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Abstract
Elliptic Newton flows are generated by a continuous, desingularized Newton method for doubly periodic meromorphic functions on the complex plane. In the special case, where the functions underlying these elliptic Newton flows are of second-order, we introduce various, closely related, concepts of structural stability. In particular, within the class of all (second order) elliptic Newton flows, structural stability turns out to be a generic property. Moreover, the phase portraits of all such structurally stable flows are equal, up to conjugacy. As an illustration, we treat the elliptic Newton flows for the Jacobi functions sn, cn and dn.
†Dedicated to H.Th. Jongen on the occasion of his 60th birthday.
Acknowledgement
F. Twilt likes to express his gratitude to Bert Jongen–friend and teacher–for drawing his attention, a long time ago, to the beauty of Newton flows.
Notes
†Dedicated to H.Th. Jongen on the occasion of his 60th birthday.
Notes
1. That is, , where I
2 stands for the 2 × 2-unit matrix.
2. An equilibrium for a C 1-vector field on ℝ2 is called hyperbolic, if the Jacobi matrix at this equilibrium has only eigenvalues with non-vanishing real parts, (cf. Citation12).
3. Two C 1-vector fields on ℝ2 are called conjugate (or topological equivalent) if a homeomorphism from ℝ2 to itself exists, which maps the maximal trajectories of one of these vector fields onto those of the other, preserving their orientations.
4. In fact, ζ and σ are quasi-periodic in the sense that ζ(z + ω j ) − ζ(z) = η j and σ (z + ω j ) = c j σ(z), for all z, with ηj and c j appropriate constants, j=1, 2.
5. If Order(f) = r and f has only simple zeros/poles, then f* admits 2r critical points (counted by multiplicity).
6. Note that if [f] = [g], then the Newton flows and
are conjugate because at each point z the vectors
and
are equal up to strictly positive multiplicative factors.
7. Note that the quotient topology on ℂ/Λ is compatible with the holomorphic structure on T.
8. Compare also Section 1. For a similar result in case of rational Newton flows, see Theorem 9.4.12 in Citation13.
9. The canonical regions for are the connected components of what is left after deleting from T the topological closures of the stable and unstable manifolds at the saddles for
.
10. is a realization in T of an abstract, oriented, connected one-dimensional cell complex, cf. Citation22.
11. Compare Definition 4.2 in Citation22.
12. Here we also use that the (holomorphic) functions σω1, ω2 depend continuously on ω1, ω2.
13. One might argue that the appearance of the square root gives rise to some ambiguity. However, it is possible (cf. Citation18) to express (℘(z/g) − e
j
), j = 1, 2, 3, and also (e
1 − e
3), (e
2 − e
3)/(e
1 − e
3), (e
1 − e
2)/(e
1 − e
3) as squares θ2 of quotients of theta functions. Then, the possible ambiguity can be avoided by defining .
14. W.r.t. the side through , use that if z is on this side, then
is on the side through s**;
, in particular
.
15. In fact, by a careful analysis of the equi-anharmonic case, it can be proved that if the angle between the vectors K and iK′ is less (greater) than 1/2π, the saddle in K is connected to the source in iK′ (in 2K + iK′), and hence the unstable manifold at K + iK′ connects 2K with 2iK′, (0 with 2K + 2iK′). Here we will not dwell on this proof, since we are mainly interested in Newton flows up to conjugacy.
16. Here we also use that in the real rectangular case, ℘(z) is also real valued on the lines midway between the edges of its period parallelogram.