Newton flows for elliptic functions II Structural stability: Classification & Representation

Abstract: In our previous paper we associated to each non-constant elliptic function $f$ on a torus $T$ a dynamical system, the elliptic Newton flow corresponding to $f$. We characterized the functions for which these flows are structurally stable and showed a genericity result. In the present paper we focus on the classification and representation of these structurally stable flows. The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded, graph $\mathcal{G}(f)$ on a torus $T$ with $r$ vertices, 2$r$ edges and $r$ faces that fulfil certain combinatorial properties ( Euler, Hall) on some of its subgraphs. The graph $\mathcal{G}(f)$ determines the conjugacy class of the flow. [classification] A connected, cellularly embedded toroidal graph $\mathcal{G}$ with the above Euler and Hall properties, is called a Newton graph. Any Newton graph $\mathcal{G}$ can be realized as the graph $\mathcal{G}(f)$ of the structurally stable Newton flow for some function $f$. This leads to: up till conjugacy between flows and (topological) equivalency between graphs, there is a one to one correspondence between the structurally stable Newton flows and Newton graphs, both with respect to the same order $r$ of the underlying functions $f$.[representation] Finally, we clarify the analogy between rational and elliptic Newton flows, and show that the detection of elliptic Newton flows is possible in polynomial time.