POINTS PÉRIODIQUES D’APPLICATIONS

Periodic points of birational maps of P2 Let f: P2 99K P2 be a birational map of P2 defined by three homogeneous polynomials of degree d with no common factors. We define f to be generic iff ∀n ≥ 0, f−n(I(f)) is finite where I(f) is the set of points of indeterminacy of f. In dynamics, the number of periodic points and more precisely its growth rate with respect to the period is very interesting related in particular to the entropy of the system. We are interested here in studying the case of generic birational transformations of P2. The principal tool is the well-known theorem of Bezout. To use it in its full strength it’s necessary to define precisely the multiplicity of a solution of a system which we recall in the first part. We study the multiplicity at the indeterminacy points and we are then able to prove the following central theorem: Theorem 0.1. Let f: P2 99K P2 be a generic birational mapping of P2, and Fixn the number of non-critical periodic points of fn (counted with multiplicity). If Fixn is finite for all n ≥ 1, there exists a constant D> 0, s.t. ∀n ≥ 1 dn + 2 ≥ Fixn ≥ dn −D. We deduce from that result that we can canonically associate to f a (1, 1) positive closed current T+. It’s defined as the limit of the following sequence of currents T+n:= 1 dn (fn)∗ω where ω is the usual kahlerian form on P2. These currents have only a finite number of singularities, so that we can define their self-intersections. We then have: Theorem 0.2. Let f: P2 99K P2 be a generic birational map of P2. Let us denote by I∞+ the increasing union of the sets of indeterminacy of the succesive iterates of f. I∞+ is at most countable. There exists a numerical function: µ ∞ :