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)
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
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
in symbols:
On the other hand it is trivial to see that the base change to Kp of Jr(Xp) is
KP)'I i n
®KP = j;(Xp 0 Kp).
By Equations 0.27 and 0.28 we see that
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
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