Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 235
natural assumptions on dynamical degrees, we prove that the measure of maximal
entropy is non-uniformly hyperbolic and we study its sharp ergodic properties.
2.1 Examples, Degrees and Entropy
Let V be a convex open set in C k and U ⋐ V an open subset. A proper holomorphic
map f : U → V is called a polynomial-like map. Recall that a map f : U → V is
proper if f −1 (K) ⋐ U for every compact subset K of V.Themapf sends the boundary
of U to the boundary of V; more precisely, the points near ∂U are sent to points
near ∂V . So, polynomial-like maps are somehow expansive in all directions, but the
expansion is in the geometrical sense. In general, they may have a non-empty critical
set. A polynomial-like mapping f : U → V defines a ramified covering over V.
The degree dt of this covering is also called the topological degree. It is equal to the
number of points in a generic fiber, or in any fiber if we count the multiplicity.
Polynomial-like maps are characterized by the property that their graph Γ in
U ×V is in fact a submanifold of V ×V,thatis,Γ is closed in V ×V.So,anysmall
perturbation of f is polynomial-like of the same topological degree dt, provided that
we reduce slightly the open set V. We will construct large families of polynomiallike
maps. In dimension one, it was proved by Douady-Hubbard [DH] that such
a map is conjugated to a polynomial via a Hölder continuous homeomorphism.
Many dynamical properties can be deduced from the corresponding properties of
polynomials. In higher dimension, the analogous statement is not valid. Some new
dynamical phenomena appear for polynomial-like mappings, that do not exist for
polynomial maps. We use here an approach completely different from the one dimensional
case, where the basic tool is the Riemann measurable mapping theorem.
Let f : C k → C k be a holomorphic map such that the hyperplane at infinity is
attracting in the sense that � f (z)�≥A�z� for some constant A > 1andfor�z� large
enough. If V is a large ball centered at 0, then U := f −1 (V ) is strictly contained
in V. Therefore, f : U → V is a polynomial-like map. Small transcendental perturbations
of f , as we mentioned above, give a large family of polynomial-like maps.
Observe also that the dynamical study of holomorphic endomorphisms on P k can
be reduced to polynomial-like maps by lifting to a large ball in C k+1 .Wegivenow
other explicit examples.
Example 2.1. Let f =(f1,..., fk) be a polynomial map in Ck , with deg fi = di ≥ 2.
Using a conjugacy with a permutation of coordinates, we can assume that
d1 ≥ ··· ≥ dk. Letf +
i denote the homogeneous polynomial of highest degree in
fi.If{ f + 1 = ···= f + k = 0} is reduced to {0},thenfis polynomial-like in any large
ball of center 0. Indeed, define d := d1 ...dk and π(z1,...,zk) := (z d/d1
1 ,...,z d/dk k ).
Then, π ◦ f is a polynomial map of algebraic degree d which extends holomorphically
at infinity to an endomorphism of Pk . Therefore, �π ◦ f (z)� � �z�d for
�z� large enough. The estimate � f (z)� � �z�dk near infinity follows easily. If we
consider the extension of f to Pk , we obtain in general a meromorphic map which
is not holomorphic. Small pertubations fε of f may have indeterminacy points in
Ck and a priori, indeterminacy points of the sequence ( f n ε )n≥1 may be dense in Pk .