as desired. Having made the necessary choices to ensure that (25) will hold, we now choose any extension of χ ′ from (L × /µ(L)) ⊗ Q to a Q-linear injection χ ′ : K × /µ ֒→ C ′ . Note that K × /µ is a Q-vector space. Hence we have found a character χ ∈ Hom E f (K × , C × ) satisfying (25). Thus we are done in the case where K = K 0 . In fact, we have shown that the characters χ with finite kernel in pr −1 X (η) are dense in the systemX(C) of Theorem 9 where we only assume (ker P × ) tors to be finite. We now prove this stronger statement in case K is any radically closed Galois extension of K 0 . This will imply the theorem for such K. For • X(C) constructed as in Theorem 7 using K 0 resp. K we write • X K 0 (C) and We claim that the natural map res : is continuous and surjective. Since res maps pr −1 (η K 0 ) to pr −1 (η K ) the proven denseness of pr −1 (η K 0 ) in • X K 0 (C) will then imply denseness of pr −1 (η K ) in • X K (C). If P n → P then res P n → res P by the definition of pointwise convergence. The spaces being metrizable it follows that res is continuous. Let (p, P × ) be a point of . Let R 0 be the integral closure of R 0 in K 0 . The extension R 0 ⊃ R is integral and hence there is a prime ideal p of R 0 with p = R ∩ p. We have an exact sequence It splits because κ(p) × is divisible. Moreover the quotient L is uniquely divisible, since κ(p) × is divisible and µ(κ(p)) = µ(κ(p)). Consider the map P × : κ(p) × → C × . The subgroup exp −1 (im P × ) generates a Q-vector space V in C of at most countable dimension. Hence there is a Q-linear injection ω of the Q-vector space L into C whose image meets V only in 0. The resulting homomorphism prolongs P × and the torsion subgroup of ker P × = ker P × is finite. Explicitely Here α p : κ(p) × → κ(p) × is a chosen splitting of (31). Hence res is surjective.
We now investigate connectedness properties of our spaces. Let V be a countable Q-vector space and let H = Hom(V, C × ) be the set of group homomorphisms ϕ : V → C × . We give H the topology of pointwise convergence i.e. the subspace topology of the inclusion Let H inj ⊂ H be the subspace of injective ϕ's.
Lemma 14. For any ϕ 0 , ϕ 1 ∈ H and neighborhoods U 0 of ϕ 0 and U 1 of ϕ 1 there are ψ 0 , ψ 1 ∈ H inj with ψ 0 ∈ U 0 and ψ 1 ∈ U 1 that can be connected by a continuous path in H inj .
Proof. Choose a finite set ∅ = S ⊂ V and some ε > 0 such that The open sets U ′ i are non-empty since ϕ i ∈ U ′ i for i = 0, 1. Choose a Zbasis b = {v 1 , . . . , v n }, n ≥ 0 of the finitely generated torsion free (hence free) subgroup of V generated by S. For small enough δ > 0 any ϕ ∈ H with ϕ(v) ∈ U δ (ϕ 0 (v)) for v ∈ b satisfies ϕ(s) ∈ U ε (ϕ 0 (s)) for all s ∈ S, and similarly for ϕ 1 instead of ϕ 0 . This implies that for small δ > 0 and i = 0, 1 Choose a Q-vector space complement C 0 ⊂ C to the countable Q-vector space generated by exp −1 (im ϕ 0 ). Then C 0 is dense in C since it has to contain two Rlinear independent vectors. Hence there is an injective Q-linear map ψ ′ For a continuous path α : [0, 1] → C with α(0) = 0, α(1) = 1 set We want ψ t to be in H inj for all 0 ≤ t ≤ 1. This means that ψ t (v) = 1 or equivalently ψ ′ t (v) / ∈ 2πiZ for all 0 = v ∈ V and 0 ≤ t ≤ 1. In other words, α has to avoid the set by the construction of ψ ′ 0 . Hence Ω consists of the complex numbers Thus Ω ⊂ C is countable. By construction, α(0) = 0 and α(1) = 1 are not in Ω. The complement of any countable set in C is path connected. In fact, given two points z 1 = z 2 in the complement, for all but countably many slopes the lines through them lie in the complement. Taking two such lines through z 1 resp. z 2 which intersect we get a path from z 1 to z 2 . Hence a path α avoiding Ω exists. The corresponding continuous path t → ψ t joins ψ 0 with ψ 1 in H inj .
We call a topological space Z almost path-connected if for any two points z 0 , z 1 ∈ Z and any two neighborhoods U 0 of z 0 and U 1 of z 1 there are two pointsz 0 ∈ U 0 andz 1 ∈ U 1 and a continuous map γ : Let X 0 be an arithmetic scheme. The group Q >0 acts onX 0 (C) via the homeomorphisms F q for q ∈ Q >0 . We give the quotient topology ofX 0 (C) × R >0 . Then the R >0 -action is continuous. The continuous projectioň is open since Q >0 acts by homeomorphisms on the left. Set Then d µ = 1 if and only if K 0 is disjoint from the maximal cyclotomic extension of Q in K. Consider: Theorem 15. The space X 0η 0 has d µ connected components. Each of them is almost path-connected.
Proof. Let Hom f (µ(K), C × ) = Hom f (µ(K), S 1 ) ⊂ (S 1 ) µ(K) be the space of homomorphisms µ(K) → C × with finite kernel, equipped with the topology of pointwise convergence. Fix an isomorphism i : µ(K) ∼ − → µ(C) and consider the induced topological isomorphism Here, if ζ is an N-th root of unity, ζ a := ζ a N if a N mod N is the image of a ∈Ẑ under the projectionẐ →Ẑ/NẐ = Z/N. The character χ a is injective if and only if a ∈Ẑ × . In general, if ker χ a is finite, it is cyclic since µ(K) ⊂ K × . For | ker χ a | = N let χ a : µ(K)/µ N (K) ֒→ C × be induced by χ a . Let χ be the composition i.e. χ = χ a • ( ) −N . Then χ is injective and hence of the form χ = χ b for some b ∈Ẑ × . By construction we have χ a = χ N and hence a = Nb. On the other hand, if a has this form, then χ a has finite kernel. Hence the isomorphism (32) restricts to a homeomorphism: Here NẐ × ⊂Ẑ carries the subspace topology. Note that N ∩Ẑ × = {1} inẐ. Consider the restriction map It is continuous for the topologies of pointwise convergence, and surjective by Lemma 3. Choose a splitting α : K × → µ(K) of the exact sequence Given a character χ µ ∈ Hom f (µ(K), C × ), the following map is a homeomorphism onto its image Here we view ψ = χ(χ −1 µ • α) which is trivial on µ(K) as a character on K × /µ(K). The inverse map f −1 defined on im f sends ψ : K × /µ(K) → C × to the character χ defined by χ(x) = ψ(x)χ µ (α(x)) for x ∈ K × . The continuity of f and f −1 can be checked on pointwise convergent sequences of characters. Moreover, we claim that f restricts to a homeomorphism between the subspace of characters with finite kernels within r −1 (χ µ ) (recall E f ⊂ E) and Hom inj (K × /µ(K), C × ). For this it suffices to check that the projection map π : For x ∈ ker χ we have ψ(x) = χ −1 µ (α(x)) ∈ µ(C). Hence ψ(x) N = 1 for some N ≥ 1, i.e. x N ⊗1 = x⊗N ∈ ker ψ ⊗Q, hence x⊗1 ∈ ker ψ ⊗Q. Thus we obtain the map π ⊗ id of (36). If x ⊗ q is zero, either q = 0 or x = 1 i.e. x ∈ µ(K) i.e. x N = 1 and hence x ⊗ N = 0. In both cases x ⊗ q = 0. Hence π ⊗ id is injective. Given x ∈ Ker ψ we have χ(x) = χ µ (α(x)) and hence χ(x) N = 1 for some N ≥ 1. Hence x N ⊗ 1 = x ⊗ N ∈ ker χ ⊗ Q and hence x ⊗ 1 ∈ ker χ ⊗ Q. Thus x ⊗ q ∈ ker χ ⊗ Q is mapped to x ⊗ q and therefore π ⊗ id is surjective. Lemma 14 and the preceding facts about the map f in (35) show that the fibres of the restriction map r are almost path connected. We have: .
Using (33) we may therefore view r as an N -equivariant continuous surjection with almost path-connected fibres Passing to the colimit over N we obtain a Q >0 -equivariant continuous surjection with almost path-connected fibres Hence the maps (37) and (38) are G-equivariant as well. The G-action on the right is given by multiplication with the cyclotomic character κ of G. Here, κ is the composition, where µ = µ(K): The quotientẐ × /κ(G) = Gal(K 0 ∩ Q(µ)/Q) has order d µ . Passing to the orbit spaces mod G in (38) we obtain a map whose fibres are continuous images of the fibres of r in (38) and are therefore almost path connected. Incidentally, note that the quotient topology on pr −1 X 0 (η 0 ) = pr −1 X (η)/G equals the subspace topology withinX 0 (C), the latter being equipped with the quotient topology viaX 0 (C) =X(C)/G.
Then F is almost path connected and Hence the continuous map is a bijection and therefore (X 0η 0 ) a is almost path connected. The composition is continuous and its fibres are the spaces (X 0η 0 ) a . Since the groupẐ × /κ(G) is finite and discrete, it follows that the connected components of X 0η 0 are the d µ spaces (X 0η 0 ) a for a ∈Ẑ × /κ(G).
Remark. For K = K 0 , consider the map r of (34). Let Hom inj (µ(K), C × ) be the set of injective homomorphisms µ(K) ֒→ C × . Then the Galois group G acts without fixed points on This follows from [KS16] Lemma 4.12 (the condition F ⊃ µ ∞ is not necessary).
with the quotient topology. The space Given a multiplicative map P × : κ(x) × → C × with kernel of class E pullback by f ♯ τ,x gives a multiplicative map The kernel of f τ (P × ) is contained in the one of P × and hence it is of class E as well. Thus we get an N -equivariant map One can show that it induces an N -equivariant map of orbit spaces which is independent of the choice of τ . For C the field of complex numbers, the maps • f τ and • f 0 are both continuous. From • f 0 we get mapsf 0 :X 0 (C) →X ′ 0 (C) and f 0 : X 0 → X ′ 0 which commute with the Q >0 -resp. R >0 -actions. Our constructions therefore give functors from the category of arithmetic schemes with dominant morphisms to the categories of N -resp. Q >0 -resp. R >0 -dynamical systems with equivariant (continuous) maps.
Let us look at the affine case X 0 = spec R 0 . Here • X(C) consists of the multiplicative maps P : R → C described after Definition 2. Since P (0) = 0, the map P is uniquely determined by its restriction to R ♦ = R \ 0. Let ZR ♦ denote the monoid algebra of (R ♦ , ·). The multiplicative map P : R ♦ → C extends uniquely to a ring homomorphism P : ZR ♦ → C. This gives an inclusion • X(C) ⊂ Hom(ZR ♦ , C) = (spec ZR ♦ )(C) .
Passing to the orbit spaces under the G-action, we get an inclusion Here we used [KS16,Lemma 4.9]. For a review of the rational Witt vectors of a ring, we refer to [KS16,4.1]. Arguing as in [KS16,Remark 4.3], one sees that there is a canonical isomorphism W rat (R 0 ) = (ZR ♦ ) G .

Hence we find that
Proof. We view r 0 as a function on • X(C) via r 0 (P ) = P (r 0 ) and let V (r 0 ) = spec R/r 0 R ⊂ X be the zero-set of r 0 in X. Because of the commutative diagram (4) it suffices to show that {P ∈ • X(C) | r 0 (P ) = 0} = pr −1 X (V (r 0 )) .
In particular, for R 0 the ring of integers in a number field K 0 , an algebraic number r 0 ∈ R 0 viewed as a complex valued function on • X 0 (C) has zeroes precisely in the N -orbits pr −1 X 0 (p 0 ) for the prime divisors p 0 of r 0 .

Miscellanious remarks and questions
If the Hasse-Weil zeta function X 0 is to be the Ruelle zeta function of X 0 , then the "Fuller indices" of the compact packets Γ x 0 of periodic orbits for the closed points x 0 of X 0 all ought to be 1. Is there a natural extension of the Fuller index to spaces like X 0 which gives the value 1 to each Γ x 0 ? Given an arithmetic scheme X 0 there are natural flat complex line bundles C(s) for s ∈ C on the associated R >0 -dynamical system X 0 =X 0 (C) × Q >0 R >0 . Let C(s) be C as a vector space, and equipped with the Q >0 -action where q ∈ Q >0 acts by multiplication with q s . We get a left action of Q >0 onX 0 (C)×R >0 ×C(s) by setting q(P 0 , u, z) := (F −1 q (P 0 ), qu, qz) . The quotient by Q >0 is the total space of C(s) C(s) = (X 0 (C) × R >0 × C(s))/Q >0 .
Together with the canonical projection π : C(s) −→ X 0 we obtain the line bundle C(s). For s, s ′ ∈ C we have canonical isomorphisms C(s) ⊗ C(s ′ ) = C(s + s ′ ) .
Similarly we obtain flat real resp. rational line bundles R(s) for s ∈ R and Q(s) for s ∈ Q.
In view of the considerations in [Den00], [Den98], [Den01], [Den08],the cohomologies H i (X 0 , C(s)) may contain interesting arithmetic information about X 0 . Are the groups H i (X 0 , Q(n)) related to motivic cohomology? We also draw attention to the foliation cohomologies H i (X 0 , R) and H i (X 0 , C) (or their maximal Hausdorff quotients) equipped with the induced R >0 -action. Here R resp. C is the sheaf of germs of real resp. complex valued functions f on X 0 such that the map (P 0 , u) → f ([P 0 , u]) has continuous partial derivatives in u ∈ R >0 of all orders and locally constant in P 0 ∈X 0 (C) if u is fixed. Assume that Q >0 acts properly discontinuously onX 0 (C) × R >0 . We have not checked this yet but it seems possible since the stabilizers of Q >0 onX 0 (C) are trivial or of the form Nx Z 0 and the quotient R >0 /Nx Z 0 is harmless. If θ = u ∂ ∂u denotes the infinitesimal generator of the R >0 -action, we then obtain a short exact sequence of sheaves on X 0 for any s ∈ C: For surjectivity, given a local section f of C, choose some u 0 such that the following integral is locally defined: Here θ also denotes the map on cohomology induced by the sheaf map θ : C → C.
On H i (X 0 , C) it can be viewed as the infinitesimal generator of the induced R >0action. Its spectrum could be interesting.
One prediction in [Den98] was that X 0 should have a "compactification" X 0 corresponding to an Arakelov compactification X 0 of X 0 . The fixed points of the R-action on X 0 should be the set X 0 (C)/F ∞ where F ∞ : X 0 (C) → X 0 (C) is induced by complex conjugation on C. In this context, we make the following observation. The isotropy groups in Q >0 of the classical points X 0 (C) ⊂X 0 (C) ( p l ) = ( l p )? Is there a natural leafwise metric on X 0 or on a sub-dynamical system which transforms by the factor u unter the action of u ∈ R >0 , i.e. by e t under the flow φ t ? This may happen only on the level of cohomology. The complex points of a Shimura variety can be described group theoretically Is there a group theoretical description of • Sh(G, X) Z (C)?
For C p =Q p the Q >0 -dynamical systemX 0 (C p ) may be interesting for studying the arithmetic geometry of X 0 from a p-adic point of view. The constructions ofX 0 (C) andX 0 (C p ) are extrinsic in that they use the Galois action on the infinite ramified covering X of X 0 . There should be an intrinsic description of these systems and even of a more fundamental objectX 0 with Q >0 -action whose C-resp. C p -valued points would beX 0 (C) resp.X 0 (C p ) but which exists also if X 0 is an F p -scheme. It might then be possible to take the C ∞ = F p ((t)) points ofX 0 for the purposes of function field arithmetic.