New invariants and attracting domains for holomorphic maps in $\mathbf{C}^2$ tangent to the identity

Abstract

We study holomorphic maps of C 2 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.

Domain of Attraction for Maps Tangent to the Identity in $$\mathbb {C} ^2$$ C 2 with Characteristic Direction of Higher Degree

“…b The example given that satisfies these conditions was f (z, w) = (z + w 2 , w). c The main theorem in [Ro2] applied only to apparent characteristic directions and more specifically required t = k, whereas Theorem A applies to apparent (and other) characteristic directions for t ≥ k.…”

“…Theorem A is an extension of the following theorem due to Rong that, among other things, assumes t = k and, as a consequence of its assumptions, the characteristic direction is apparent [Ro2].…”

“…In the irregular case, Vivas and the author (for a unique non-degenerate characteristic direction) independently showed that there always is a domain of attraction along an irregular characteristic direction (see [L1, L2, V2]). In the apparent case, Rong and Vivas independently showed sufficient conditions for the existence (or non-existence) of a domain of attraction along an apparent characteristic direction (see [Ro2,V2]). Theorem A extends what is known about the existence of a domain of attraction when the direction is Fuchsian, apparent, or dicritical.…”

A Brief Survey on Local Holomorphic Dynamics in Higher Dimensions

“…Now assume that µ < ∞ and rewrite f as Theorem 4.2 (Rong,[R7]). Let f be a holomorphic map in C 2 , tangent to the identity at the origin.…”

Attracting domains for quasi-parabolic analytic transformations of ℂ2

“…the linear part of the transformation is the identity). See, for example, [1] and the references therein for a survey on known results and [8][9][10] for more recent advancements. In [3][4][5][6][7], some of these results have been extended to quasi-parabolic transformations.…”