THE NON-DICRITICAL ORDER AND ATTRACTING DOMAINS OF HOLOMORPHIC MAPS TANGENT TO THE IDENTITY

Abstract

We study the local dynamics of holomorphic maps f in Cn tangent to the identity at a fixed point p with a non-degenerate characteristic direction [v]. In [M. Hakim, Analytic transformation of (Cp, 0) tangent to the identity, Duke Math. J.92 (1998) 403–428], n - 1 invariants αj, 1 ≤ j ≤ n - 1, called the directors, were associated to [v] and it was shown that if Re αj > 0 for all j then f has an attracting domain at p tangent to [v]. In this paper, we study the case Re αj = 0 for some j. With the help of a new invariant μ called the non-dicritical order, we show that f has an attracting domain at p tangent to [v] if μ ≥ 1. We also study the "spiral domains" when μ = 0. For n = 2, we show that f has an attracting domain at p tangent to [v] if and only if either the director α > 0 or μ ≥ 1.