Generic Antiholomorphic Polynomial Vector Fields

TL;DR

This work provides a complete moduli-type classification of generic antiholomorphic polynomial vector fields $\dot z = \overline{P(z)}$ by combining a combinatorial invariant (noncrossing trees arising from the separatrix graph) with an analytic invariant given by line integrals across sepal zones. A Realization Theorem proves that every admissible pair $(G,\eta)$, with $G$ a $(k+2)$-noncrossing tree and $\eta\in\mathbb{H}^k$, is realized by some monic centered polynomial $P$ of degree $k+1$, yielding a moduli space of generic fields homeomorphic to $\mathcal{A}_k \times \mathbb{H}^k$ and counting the generic strata by the number $A(k+2)$ of noncrossing trees. The paper also treats non-generic cases with maximal heteroclinic connections via a ternary-tree invariant and provides a detailed bifurcation diagram for the quadratic family $\dot z = \overline{z^2 - \varepsilon}$, where heteroclinic transitions occur along three rays in parameter space. Overall, the results give a concrete, realizable, and highly structured description of the dynamics and parameter-space stratification for generic and near-generic antiholomorphic polynomial vector fields, with explicit combinatorial and analytic data controlling equivalence classes and bifurcations.

Abstract

An analytic classification of generic anti-polynomial vector fields $\dot z = \overline{P(z)}$ is given in term of a topological and an analytic invariants. The number of generic strata in the parameter space is counted for each degree of $P$. A Realization Theorem is established for each pair of topological and analytic invariants. The non-generic case of a maximal number of heteroclinic connections is also given a classification. The bifurcation diagram for the quadratic case is presented.

Figures (13)

• Figure 1: Phase portrait of $\dot z = \overline{P(z)}$, with $\deg P = 5$
• Figure 2: Points $Z,W$ with $\omega$-limit in different ends, and three boundaries in dashed lines
• Figure 3: Incoming graph obtained from a separatrix graph for $k = 2$.
• Figure 4: A cycle in an incoming graph prevents any outgoing separatrix at $z_j$ to reach the circle at infinity.
• Figure 5: There is a bijection between $n$-nc trees such that $\deg 1 = 1$ having $(1,j)$ as an edge and the set of pairs $(G_1,G_2)$ of nc trees on $j-1$ and $n+1-j$ vertices.

Theorems & Definitions (44)

• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Definition 2.5
• proof : Proof of Proposition \ref{['prop:graphe sep']}
• Lemma 2.7
• proof
• Lemma 2.8
• proof
• Lemma 2.9