Discrete Holomorphic Local Dynamical Systems

32 Marco Abate
Theorem 4.24 (Naishul, 1983 [N]). Let f , g ∈ End(C,O) be two holomorphic local
dynamical systems with an elliptic fixed point at the origin. If f and g are topologically
locally conjugated then f ′ (0)=g ′ (0).
See [P7] for another proof of this result.
5 Several Complex Variables: The Hyperbolic Case
Now we start the discussion of local dynamics in several complex variables. In this
setting the theory is much less complete than its one-variable counterpart.
Definition 5.1. Let f ∈ End(Cn ,O) be a holomorphic local dynamical system at
O ∈ Cn , with n ≥ 2. The homogeneous expansion of f is
f (z)=P1(z)+P2(z)+···∈C0{z1,...,zn} n ,
where Pj is an n-uple of homogeneous polynomials of degree j. In particular, P1 is
the differential dfO of f at the origin, and f is locally invertible if and only if P1 is
invertible.
We have seen that in dimension one the multiplier (i.e., the derivative at the origin)
plays a main rôle. When n > 1, a similar rôle is played by the eigenvalues of the
differential.
Definition 5.2. Let f ∈ End(Cn ,O) be a holomorphic local dynamical system at
O ∈ Cn , with n ≥ 2. Then:
– if all eigenvalues of dfO have modulus less than 1, we say that the fixed point O
is attracting;
– if all eigenvalues of dfO have modulus greater than 1, we say that the fixed
point O is repelling;
– if all eigenvalues of dfO have modulus different from 1, we say that the fixed
point O is hyperbolic (notice that we allow the eigenvalue zero);
– if O is attracting or repelling, and dfO is invertible, we say that f is in the
Poincaré domain;
– if O is hyperbolic, dfO is invertible, and f is not in the Poincaré domain (and
thus dfO has both eigenvalues inside the unit disk and outside the unit disk) we
say that f is in the Siegel domain;
– if all eigenvalues of dfO are roots of unity, we say that the fixed point O is
parabolic; in particular, if dfO = id we say that f is tangent to the identity;
– if all eigenvalues of dfO have modulus 1 but none is a root of unity, we say that
the fixed point O is elliptic;
– if dfO = O, we say that the fixed point O is superattracting.
Other cases are clearly possible, but for our aims this list is enough. In this survey
we shall be mainly concerned with hyperbolic and parabolic fixed points; however,
in the last section we shall also present some results valid in other cases.