Discrete Holomorphic Local Dynamical Systems

Dynamics of Rational Surface Automorphisms 89
Corollary 2.9. If η is an eigenvalue of f ∗ , then either η = λ ,λ −1 ,or|η| = 1.
Proof. We have seen that λ ( f ) is the only eigenvalue of modulus > 1. Now we know
that ( f ∗ ) −1 =(f −1 ) ∗ ,soifη is an eigenvalue of f ∗ ,thenη −1 is an eigenvalue of
( f −1 ) ∗ . Applying the Theorem to f −1 , we conclude that λ is the only eigenvalue
for ( f −1 ) ∗ which is > 1. ⊓⊔
Let χ f denote the characteristic polynomial of f ∗ . It follows that χ f is monic, and
the constant term (the determinant of f ∗ , an invertible matrix) is ±1. Let ψ f denote
the minimal polynomial of λ . By the Theorem, we see that except for λ and 1/λ ,
all zeros of χ f (and thus all zeros of ψ f ) lie on the unit circle. Such a polynomial ψ f
is called a Salem polynomial,andλ is a Salem number. We may factor χ f = C · ψ f ,
where C is a polynomial whose coefficients belong to Z, and the roots of C lie in the
unit circle. It follows by elementary number theory that the zeros of C are roots of
unity.
The following is a heuristic argument for the existence of a positive, closed invariant
1,1-current. Suppose that X is a Kähler surface and that f ∈ Aut(X) has
λ ( f ) > 1. Let ω + be a positive, smooth cohomology class which is an eigenvector
of λ . By this we mean that there is a smooth form ω + such that the cohomology
class is expanded
{ f ∗ ω + } = f ∗ {ω + } = λ {ω + }.
Thus f ∗ ω + − λω + is cohomologous to zero, and thus there is a smooth function γ +
such that
1
λ f ∗ ω + − ω + = dd c γ + .
If we can take ω + to be poisitive, then f ∗ ω + will be positive, and we see that γ +
is essentially pluri-subharmonic, i.e., dd c γ + + ω + ≥ 0. Applying λ −1 f ∗ repeatedly,
we find
If we define
1
λ 2 f ∗2 ω + − 1
λ f ∗ ω + = 1
λ ddc f ∗ γ + = 1
λ ddcγ + ◦ f
1
λ n f ∗n ω + − 1
λ n−1 f ∗(n−1) ω + = 1
λ n−1 ddcγ + ◦ f n−1 .
g + =
∞ γ
∑
n=0
+ ◦ f n
λ n , T + = ω + + dd c g + ,
then we see that g + is continuous, since the defining series converges uniformly.
Further, T + is a positive, closed current with the invariance property f ∗ T + = λ T + .
We may apply the same argument with f −1 and the eigenvalue λ −1 obtain a positive,
closed current T − with the property that ( f −1 ) ∗ T − = λ T − . We may obtain an
invariant measure μ := T + ∧ T − .
Notes The currents T ± and measure μ have been shown to have many of the same
properties that were found for the Hénon family (see [C2,Du]). Much of this theory
may be carried over to more general meromorphic surface mappings; one recent
work in this direction is [DDG1–3].