Newton flows for elliptic functions II

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 G( f ) on a torus T with r vertices, 2r edges and r faces that fulfil certain combinatorial properties (Euler, Hall) on some of its subgraphs. The graph G( f ) determines the conjugacy class of the flow [classification]. A connected, cellularly embedded toroidal graph G with the above Euler and Hall properties, is called a Newton graph. Any Newton graph G can be realized as the graph 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. The proofs of the above results rely on Peixoto's characterization/classification theorems for structurally stable dynamical systems on compact 2-dimensional manifolds, Stiemke's theorem of the alternatives, Hall's the-B Gerard F. Helminck

Newton flows for elliptic functions III & IV

“…Argueing basically as in the proof of Theorem 4.1 from our paper [3] , it can be shown that X(G)) is equivalent with an elliptic Newton flow generated by a function on three Altogether, we find:…”

“…Following [10], G ∧ G * determines a C 1 -structurally stable flow X(G) on T . In [3] we proved that, if the A-property holds as well, X(G) is topologically equivalent with a structurally stable elliptic Newton flow of order r.…”

“…2 Occasionally, we will refer to N (f ) as to a toroidal (elliptic) Newton flow. 3 In fact λ 0 = −η(f (γ 2 ))ω 1 + η(f (γ 1 ))ω 2 , where η(f (·)) stands for winding numbers of the curves f (γ 1 ) and f (γ 2 ) with γ i the sides of Pω 1 , ω 2 spanned by ω i , i = 1, 2, (cf. [10]).…”

“…The main results obtained in [3] are: -If N (f ) and N (g) are structurally stable and of the same order, then:…”

“…This graph admits a cellular embedding in T (cf. [3]), which will be referred to as to the "distinguished" graph G ∧ G * with as faces the so-called canonical regions (compare Fig.3). Following [10], G ∧ G * determines a C 1 -structurally stable flow X(G) on T .…”