Abstract
Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy between the rational (complex) and the elliptic (i.e. doubly periodic meromorphic) functions, a theory on the class of so-called Elliptic Newton flows is developed. With respect to an appropriate topology on the set of all elliptic functions f of fixed order we prove: For almost all functions f, the corresponding Newton flows are structurally stable i.e. topologically invariant under small perturbations of the zeros and poles for f [genericity]. They can be described in terms of nondegeneracy-properties of f similar to the rational case [characterization].
1. Meromorphic Newton flows
In this section, we briefly explain the concept of meromorphic Newton flow. For details and historical notes, see [Citation1–Citation6]. In the sequel, let f stand for a non-constant meromorphic function on the complex plane. So, f(z) is complex analytic for all z in with the possible exception of (countably many) isolated singularities: the poles for f.
The (damped) Newton method for finding zeros of f (with starting point ) is given by
(1)
Dividing both sides of (Equation1(1) ) by the ‘damping factor’
and choosing
smaller and smaller yields an ‘infinitesimal version’ of (Equation1
(1) ), namely
(2)
Conversely, Euler’s method applied to (Equation2(2) ) gives rise to an iteration of the form (Equation1
(1) ). A dynamical system of type (Equation2
(2) ) is denoted by
. For this system, we will interchangeably use the following terminologies: vector field [i.e. the expression on its r.h.s.], or (Newton-)flow [when we focus on its phase portrait(=family of all maximal trajectories as point sets)].
Obviously, zeros and poles for f are removable singularities for and turn into isolated equilibria for
. Special attention should be paid to those points z where
and
. In these (isolated!) so-called critical points , the vector field
is not well defined. We overcome this complication by introducing an additional ‘damping factor’
(
) and considering a system
of the form
(3)
Clearly, may be regarded as another ‘infinitesimal version’ of Newton’s iteration (Equation1
(1) ). Note that, where both
and
are well defined, their phase portraits coincide, including the orientations of the trajectories (cf. Figure ). Moreover,
is a smooth, even real (but not complex) analytic vector field on the whole plane. For our aims, it is enough that
is of class
. We refer to
as to a desingularized Newton flow for f on
.
Integration of (Equation2(2) ) yields:
(4)
where z(t) denotes the maximal trajectory for through a point
. So we have
(5)
It is easily verified that these Newton flows fulfil a duality property which will play an important role in the sequel:(6)
As a consequence of (Equation5(5) ), (Equation6
(6) ) and using properties of (multi-)conformal mappings, we picture the local phase portraits of
and
around their equilibria. See the comments on Figure , where N(f), P(f) and C(f) stand for, respectively, the set of zeros, poles and critical points of f.
Comments on Figure : Figure (a) and (b) In case of a k-fold zero (pole), the Newton flow exhibits an attractor (repellor) and each (principal) value of argf appears precisely k times on equally distributed incoming (outgoing) trajectories. Moreover, as for the (positively measured) angle between two different incoming (outgoing) trajectories, they intersect under a non vanishing angle , where
stands for the difference of the argf values on these trajectories, i.e. these equilibria are star nodes. In the sequel we will use: If two incoming (outgoing) trajectories at a simple zero (pole) admit the same argf value, those trajectories coincide.
Figure (c) and (d): In case of a k-fold critical point (i.e. a k-fold zero for , no zero for f) the Newton flow exhibits a k-fold saddle, the stable (unstable) separatrices being equally distributed around this point. The two unstable (stable) separatrices at a onefold saddle, see Figure (c), constitute the ‘local’ unstable (stable) manifold at this saddle point.
In the sequel, we shall need:
Remark 1.1:
Let be either a simple zero, pole or critical point for f. Then
is a hyperbolicFootnote1 equilibrium for
. (In case of a zero or critical point for f, this follows by inspection of the linearization of the r.h.s. of
: in case of a pole use (Equation6
(6) ).)
Remark 1.2:
[Desingularized meromorphic Newton flows in -setting] If we put
, the desingularized Newton flow
takes the form
(7)
where stands for transpose, and
for the co-factor (adjoint) matrixFootnote2 of the Jacobi matrix
of F. (The r.h.s. of (Equation7
(7) ) vanishes at points corresponding to poles of f)
We end up with the Figures and illustrating the above explanation.
2. Rational Newton flows; the purpose of the paper
Here we present some earlier results on meromorphic Newton flows in the special case of rational functions. Throughout this section, let f be a (non-constant) rational function. By means of the transformation , we may regard f as a function on the extended complex plane
. As usual, we identify the latter set with the sphere
(as a Riemann surface) and the set
of extended functions f with the set of all (non-constant) meromorphic functions on
. The transformation
turns the ‘planar rational Newton flow’
,
, into a smooth vector field on
, denoted
, cf. [Citation7,Citation8]. In the theory on such vector fields the concept of structural stability plays an important role, see e.g. [Citation9] or [Citation10]. Roughly speaking, structural stability of
means ‘topological invariance of its phase portrait under sufficiently small perturbations of the problem data’. Here we briefly summarize the results as obtained by Jongen et al., Shub (cf. [Citation4,Citation5,Citation7,Citation11]):
Theorem 2.1:
[Structural stability for rational Newton flows] Let , then:
(i) | Characterization: The flow
| ||||
(ii) | Genericity: For ‘almost all’ functions f in | ||||
(iii) | Classification: The conjugacy classes of the structurally stable flows | ||||
(iv) | Representation: Up to conjugacy for flows and (topological) equivalency for graphs, there is a 1-1- correspondence between the set of all structurally stable flows |
The purpose of the present paper and its sequel [Citation12] is to find out whether similar results hold for elliptic Newton flows (i.e. meromorphic Newton flows in the case of elliptic functions), be it that, especially in the cases (iii) Classification and (iv) Representation, the proofs are much harder, see [Citation12]. In the present paper, we focus on the first two properties mentioned in Theorem 2.1: characterization and genericity.
Phase portraits of rational Newton flows (even structurally stable) on are presented in Figures and . The simplest example of a spherical rational Newton flow is the so-called North–South flow, given by
, see Figure ; structurally stable if
. Intuitively, it is clear that the phase portraits of
and
are topologically equivalent (i.e. equal up to conjugacy), see Figures and .
One of the first applications of Newton flows was Branin’s method for solving non linear problems, see [Citation1,Citation2,Citation4]. It was Smale, see [Citation6], who stressed the importance for complexity theory of classifying Newton graphs on the sphere that determine the desingularized rational Newton flows. This was done for a class of polynomials in [Citation5] and in general in [Citation11]. Also in the elliptic case, the classification of so-called elliptic Newton graphs on the torus, which determine the desingularized elliptic Newton flows has implications for complexity theory, see [Citation12].
3. Elliptic Newton flows: definition
Throughout this section, let f be a (non-constant) elliptic, i.e. a meromorphic and doubly periodic function of order r with
as a pair of basic periods.Footnote3 We always assume that
The associated period lattice is denoted by
, and
stands for the ‘half open/half closed’ period parallelogram
On
, the function f has r zeros and r poles (counted by multiplicity).
By Liouville’s Theorem, these sets of zeros and poles determine f up to a multiplicative constant , and thus also the class [f] of all elliptic functions of the form
Consider the Riemann surface . It can be obtained from
by identifying opposite sides in the boundary of this parallelogram. The planar, desingularized Newton flow
is doubly periodic on
with respect to the lattice
periods
. Hence, this flow may be interpreted as a
-(even smooth, but nowhere meromorphic) vector field, say
, on
; its trajectories correspond to the lines
constant, cf. (Equation5
(5) ). It will be treated as such. We refer to
as the (desingularized) elliptic Newton flow for f on
.
If g is another function in [f], the planar flows and
have equal phase portraits, as follows by inspection of the expressions of these flows in Section 1; see also Figure . Hence, the flows
and
, both defined on
, have equal phase portraits.
Definition 3.1:
If f is an elliptic function of order r, then:
(1) | The elliptic Newton flow for f, denoted | ||||
(2) | The set of all elliptic Newton flows of order r with respect to a given period lattice |
Regarding flows on with the same phase portraits as equal, compare the ‘desingularization’ step leading from (Equation2
(2) ) to (Equation3
(3) ) and see also Figure , we may interprete the elliptic Newton flow
as a smooth vector field on the compact torus
. Consequently, it is allowed to apply the theory for
-vector fields on compact two-dimensional differential manifolds. For example: Since there are no closed orbits by (Equation4
(4) ), and applying the Poincaré–Bendixson–Schwartz Theorem, cf. [Citation3,Citation4,Citation9], we find:
Lemma 3.1:
The limiting set of any (maximal) trajectory of tends – for increasing t – to either a zero or a critical point for f on
, and – for decreasing t – to either a pole or a critical point for f on
.
We also have:
Remark 3.1:
Hyperbolic equilibria for correspond to such equilibria for
.
4. Elliptic Newton flows: representation
Let f be as in Section 3, i.e. an elliptic function of order r with
,
, an (arbitrary) pair of basic periods generating a period lattice
. The set of all such functions is denoted by
.
Let the zeros and poles for f on be
, resp.
(counted by multiplicity). Then we have: (cf. [Citation13])
(8)
We consider f as a meromorphic function on the quotient space . The zeros and poles for f on
are given by, respectively,
and
, where
stands for the congruency class
of a number in
. Apparently, from (Equation8
(8) ) it follows:
(9)
Moreover, a parallelogram of the type contains one representative of each of these classes: the r zeros/poles for f on this parallelogram.
An elliptic Newton flow corresponds uniquely to the class [f]. So we may identify the set
with the set
.
On its turn, the class [f] is uniquely determined (cf. Section 3) by sets of zeros/poles, say /
, both situated in some
. Thus, the sets
fulfil the conditions (Equation9(9) ). Conversely, we have:
Let two sets of classes
(repetitions permitted!), fulfilling conditions (Equation9
(9) ), be given. Assume that the representatives,
and
, of these classes are situated in a half open/half closed parallelogram spanned by an (arbitrary) pair of basic periods of
. We put
Consider functions of the form(10)
where is an arbitrary constant, and
stands for Weierstrass’ sigma function corresponding to
(cf. [Citation13]). Since
is a holomorphic, quasi-periodic function with only simple zeros at the lattice points of
, a function given by (Equation10
(10) ) is elliptic of order r; the zeros and poles are
resp.
(cf. [Citation13]). Such a function determines precisely one element of
. (Note that if we choose
in any other period parallelogram, we obtain a representative of the same Newton flow, cf. Section 3; moreover, the incidental role of
does not affect generality)
Altogether, we have proved:
Lemma 4.1:
The flows in are given by all ordered pairs
,
of sets of classes
that fulfil (Equation9
(9) ).
Remark 4.1:
Interchanging the roles of and
reflects the duality property, cf. (Equation6
(6) ). In fact, we have
.
On the subset in
of pairs
, that fulfil condition (Equation9
(9) ), we define an equivalence relation
:
The topology
on
Clearly, the set may be identified with a representation space for
and thus for
. This space can be endowed with a topology which is successively induced by the quotient topology on
, the product topology on
, the relative topology on
as a subset of
and the quotient topology w.r.t. the relation
.
Finally, we endow with the weakest topology, say
, making the mapping
continuous.
The topology on
is induced by the Euclidean topology on
, and is natural in the following sense: Given f in
and
sufficiently small, a
-neighbourhood
of f exists such that for any
, the zeros (poles) for g are contained in
-neighbourhoods of the zeros (poles) for f.
Until now, we dealt with elliptic Newton flows with respect to an arbitrary, but fixed, lattice, namely the lattice
for f. Now, we turn over to a different lattice, say
, i.e. pairs of basic periods for
and
are not necessarily related by a unimodular transformation. Firstly, we treat a simple case: For
, we define
. Thus
is an elliptic function, of order r, with basic periods
generating the lattice
.
W.r.t. the transformation , there holds the following lemma:
Lemma 4.2:
The transformation induces a homeomorphism from the torus
onto
, such that the phase portraits of the flows
and
correspond under this homeomorphism, thereby respecting the orientations of the trajectories.
Proof Under the transformation , the flow
given by (3), transforms into:
Since , this
-flow on
has the same phase portraits as
. The assertion follows because the transformation
induces a homeomorphism between the tori
and
.
In other words: from a topological point of view, the Newton flows and
may be considered as equal.
More general, we call two Newton flows and
equivalent (
) if they attain representatives, say
, respectively,
, and there is a homeomorphism
, induced by the linear (over
) basis transformation
, such that their phase portraits correspond under this homeomorphism, thereby respecting the orientations of the trajectories.
From now on, we choose for a pair of so-called reducedFootnote4 periods for f, so that the quotient
satisfies the conditions:
(11) (Such a choice is always possible (cf. [Citation14]). Moreover,
is unique in the following sense: if
is another pair of reduced periods for f, such that
also satisfies the conditions (Equation11
(11) ), then
).
Together with Lemma 4.2 this yields:
More generally, we have:
Lemma 4.3:
Let f be – as before – an elliptic function of order r with as period lattice, and let
be an arbitrary lattice. Then, there exists a function, say
, with
, such that
.
Proof Choose and
,
as basic periods for
resp.
and let H be a linear basis transformation from
to
. The zeros and poles for f are represented by the tuples
resp.
in
that fulfil (Equation8
(8) ). Under H these tuples turn into tuples
,
in
, satisfying (Equation8
(8) ) with
in the role of
. The latter pair of tuples determines a function
in
and thus, by Lemma 4.1, a Newton flow
in
. Now, the chain rule applied to the
-versions (cf. (Equation7
(7) )) of
and
yields the assertion. (Use that, by assumption,
).
Remark 4.2:
Note that if (thus the basic periods for
, and
are related by unimodular transformations), then:
. Also we have:
implies
.
We summarize the homogeneity results, specific for continuous elliptic Newton flows, as obtained in this and the preceding section (choose )
Theorem 4.1:
[The canonical form for elliptic Newton flows] Given an arbitrary elliptic Newton flow, say , on
, there exists a function
in
, of the form (Equation10
(10) ) with
and
in the parallellogram
and
, such that
In particular, it is always possible to choose ,
as in (Equation11
(11) ) and to apply the linear transformation
. So that we even may assume that (1, i) is a pair of reduced periods for the corresponding elliptic function on
.
Hence, in the sequel, we suppress – unless strictly necessary – references to: and write
instead of
.
We end up by presenting two pictures of Newton flows for , where
stands for a Jacobian function. This is a second-order elliptic function, attaining only simple zeros, poles and critical points, characterized by the basic periods
. So does the phase portrait of its Newton flow. Here
are two parameters defined in terms of the Weierstrass function
.
It turns out that for the phase portrait there are – up to topological equivalency – only two possibilities, corresponding to the form (rectangular or not) of the parallelogram with
and
In fact, the crucial distinction between these two cases is whether there occur so-called ‘saddle-connections’ (i.e. (un)stable manifolds connecting saddles) or not (cf. [Citation15]). Hence, it is sufficient to select for each possibility one suitably chosen example. See Figure [non-rectangular, equiharmonic subcase, given by
] and Figure [rectangular subcase given by
]. For a detailed argumentation, see our previous work [Citation15]; compare also the Remarks 2.14 and 2.15 in the forthcoming [Citation12].
Note that in Figures and the points, labelled by and
correspond to the same toroidal zero for
(denoted by
), whereas both 2K and
correspond to the other zero (denoted by
).
Similarly, stands for a pole (denoted by
) on the torus, the pair
for the other pole (denoted by
). The four torodial critical points (denoted by
) are represented by, respectively, the pairs
and the points
and
; see e.g. [Citation16] or [Citation13].
It is well known that the periods are not independent of each other, but related via a parameter
, see e.g. [Citation16]. In the situation of Figure : if
, then
and the phase portraits of
turn into that of
; if
, then
and the phase portraits of
turn into that of
; compare also Figures and .
In part II (cf. [Citation12]) of our serial on elliptic Newton flows, it has been proved – that up to conjugacy – there is only one second-order structurally stable elliptic Newton flow. So that, in a certain sense, the pictures in Figure represent all examples of possible structurally stable Newton flows of order 2. On the other hand, in the case of order , there are several different possibilities as is explained in part III (cf. [Citation17]) of our serial.
5. Structural stability: characterization and genericity
Adopting the notations introduced in the preceding section, let f be a function in the set and
its associated Newton flow (as a smooth vector field on the torus
).
By X(T) we mean the set of all -vector fields on T, endowed with the
-topology (cf. [Citation18]). We consider the map:
The topology on
and the
-topology on X(T) are matched by:
Lemma 5.1:
The map is
continuous.
Proof In accordance with Theorem 4.1 and (Equation10(10) ), we assume
Put and
. Then, the planar version
of the flow
takes the form: (cf. (Equation3
(3) ))
(12)
The expression in the r.h.s. is well defined (since for all z) and depends – as function (F) on
– continuously differentiable on x(=Re z) and y(=Im z). So does the Jacobi matrix (DF) of F. Analogously, a function
chosen
-close to f , gives rise to a system
and a function G with Jacobi matrix DG. Taking into account the very definition of
-topology on X(T) , the mapping
is continuous as a consequence of the following observation: If -w.r.t. the topology
- the function g approaches f , i.e. the zeros and poles for g approach those for f, then G and DG approach F, respectively, DF on every compact subset of
.
Next, we make the concept of structural stability for elliptic Newton flows precise.
Definition 5.1:
Let f, g be two functions in . Then, the associated Newton flows are called conjugate, denoted
, if there is a homeomorphism from T onto itself, mapping maximal trajectories of
onto those of
, thereby respecting the orientation of these trajectories.
Note that the above definition is compatible with the concept of ‘equivalent representations of elliptic Newton flows’ as introduced in Section 4; compare also (the comment on) Definition 3.1.
Definition 5.2:
The flow in
is called
-structurally stable if there is a
-neighbourhood
of f, such that for all
we have:
.
The set of all structurally stable Newton flows is denoted
.
From Lemma 5.1 it follows:
Corollary 5.1:
If , as an element of
, is
-structurally stable [Citation9], then this flow is also
-structurally stable.
So, when discussing structural stability in the case of elliptic Newton flows, we may skip the adjectives and
.
Definition 5.3:
The function f in is called non-degenerate if:
All zeros, poles and critical points for f are simple;
No two critical points for f are connected by an
-trajectory.
Note: If f is non-degenerate, then also, and these functions share their critical points; moreover, the derivative
is elliptic of order 2r (=number, counted by multiplicity, of the poles for f on T). Since all zeros for f are simple, the 2r zeros for
(on T) are the critical points (all simple) for f, and we find that f, as a function on T, has precisely 2r different critical points. Compare also the forthcoming Lemma 5.2 (Case
).
The main result of this section is:
Theorem 5.1:
[Characterization and genericity of structural stability]
(1) |
| ||||
(2) | The set |
Proof Will be postponed until the end of this section.
We choose another lattice, say . The functions f and g in
determine, respectively, functions
and
in
, compare Lemma 4.3. It is easily verified:
if and only if
is structurally stable if and only if
is structurally stable.
f is non-degenerate if and only if
is non-degenerate.
Steady streams
We consider a stream on (cf. [Citation19]) with complex potential
(13)
The stream lines are given by the lines constant, and the velocity field of this stream by
. Zeros and poles for f of order n resp. m , are just the sinks and sources of strength n, resp. m. Moreover, it is easily verified that the so-called stagnation points of the steady stream (i.e. the zeros for
) are the critical points of the planar Newton flow
. Altogether, we may conclude that the velocity field of the steady stream given by w(z) and the (desingularized) planar Newton flow
exhibit equal phase portraits.
From now on, we assume that f has – on a period parallelogram P – the points and
as zeros, resp. poles, with multiplicities
, resp.
We even may assumeFootnote5 that all these zeros and poles are situated inside P (not on its boundary), cf. Figure .
Since f is elliptic of order r, we have:(14)
Note that there is an explicit formula for . In fact, we have:
where and
are reduced periods for
, and
stands for winding numbers of the curves
and
(compare Figure and [Citation13]).
The derivative is an elliptic function of order
, and
By (Equation10(10) ) we have:
(15)
So, by (Equation13(13) )
where stands for the Weierstrass zeta function (cf. [Citation13]) associated with the lattice
. Since
is a quasi-periodic, meromorphic function with only poles (all simple!) in the points of the lattice
, the function
is elliptic of order
with K (simple) poles given by:
, situated in its period parallelogram P. It follows that
has also K zeros (counted by multiplicity) on P. These zeros correspond with the critical points for
on the torus T.
Since , where
stands for the (elliptic!) Weierstrass
-function (cf. [Citation13]) associated with
, we find (use also
):
(16)
From (Equation14(14) ) and (Equation15
(15) ), it follows:
is determined by
. In mutually disjoint and suitably smallFootnote6 neighbourhoods, say
and
, of respectively
we choose arbitrary points
and put, with fixed values
:
(17)
In this way are close to, respectively,
and
is close to
. Finally, we put
(18)
Perturbating into respectively
, and putting
, we consider functionsFootnote7
on the product space
given by: (compare (Equation15
(15) ))
(19)
Then, for each in
, the function
is elliptic in z, of order r. The points
, resp.
are the zeros and poles for
on P ( of multiplicity
resp.
. Moreover,
has
critical points on P (counted by multiplicity). Note that the Newton flow
on T is represented by the pair
in
, i.e. by
but also by the pair (a, b) in the quotient space as introduced in Section 4, i.e. by
In particular,
If
, then
and thus
;
If
(thus
), i.e. all zeros, poles are simple, then
Lemma 5.2:
Let . Then:
Under suitably chosen – but arbitrarily small– perturbations of the zeros and poles for f, thereby preserving the multiplicities of these zeros and poles, the Newton flow turns into a flow
with only (K different)onefold saddles.
Proof We consider and write:
These are meromorphic functions in each of the variables .
Define:
Since the and
, are poles for
as an elliptic function in z, we have: If
, then:
The subset of elements
that fulfils these inequalities is open in
. On this set
(that contains the critical set
), the function
is analytic in each of its variables. (Thus
is a closed subset of
). For the partial derivatives of
on
we find: (use (Equation13
(13) ), (Equation19
(19) ), compare also (Equation16
(16) ) and Footnote 7)
By the Addition Theorem of the -function [cf. [Citation13]], we have:
So, let in
, then
From this follows:
If , then at most one of
vanishes. By a similar reasoning, we even may conclude:
(20)
The latter conclusion cannot be drawn in case ; however, see the forthcoming Remark 5.1. Note that always
.
Under the assumption that let
be the different critical points for
with multiplicities
. If
tends to
, then
of the K critical points for
(counted by multiplicity) tend to the
-fold saddle
for
. It follows that, if
is sufficiently close to
, then
critical points for
(counted by multiplicity) are situated in, suitably small, disjunct neighbourhoods, say
, around
We choose
so close to
that this condition holds. If all the critical points for f, i.e. the saddles of
are simple, there is nothing to prove. So, let
, thus
, i.e.
. Without loss of generality, we assume (see (Equation20
(20) )) that
. According to the Implicit Function Theorem a local parametrization of
around
exists, given by:
where Thus, at
we have:
Since and
, it follows that
Note that , depends complex differentiable on z. So the zeros for
are isolated. Thus, on a reduced neighbourhood of
, say
, neither
nor
vanish. If z tends to
, then:
tends to along
, and we cross
, meeting elements
, such that
Thus,
Hence, we have: . So, the
critical points for
that approach
via the ‘curve’
are all simple, whereas the critical points for approaching
are still situated in, respectively
. If
, we repeat the above procedure with respect to
, etc. In a finite number of steps, we arrive at a flow
with only simple saddles and
arbitrary close to
.
Remark 5.1:
The case .
If , then the function f has – on T – only one zero and one pole, both of order r; the corresponding flow
is referred to as to a nuclear Newton flow. In this case, the assertion of Lemma 5.2 is also true. In fact, even a stronger result holds:
Nuclear Newton flows will play an important role in the creation of elliptic Newton flows, but we postpone the discussion on this subject to a sequel of the present paper, see [Citation20].
We end up by presenting the (already announced) proof of Theorem 5.1
Proof of Theorem 5.1:
The ‘density part’ of Assertion 2: Let be an arbitrarily small
-neighbourhood of a function f as in Lemma 5.2. We split up each of the A different zeros and B different poles for f into
resp.
mutually different points, contained in disjoint neighbourhoods
resp.
(compare Figure ) and take into account relation (Equation8
(8) ). In this way, we obtain 2r different points, giving rise to an elliptic function of the form (Equation10
(10) ), with these points as the r simple zeros /r simple poles in P. We may assume that this function is still situated in
, see the introduction of the topology
in Section 4. Now, we apply Lemma 5.2 (case
) and find in
an elliptic function, of order r with only simple zeros, poles and critical points. This function is non-degenerate if the corresponding Newton flow does not exhibit trajectories that connect two of its critical points. If this is the case, none of the straight lines connecting two critical values for our function, passes through the point
. If not, then adding an arbitrarily small constant
to f does not affect the position of its critical points, and yields a function – still inFootnote8
- with only simple zeros and poles. By choosing c suitably, we find a function, renamed f, such that none of the straight lines connecting critical values (for different critical points) contains
. So, we have:
, i.e.
is dense in
.
The ‘if part’ of Assertion 1: Let . Then all equilibria for the flow
are hyperbolic (cf. Remarks 1.1 and 3.1). Moreover, there are neither saddle-connections nor closed orbits (compare (Equation4
(4) )) and the limiting sets of the trajectories are isolated equilibria (cf. Lemma 3.1). Now, the Baggis–Peixoto Theorem for structurally stable
-vector fields on compact two-dimensional manifolds (cf. [Citation9,Citation21]) yields that
is
-structurally stable, and is by Corollary 5.1 also
-structural stable.
The ‘only if part’ of Assertion 1: Suppose that , but
in
. Then there is a
-neighbourhood of f, say
, such that for all
:
. Since
is dense in
(already proved), we may assume that
. So,
has precisely r hyperbolic attractors/repellors and does not admit ‘saddle connections’. This must also be true for
, in contradiction with
.
The ‘openess part’ of Assertion 2: This a direct consequence of the Assertion 1 (which has already been verified above).
Acknowledgements
The authors like to thank the referees for valuable comments and M.V. Borzova and S.V. Polenkova for making all the Figures in this paper and its sequels [Citation12,Citation17] and [Citation20].
Notes
No potential conflict of interest was reported by the authors.
1 An equilibrium for a - vector field on
is called hyperbolic if the Jacobi matrix at this equilibrium has only eigenvalues with non vanishing real parts (cf. [Citation10]).
2 i.e. , where
stands for the
-unit matrix.
3 I.e. each period is of the form In particular,
(cf. [Citation13,Citation14]).
4 The pair of basic periods for f is called reduced or primitive if
is minimal among all periods for f , whereas
is minimal among all periods
for f with the property
(cf. [Citation14]).
5 If this is not the case, an (arbitrary small) shift of P along its diagonal, is always possible such that the resulting parallelogram satisfies our assumption (cf. Figure and [Citation13]).
6 Choose these neighbourhoods so that they are contained in the period parallelogram P, cf. Figure .
7 Note that, when perturbing () in the indicated way, the winding numbers
,
, and thus also
, remain unchanged.
8 Note that at simple zeros an analytic function is conformal. In case of a pole, use also (Equation6(6) ).
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