xxvi INTRODUCTION

as z moves in P 1 . The z's for which the orbit is "unstable" form a closed set called

the Julia set. The complement of the Julia set is called the Fatou set. Points that

eventually return to themselves after a number of iterations are called periodic. Up

to finitely many points the Julia set turns out to coincide with the closure of the set

of periodic points and usually looks like a "fractal". A point z E C is critical if the

tangent map of s at z vanishes. The orbits of critical points hold the key to many of

the dynamical properties of s. In particular the maps s for which all critical points

have finite orbits have been closely investigated by Thurston [51]; they are called

post critically finite and one can attach orbifolds and Euler characteristics to them.

Postcritically finite maps with Euler characteristic 0 have been classified and they

are essentially those that admit an "analytic uniformization" by the complex plane

C. Let us call them here, for simplicity, flat maps. Flat maps can be explicitly

described as being either multiplicative maps, or Chebyshev polynomials or Lattes

maps (the latter being maps induced by endomorphisms of elliptic curves). The

Julia sets of flat maps are "smooth" (i.e. not "fractals"). For multiplicative maps

the Julia set is a circle. For Chebyshev maps the Julia set is a segment. For Lattes

maps the Julia set is the whole of P 1 . (Note however that flat maps are not the

only maps that have smooth Julia sets !)

Flat maps play a key role in our theory. Indeed let us attach to any rational map

s a correspondence X* = (P 1 , P 1 , id, s) in complex algebraic geometry. Assume,

for simplicity, that s has rational coefficients. Fix a prime p. Then, as explained

earlier, we can (roughly speaking) attach to X a correspondence X^ in S—geometry

at p. What we will prove in this book is, roughly speaking, that the categorical

quotient of X$ in S—geometry is non-trivial for almost all p if and only if s is a flat

map.

More generally the concept of postcritical finiteness (and a suitable generaliza-

tion of it to the case of correspondences that do not come from dynamical systems)

plays a key role in the present book. This will be explained in detail in Chapters 1

and 2.

We close our discussion here by noting that complex dynamics has an interesting

p—adic analogue. For instance, the Lubin-Tate theory of formal groups [92], [93]

can be viewed as an early incarnation of a non-archimedian dynamics. For more

recent work see, for instance [93], [5], [6], [91], and the bibliographies therein.

Although our theory lives in the p—adic world and although we will use formal

groups at periodic points in our theory, we will not need to use, in our book, the

"genuine" theory of p—adic dynamical systems in the above cited papers. Here, by

"genuine" we mean "dealing with the p—adic Julia set, p—adic wandering Fatou

domains, etc." (On the other hand, as already mentioned, we do need to use the

theory of complex dynamical systems through the orbifold characterization of flat

maps.) Needless to say it would be very interesting to make our theory interact

with the "genuine" theory of p—adic dynamical systems.

0.3.5. C o n n e s ' n o n - c o m m u t a t i v e g e o m e t r y . The approach in the present

book seems to be, in some sense, perpendicular to Connes' non-commutative geo-

metric approach to quotient spaces; cf. our previous remarks about invariant versus

groupoid ideology. Nevertheless, since some examples occur in both theories it is

not unreasonable to expect interactions between the two approaches. In what fol-

lows we shall discuss some general principles of non-commutative geometry and we