NEW INVARIANTS AND ATTRACTING DOMAINS FOR HOLOMORPHIC MAPS IN C 2 TANGENT TO THE IDENTITY

We study holomorphic maps of C2 tangent to the identity at a fixed point which have degenerate characteristic directions. With the help of some new invariants, we give sufficient conditions for the existence of attracting domains in these degenerate characteristic directions. 2010 Mathematics Subject Classification: 32H50.


Introduction
A holomorphic map f in C n is said to be tangent to the identity at a fixed point p, if the Jacobian df p is equal to the identity matrix. Choose local coordinates (z, w) with p as the origin O. Let f (z, w) = (z + ∞ i=2 p i (z, w), w + ∞ i=2 q i (z, w)) be the homogeneous expansion of f , where p i (z, w) and q i (z, w) are homogeneous of degree i. The order k of f at O is by definition the minimum of i such that either p i (z, w) ≡ 0 or q i (z, w) ≡ 0. A direction [z : w] is called a characteristic direction of f , if there exists λ ∈ C such that p k (z, w) = λz and q k (z, w) = λw. If λ = 0, then [z : w] is said to be non-degenerate, otherwise degenerate.
The study of local dynamics of holomorphic maps tangent to the identity has been centered on generalizing the well-known Leau-Fatou Flower Theorem in the one-dimensional case (see e.g. [4], [7]). In [11], Hakim proved such a generalization for generic maps of C n , i.e. those with non-degenerate characteristic directions. In [2], Abate obtained a full generalization in dimension two. In particular, Abate introduced a "residual index" to each characteristic direction of a map tangent to the identity and showed that there exist "parabolic curves" in directions with residual index not in Q + ∪ {0}. In [13], Molino showed the existence of parabolic curves in directions with non-zero residual index, under a mild extra assumption. In [12], Hakim provided sufficient conditions for the existence of attracting domains in nondegenerate characteristic directions. In [17], the author introduced a new invariant, called the "non-dicritical order", associated to each non-degenerate characteristic direction and provided further sufficient conditions for the existence of attracting domains in such directions. More recently, there have been some studies on the local dynamics of so called "one-resonant" and "multi-resonant" biholomorphisms (see e.g. [8,9,10,14]).
The general theory for local dynamics in degenerate characteristic directions with zero residual index is yet to be developed. Characteristic directions of a holomorphic map in C 2 can be divided into three types: irregular, Fuchsian and apparent (cf. [5]). In [18,19], Vivas gave sufficient conditions for the existence of attracting domains in the irregular and Fuchsian directions. However, no general sufficient conditions are known for the existence of attracting domains in the apparent directions.
In this paper, we define some new invariants associated to a degenerate characteristic direction of holomorphic maps in C 2 and give sufficient conditions for the existence of attracting domains in such a direction. The currently known invariants such as non-degeneracy and residual index are defined using only the leading nonlinear terms of the homogeneous expansion of a map. However, examples show that sometimes the higher order nonlinear terms also play an important role in the local dynamics of a map. Our new invariants capture exactly this feature.
Our main result is the following In section 2, we will define our new invariants and explain the terms essentially non-degenerate and transversally attracting. In section 3, we prove Theorem 1.1.
The author would like to thank Berit Stensønes for many stimulating discussions, and the referee for very useful comments.

New invariants
Let f be a holomorphic map in C 2 , tangent to the identity at a fixed point p, with order t + 1, t ≥ 1. Assume that [v] is a degenerate characteristic direction of f . In suitable local coordinates (z, w) around p, we can assume that p is the origin where P t (z, w) and Q t (z, w) are homogeneous of degree t. We say that f is gener- Remark 2.1. A holomorphic map f of order t + 1 at the origin, with a characteristic direction [v] in suitable local coordinates (z, w), can be written as It is easy to check that ordR Φ (Z) = t for every Φ ∈ G, thus being generically degenerate is well defined. We define the virtual order to be and set ν := s − t, which we call the weight. Let H be the subgroup of G consisting of Φ's with ordQ Φ (Z) = s. Then we define the essential order to be Obviously both the virtual order and the essential order are invariants associated to f in the degenerate characteristic direction [v]. Finally, we say that [v] is an essentially non-degenerate characteristic direction of f if µ < s.
Remark 2.4. For an essentially non-degenerate characteristic direction, we certainly have µ < ∞. However, the case µ = ∞ is also interesting. For instance, if µ = s = ∞ then there exists a line of fixed points through p. For the dynamics in this case, see e.g. [6].
For later convenience, set r := µ − 1. Rewrite (2.1) as (2.4) z 1 = z + az r+1 + P (z) + wS(z, w), It is then easy to see that the director is well-defined and the condition (2.5) does not depend on the choice of Φ.
Remark 2.5. The power function x −t/r is a multi-valued function with r/gcd(r, t) different values. Hence α has r/gcd(r, t) different values. The condition (2.5) should be interpreted as one of these values has positive real part. Such a condition is quite weak. In fact, one readily checks that the only case when this condition fails is when t/r = 1/2 and α is purely imaginary.
Remark 2.6. Although above definitions were given under the assumption that [v] is a degenerate characteristic direction for f , it is easy to see that the same definitions can be made replacing generically degenerate by non-degenerate. In the latter case, a non-degenerate characteristic direction is essentially non-degenerate and we have r = t. Then our director minus one is exactly the director of f in the non-degenerate characteristic direction defined in [11]. In this sense, our main theorem is a natural generalization of Hakim's results in [11] to the degenerate case.
For z small enough, we have For n large, we have This completes the proof of Theorem 1.1.
Remark 3.1. For r = 2 and t = 1, a special case of maps as in (2.4) was studied in [15]. (As noted there, we do not get a "flower" of attracting domains, but only some "petals".) Such maps also show up in the study of the local dynamics of holomorphic maps of C 2 with a Jordan fixed point (cf. [16], see also [1,3]).