Discrete Holomorphic Local Dynamical Systems

Dynamics of Rational Surface Automorphisms 83
which would be exactly what you would obtain G + and G − were smooth and you
integrate by parts: � χdd c G + ∧ dd c G − = � χdd c (G + ∧ μ − )= � (dd c χ) ∧ (G + ∧
μ − ). This defines μ as a distribution. But since μ ± are a positive currents, we see
that (*) defines μ as a positive distribution, and thus a measure.
A variant of the convergence theorem above deals with the normalized pullbacks
d −n f n∗ (χμ + )=χ( f n )μ + :
Theorem 1.39. Let χ be a test function, and let c = � χμ.Then
lim
n→∞ d−n f ∗n (χμ + )= lim
n→∞ χ( f n )μ + = cμ + .
A measure ν is said to be mixing with respect to a transformation f if limn→∞ ν(A∩
f −n B)=ν(A)ν(B) for all Borel sets A and B. For this, it suffices to show that for
every pair of smooth functions ϕ and ψ,wehave
�
lim
n→∞
ϕ( f n �
)ψμ=
�
ϕμ
ψμ
Corollary 1.40. The measure μ is mixing (and thus ergodic).
Proof. We have � ϕ( f n )ψμ= � (ϕ( f n )μ + ) ∧ (ψμ − ),soasn → ∞, the right hand
integral converges to � (cμ + ) ∧ ψμ − = c � ψμ + ∧ μ − = c � ψμ, and the constant is
c = � ϕμ. ⊓⊔
2 Rational Surfaces
2.1 Blowing Up
In the second section of these notes, we will consider automorphisms of a compact,
rational surface M. Given an automorphism f ∈ Aut(M), we can look at its pullback
on cohomology f ∗ ∈ GL(H 2 (M)). The dynamical degree is then defined as
λ ( f ) := lim
n→∞ || f n∗ || 1/n .
A surface M ′ that is birationally equivalent to M may be topologically different:
for instance, H 2 (M ′ ) and H 2 (M) can have different dimensions, and so the induced
maps on cohomology will not be conjugate. However, in [DF]itisshownthatλ ( f )
is the same for all birationally equivalent maps.
Here we will focus on automorphisms for which λ ( f ) > 1. Although we will not
discuss entropy here, we note that in this case the entropy is log(λ ( f )) > 0. Much of
the theory for Hénon maps can be carried over to the case of automorphisms of compact
surfaces. However, the Hénon family of diffeomorphisms themselves will not
be part of this section: for a Hénon map H, there is no compact, complex manifold
M which compactifies C 2 in such a way that H becomes a homeomorphism of M.