0.3. COMPARISON WITH OTHE R THEORIES

XXV

is the unique ring endomorphism prolonging (f) : K — K and sending j)T = T^,

pT^ = T^ ,... e.t.c. A basic difference between difference algebra and differential

algebra is that the fibers of J^(X) —

Jr,~l{X)

are not affine spaces in general.

Actually it is plain that J'l(X) splits as a product of varieties

Jl(X) ~X xX* x ... x X ^ \

where X^ is obtained from X by twisting the coefficients of the defining equations

by

f%.

In particular, if X is definable over the fixed field of (p then J1(X) is just

the (r -f 1)—fold product of X with itself. In spite of this geometric difference

between difference algebra and differential algebra a lot of analogies between the

two persist. Systems of linear equations, for instance, behave similarly [62], p. 20.

At the non-linear level the geometric model theory of the two is strikingly similar;

cf. especially the work of Hrushovski and Chatzidakis explained in [69].

It is interesting to compare difference algebra and arithmetic differential alge-

bra. We refer to Remark 2.6 below for an argument suggesting that arithmetic

differential algebra could be viewed as obtained from difference algebra by adjoin-

ing certain divergent series. Here we close our discussion by indicating a possible

bridge at the level of jet spaces. Let us assume, for simplicity, that X is an affine

scheme given by the spectrum of Rp[T]/(F) as in Equation 0.3. Let Kp be the field

of fractions of Rp. Then one can consider a canonical algebraisation of the p—jet

space Jr(Xp) given as the scheme

(0.26) Jr(Xp) := Spec RP[T,T',T"', ...,T^}/(F,SF,S2F, ...,SrF).

Then clearly the p—adic completion of

Jr(Xp)

is

Jr(Xp);

in symbols:

(0.27)

Jr(Xpy

=

Jr(Xp).

On the other hand it is trivial to see that the base change to Kp of Jr(Xp) is

J4(XP

®

KP)'I i n

symbols,

(0.28)

Jr(Xp)

®KP = j;(Xp 0 Kp).

By Equations 0.27 and 0.28 we see that

Jr(Xp)

might be viewed as a "bridge"

between the jet spaces of difference algebra and arithmetic differential algebra.

Whether this bridge can actually be crossed is an open question. We can summarize

the above discussion by adjoining to the diagram in Equation 0.23 the diagram:

(2) (3)

(0.29) j j

(4) (5)

(2)=differential algebraic jet spaces; cf. Equation 0.20

(3)=arithmetic jet spaces; cf. Equation 0.4

(4)—difference jet spaces; cf Equation 0.25

(5)=canonical algebraisation of arithmetic jet spaces; cf. Equation 0.26.

0.3.4. Dynamical systems. Let us start by reviewing some aspects of the

classical iteration theory initiated by Fatou and Julia [109]. In this theory one

starts with a rational map s : P 1 — • P 1 from the complex projective line into itself

and one studies the behavior of orbits

z,s{z)1s2{z),...