Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 193
Proof. We deduce from Theorem A.14, the existence of a constant 0 < γ < 1
satisfying {ν(R,x) > 1 − γ} = E1. Indeed, the sequence of analytic sets {ν(R,x) ≥
1 − 1/i} is decreasing, hence stationary. Consider a point x ∈ X \ E1.Wehavex �∈Y
and if νn := ν(Rn,x), then∑νn ≤ 1 − γ. SinceE1 contains Y , ν0 = 0andF1 :=
g −m (x) contains exactly d mp points. The definition of ν1, which is “the multiplicity”
of d −mp ( f m )∗[Y] at x, implies that g −m (x) contains at most ν1d mp points in Y.Then
#g −m (F1 \Y)=d mp #(F1 \Y) ≥ (1 − ν1)d 2mp .
Define F2 := g −m (F1 \ Y ). The definition of ν2 implies that F2 contains at most
ν2d 2mp points in Y. Hence, F3 := g −m (F2 \ Y) contains at least (1 − ν1 − ν2)d 3mp
points. In the same way, we define F4, ..., Fn with #Fn ≥ (1 − ∑νi)d mpn . Hence, for
every n we get the following estimate:
λn(x) ≥ d −mpn #Fn ≥ 1 −∑νi ≥ γ.
This proves the lemma. ⊓⊔
End of the proof of Theorem 1.47. Let E n X denote the set of x ∈ X such that
g−ml (x) ⊂ E1 for 0 ≤ l ≤ n and define EX := ∩n≥0E n X . Then, (E n X ) is a decreasing
sequence of analytic subsets of E1. It should be stationary. So, there is n0 ≥ 0such
that E n X = EX for n ≥ n0.
By definition, EX is the set of x ∈ X such that g−mn (x) ⊂ E1 for every n ≥ 0.
Hence, g−m (EX ) ⊂ EX. It follows that the sequence of analytic sets g−mn (EX) is
decreasing and there is n ≥ 0suchthatg−m(n+1) (EX )=g−mn (EX). Sincegmn is
surjective, we deduce that g−m (EX )=EX and hence EX = gm (EX).
Assume as in the theorem that E is analytic with g−s (E) ⊂ E. DefineE ′ :=
g−s+1 (E) ∪ ...∪ E. Wehaveg−1 (E ′ ) ⊂ E ′ which implies g−n−1 (E ′ ) ⊂ g−n (E ′ ) for
every n ≥ 0. Hence, g−n−1 (E ′ )=g−n (E ′ ) for n large enough. This and the surjectiv-
ity of g imply that g−1 (E ′ )=g(E ′ )=E ′ . By Lemma 1.48, the topological degree of
(gm′ ) |E ′ is at most dm′ (p−1) for some m ′ ≥ 1. This, the identity g−1 (E ′ )=g(E ′ )=E ′
together with Lemma 1.49 imply that E ′ ⊂ E1. Hence, E ′ ⊂ EX and E ⊂ EX.
Define E ′ X := g−m+1 (EX) ∪ ...∪ EX. Wehaveg−1 (E ′ X )=g(E ′ X )=E ′ X . Applying
the previous assertion to E := E ′ X yields E ′ X ⊂ EX. Therefore, E ′ X = EX and
g−1 (EX) =g(EX) =EX. So,EX is the maximal proper analytic set in X which is
totally invariant under g.
We prove now that there are only finitely many totally invariant algebraic sets.
We only have to consider totally invariant sets E of pure dimension q. The proof
is by induction on the dimension p of X. The case p = 0 is clear. Assume that the
assertion is true for X of dimension ≤ p − 1 and consider the case of dimension p.
If q = p then E is a union of components of X. There are only a finite number of
such analytic sets. If q < p, we have seen that E is contained in EX. Applying the
induction hypothesis to the restriction of f to EX gives the result. �
We now give another characterization of EX. Observe that if X is not locally irreducible
at a point x then g −m (x) may contain more than d mp points. Let π : �X → X