table of contentspi.math.cornell.edu/~hubbard/newtonseveral.pdf · 4.7. the action of mappings fˆ...

Table of Contents

Chapter 0 Introduction 31. Introduction 32. Outline of paper 43. Acknowledgements 54. A computer tour of Newton’s method 65. Some open questions 10

Chapter 1 Fundamental properties of Newton maps 121.1. Generalities about Newton’s method 121.2. The intersection of graphs 141.3. The Russakovskii-Shiffman measure 191.4. Invariant currents 221.5. The intersection of conics 241.6. Degenerate cases 291.7. The one-variable rational functions associated to the roots 32

Chapter 2 Invariant 3-manifolds associated to invariant circles 352.1. The circles in the invariant lines 352.2. Periodic cycles on invariant circles 372.3. Unstable manifolds at infinity 422.4. The invariant manifolds of circles 452.5. The extension of Φ and the origin of “bubbles” 52

Chapter 3 The behavior at infinity when a = b = 0. 593.1. The primitive space 593.2. Newton’s method and the primitive space 60

Chapter 4 The Farey blow-up 664.1. Definition of the Farey blow-up 664.2. Naturality of the Farey blow-up 694.3. The real oriented blow-up of the Farey blow-up 704.4. Naturality and real oriented blow-ups 744.5. Inner products on spaces of homogeneous functions 754.6. Homology of the Farey blow-up 794.7. The action of mappings f(

21

) on homology 83Chapter 5 The compactification when a = b = 0 88

5.1. The tower of blow-ups when a = b = 0 885.2. Sequence spaces 925.3. The real oriented blow-up of X∞ 945.4. The homology of X1 975.7. The action of NP on homology 1005.6. The (co)homology H2(X∗∞) 1105.7. The action of N on homology 113

v

vi Contents

Chapter 6 The case where a and b are arbitrary 1186.1. A curve of order two 1196.2. The primitive space for arbitrary a and b 1206.3. Building the space X∞ 1216.4. The basins of the roots 1226.5. Real oriented blow-ups and homology 123

Bibliography 123

Abstract1

In this article we study the Newton mapN : C2 → C2 associated to two equationsin two unknowns, as a dynamical system. We focus on the first non-trivial case:two simultaneous quadratics, to intersect two conics. In the first two chapters weprove among other things

– The Russakovski-Shiffman measure does not charge the points of indetermi-nacy;

– The lines joining pairs of roots are invariant, and the Julia set of the restrictionof N to such a line has under appropriate circumstances an invariant manifold,which shares features of a stable manifold and a center manifold.

The main part of the article concerns the behavior of N at infinity. To compactifyC2 in such a way thatN extends to the compactification, we must take the projectivelimit of an infinite sequence of blow-ups. The simultaneous presence of points ofindeterminacy and of critical curves forces us to define a new kind of blow-up: theFarey blow-up.

This construction is studies in its own right in Chapter 4, where we show amongothers that the real oriented blow-up of the Farey blow-up has a topological struc-ture reminiscent of the invariant tori of the KAM theorem. We also show that thecohomology, completed under the intersection inner product, is naturally isomorphicto the classical Sobolev space of functions with square-integrable derivatives.

In Chapter 5, we apply these results to the mapping N in a particular case, whichwe generalize in Chapter 6 to the intersection of any two conics.

1Received by the editors July 22, 2002, and in revised form on January 25, 2005.Mathematics Subject Classification numbers: Primary 32H50,37H10; secondary 37F45,37E99, 65P99.Keywords and phrases Newton’s method, iteration, invariant manifold, point of indeter-minacy, infinitely many blow-ups, Farey blow-up, Sobolev space.

vii

0

Introduction

In this paper, we will study Newton’s method for solving two simultaneous qua-dratic equations in two variables.

In one dimension, if F is a polynomial, the Newton mapping is a rational func-tion and we can apply the now rather well developed theory of one-dimensionalcomplex analytic dynamics. The subject is far from completely understood, butmuch progress has been made, particularly by J. Head, Tan Lei, and M. Shishikura.More recently, [HSS] give precise results on how to find all the roots of a polynomial,based on the topology and complex analysis of the basins.

Presumably, there is no need to motivate a study of Newton’s method, in oneor several variables. The algorithm is of immense importance, and understandingits behavior is of obvious interest. It is perhaps harder to motivate the case oftwo simultaneous quadratic equations in two variables, but this is the simplest non-degenerate case, and already displays highly nontrivial dynamics.

The classical theory, due to Ostrowski and Kantorovitch, asserts that a non-degenerate zero of a C2 function f : Rn → Rn is a superattractive fixed point ofthe associated Newton map

Nf (x) = x− [Df(x)]−1f(x).For a simple presentation of this theory, see [HBH], Section 2.7; we will follow thenotation used there. Shub and Smale [ShSm] have also made an extensive study ofNewton’s method from the point of view of complexity theory.

When f : C2 → C2 is analytic, a much more detailed description of the dynamicsnear a fixed point is given in [HP], which gives a partial description of the basins ofthe roots. In that paper, we show that associated to each root x0 there is a rationalfunction gx0 : P1 → P1 constructed from the quadratic terms of the Newton mapat the root. This rational function (its Julia set, critical orbits, . . . ) is the mostimportant element in the description of the basin, and in the present paper it willalso play an important role.

What makes Nf such an intractable dynamical system is the simultaneous pres-ence of critical curves and points of indeterminacy. There is an extensive theoryfor the dynamics of birational maps P2 ∼∼> P2 [BS1-8],[FS1-3], [HO1-3],[Dil], [Br],though in no sense do they give the sort of information available for functions ofone complex variable. Of particular relevance to this paper is [HPV], where thetechnique of real oriented blow-ups is introduced. These play an important rolehere too; in [HPV], we showed something of what complications can lurk in pointsof indeterminacy (solenoids, etc.), but here the same sort of construction yields farmore elaborate structures.

There is also a reasonable theory of endomorphisms of Pn [BrD], [FS], [HP],[Jo], [Ueda], though again the descriptions provided are not nearly so precise as onemight like, and there is a drastic shortage of examples that are actually understood.

But next to nothing is known about mappings that have topological degree > 1(and in particular, critical curves), and points of indeterminacy.2 One important

2Added in proof: since this paper appeared as a preprint, V. Guedj [Gu] has written apaper that clears up many mysteries about the Russakovskii-Shiffman measure.

1

2 Hubbard and Papadopol

result is due to Russakovskii and Shiffman [RS]. There they prove that if F : Pn ∼∼> Pn is a rational mapping, and various inequalities on the algebraic degree andtopological degree are satisfied, then if you take a “generic” point, take all its inverseimages by fn, and give them all equal weight so as to form a probability measureµn, then these measures converge as n → ∞ to a measure µ independent of theoriginal point. Here generic means “chosen off a pluripolar set.”

Newton’s method for solving the equations F (x, y) = (y − f(x), x − g(y)) = 0with f a polynomial of degree p and g a polynomial of degree q does satisfy theirconditions; in fact, Russakovskii and Shiffman consider specifically the Newton mapassociated to

F (x, y) = (y − x2, x− y2).

This map, has two points of indeterminacy at infinity. Russakovskii and Shiffmancould not determine whether their measure charges the points of indeterminacy atinfinity; in this paper we prove (in considerably greater generality) that it does not(Theorem 1.3.1), following a proof of A. Douady.

2. Outline of the paper

This paper is organized into six chapters.

Chapter 0 contains, in addition to the usual outline and acknowledgements, acomputer tour of Newton’s method which presents some of the pictures which in-spired this paper.

Chapter 1 presents the results which depend neither on stable manifolds nor oninfinite chains of blow-ups.

Section 1 describes how the Newton map behaves under changes of variables;Section 2 illustrates the many simplifications which arise when the Newton

method is applied to finding intersections of graphs, i.e., applied to the equationsy = f(x), x = g(y);

Section 3 (largely due to Douady) analyzes the Russakovskii-Shiffman measurefor intersections of graphs;

Section 4 discusses some invariant currents on P2;Section 5 specializes the results of Section 2 to the case of quadratics;Section 6 gives a first look at some degenerate cases;Section 7 analyzes the rational functions of degree 2 associated to the roots.

Chapter 2 is concerned with stable and unstable manifolds of invariant subsets.One important result is Theorem 2.4.1. Each line joining a pair of roots is invariant,and the restriction of Newton’s method to such a line has a Julia set, which is thereal straight line bissecting the two roots in the complex invariant line. The questionwhether this Julia set has a stable manifold is quite subtle. It never has a genuinestable manifold, but sometimes there is a 3-dimensional manifold associated to itwhich has some features of a stable manifold and some features of a center manifold;this is the content of Theorem 2.4.1. Under more general conditions, it is an ergodicattactor [PS], and has a measure-theoretic stable manifold, the topology of whichis still pretty mysterious.

Newton in C2 3

In Chapter 3, we examine a case where the algebra simplifies: the intersectionof the parabolas y = x2, x = y2. For this map, we will analyze the behavior atinfinity, using infinitely many blow-ups.

In Chapter 4, we analyze in the abstract the process of infinitely many blow-ups used in Chapter 3. This construction, which we call the Farey blow-up, isfundamental if one wants to understand the behavior of mappings with criticalcurves and points of indeterminacy. This blow-up is intermediate between thefinite blow-ups of algebraic geometry and Noether’s “monster,” the projective limitof all finite blow-ups of the projective plane. All rational maps are well definedon the monster, and it is the space on which the Cremona group acts naturally.The monster has intrigued many mathematicians [Ma], [De], without ever reallybecoming an accepted citizen of algebraic geometry, because the topology of themonster is so wild.

In contrast, we will show that the topology of Farey blow-ups is fairly tame, andquite easy to describe in the language of real oriented blow-ups, as pioneered in[HPV]. In a rather surprising coincidence, the topology of these real oriented blow-ups is reminiscent of the topology of non-integrable Hamiltonian systems with twodegrees of freedom, with zones corresponding to rational “frequencies” separatedby tori corresponding to irrational frequencies.

We also compute the second homology groups of Farey blow-ups. This is a spacewith a natural quadratic form coming from the intersection product. This quadraticform is negative definite, and the completion of the homology with respect to thequadratic form turns out to be a very classical Sobolev space of functions with onederivative in L2. We also compute the operation of appropriate mappings on thehomology.

In Chapter 5, we perform infinitely many Farey blow-ups to construct a compactspace X∞ on which the Newton map operates, and analyze in detail the structureat infinity.

The real oriented blow-up of X∞ has a 3-dimensional boundary, and now theresemblance with the structures appearing in the KAM theorem [Ar] is uncanny.“First-order tori” corresponding to irrational numbers, and carrying irrational fo-liations, separate zones corresponding to rational numbers. Within these zones,further “second-order tori” again with irrational foliations, separate zones corre-sponding to rationals, etc., the whole thing accumulating on some Cantor set ofsolenoids.

Another way of thinking of this structure is in terms of Puiseux expansions, withthe first-order tori corresponding to certain one-term Puiseux expansions of tran-scendental curves, separating zones corresponding to Puiseux exponents of algebraiccurves. Within these zones, there are further structures corresponding to two-termPuiseux expansions, some transcendental corresponding to tori, and some algebraic,which can then be further enriched, etc. We do this in the case of the same specialNewton map as in Chapter 3, to simplify the computations.

In Chapter 6, we show that essentially all the results of Chapters 3 and 5 general-ize to any Newton map associated to a pair of quadratic equations in two variables(except some degenerate cases). One bonus is Theorem 6.4.2 that asserts thatbasins of Newton’s method are Stein. Before the blowups defined in Chapters 3-6,the Green’s function of each basin is not proper; it only becomes proper in the


infinite blow-up. We can then see that the basins before blow-ups are Stein byremoving the exceptional divisors.

3. Acknowledgments

We have many people to thank for their help in writing this paper.– Adrien Douady especially: in teaching the material of the course at Orsay, he

found an important error in the proof that the Russakovskii-Shiffman measure doesnot charge the line at infinity. He also found how to correct the proof: Section 1.3is essentially due to him.

– Jeff Diller, who helped with the computation of Section 2.1.– Robert Strichartz suggested the connection of the homology with the Sobolev

space H1.– Eric Bedford made many helpful comments about an early version of the paper.– Karl Papadantonakis contributed the computer program which made the color

pictures.– Ramin Farzaneh wrote an earlier program, which raised many of the problems

we address in this paper.– Sarah Koch and Roland Roeder read Chapters 0,1 and 2 extremely carefully,

finding innumerably many typos. In particular, the proofs of Theorems 2.2.2 and2.3.1 are theirs, and much more efficient than the earlier ones I had found. Theyalso made many suggestions to improve the presentation.

– The referee made many corrections and valuable suggestions.– A big thanks to Barbara Burke Hubbard, for innumerable many things of

course, but in particular for her help with editing, and for finding several mistakes.Peter Papadopol thanks the Department of Mathematics of Cornell University,

and the chairmen Keith Dennis, Peter Kahn, Robert Connelly and especially JohnSmillie, for many years of hospitality. From Grand Canyon University, he thanksformer President Bill Williams for his support for many years, President Gil Staffordand Chairperson Beth Dawkins for support and understanding.

4. A computer tour of Newton’s method

Before embarking on the rigorous part of this paper, we want to give an overview ofwhat computer exploration shows, emphasizing those aspects that we will be ableto explain.

The example we will examine is the Newton’s method for finding the intersectionsof two conics when the intersections are at

(00

),(

10

),(

01

), and

(23

); as we will

see in Corollary 1.5.2, this specifies a unique Newton mapping. As far as we know,there isn’t anything very special about these values of the parameters, except thatthey are real, and we can draw pictures of the basins in R2, shown in Figure 1.

Newton in C2 5

Figure 1. At left, we see the basins of the roots, marked with black dots: the basin of(00

)in red, the basin of

(10

)in green, the basin of

(01

)in blue, the basin of

(23

)in

grey. The region displayed is −4 ≤ x, y ≤ 4. Notice that the x-axis is entirely red andgreen, and the y-axis is entirely red and blue, in fact all the lines joining pairs of roots

contain only two colors. At right is a blow-up of the figure at left; notice the blue fingers

coming from above and below to touch the x-axis.

The main thing we see in these pictures is the importance of the polar locus, i.e.,

the preimage of the line at infinity. In Figure 1, it is the hyperbola, one branch ofwhich bounds the basin of

(23

), drawn in grey in the picture. The other branch is

a bit harder to see; one part bounds the basin of(

10

). The inverse image of this

polar hyperbola is a curve of degree 6, which has three branches in this picture.

It is a bit difficult to see, since two of its branches are almost superposed to the

hyperbola.

Figure 2 represents Newton’s method applied to two quadratic equations, inthe case when the roots are at

(00

),(

10

),(

01

), and

(−1−1)

. The polar curve isthe ellipse one sees emphasized in the middle. More generally, if two real conics

intersect in four real points that form a convex quadrilateral, then the polar locus

is a hyperbola, but if one intersection is in the convex hull of the others, the polar

locus is an ellipse. This time the real part of its inverse image is the 3-leafed clover, a

bit more lightly emphasized. Actually, this is only one branch of the inverse image;

there is another branch in C2.


Figure 2. Left: the basins of the roots

(00

)(red basin),

(10

)(green),

(01

)(blue),

and

(−1−1

)(grey). The polar curve is an ellipse in this case, and it together with its first

and second inverse images are drawn on the right. They appear to form the boundary of

the basins of the roots, or at least it seems that the closure of the union of the iterated

inverse images of the polar curve does.

It is not really surprising that the inverse images of the line at infinity appear

to make up the boundaries of the basins of the roots. After all, the line at infinity

itself is never in the basin of any root, so neither are its inverse images.

Since our objective is to understand the dynamics of Newton’s method in C2,

we need to find a way of making complex pictures. We will also need to find the

complex analog of the line at infinity, and its successive inverse images. Of course,

the line at infinity is still there, and so are its inverse images, but they are now of

real codimension 2, and cannot form the boundary of anything.

The most obvious picture to make is a 1-dimensional complex slice through C2,

in which we draw the basins. If we use the same Newton map with roots at

(00

),(

10

),(

01

), and

(−1−1),

and slice by the line y = (1 + i)x, then the region 0 ≤ Rex ≤ 2, | Imx| < 1 isrepresented on the left of Figure 3.

Newton in C2 7

Figure 3. The same Newton’s map shown in Figure 2, sliced by the complex line y =(1 + i)x. The different shades represent the basins of the four roots. On the right, a

blow-up of the figure on the left.

Figure 4. A sequence of “bubbles” of one domain inside another basin, converging to aboundary point of a third basin. The left hand figure is the “top bubble” of the right-hand

picture in Figure 3, and the right-hand picture is a further blow-up of that. The reader

might notice some similarity between the blue “fingers” of Figure 1 and the blue bubbles

in this figure; we will explore this similarity in Section 2.5.

What do we see? The basins of the roots are bounded by what seem to besmooth curves. Inside some of the basins there seem to be archipelagoes of islandsof a different basin, which appear to converge to a point of a boundary with a thirdbasin, where however the boundary appears perfectly smooth (see Figure 4).


In other places, the picture appears to be self similar, but the curves formingthe boundaries undulate, and these undulations do not appear to become smalleras they get closer to the center of self-similarity. These curves form a definite angleat the point of self-similarity.

Figure 5. A blow-up near the center of the left-hand side of Figure 3, and a furtherblow-up of it.

Figure 6. At right, a blow-up of the figure on the left.

The pictures seen so far are misleadingly simple. As we will see in Chapter 2,things tend to be a bit simpler when the coefficients of a Newton map are real.Figure 6 represents the line of equation y = sx. We see the basins of the rootsfor Newton’s method, where the roots are at

(00

),(

10

),(

01

)and

(αβ

), where

α = 2+2i and β = −.3. Again, we understand roughly why the drawing presents thesort of complication present, without understanding in detail just why the structureis what one sees.

Newton in C2 9

Remark. The figures shown here are chosen essentially at random; practically anyother parameters cut by any line are equally as interesting and complicated. Amuch larger gallery of pictures can be seen at web site

http://www.math.cornell.edu/˜dynamics/The site also includes Macintosh programs that the reader can download. 4

5. Some open questions

Some of the phenomena above will be explained in this paper: the apparentsmoothness of basin-boundaries, the self-similarity, the undulation of some bound-ary curves. There are many things about these pictures which we don’t understand.

– The combinatorial properties of the basins. To each root is associated a rationalfunction (see Section 1.7), and a sort of “cone” over its Julia set appears in the basin,as sketched in Section 1.4, and described in detail in [HP]. These cones must allmeet on the boundaries of the basins, since they are totally invariant, so any pointin the support of the Russakovski-Shiffman measure must be in the closure of allthe cones. Does this imply some deep relation between the four rational functionsthat arise for each quadratic Newton’s method? Any one of them determines theother three, so an algebraic relation certainly exists, but we do not see that thismeans that their dynamics “fit”.

– Is the union of the basins dense? This hides two quite different questions:– Are there any attracting cycles other than the basins?– Are there wandering domains?We conjecture that the answer to both these questions is no. But the questions

are not of equal importance: for the Newton map as applied to polynomials ofdegree greater than 2, there certainly are sometimes other attracting cycles. Butwe conjecture that there are never any wandering domains for Newton’s methodas applied to pairs of polynomials in two variables. This stands out as a centralproblem of dynamics in several variables: the tools used by Sullivan in the proof inone dimension are simply not available.

Two more questions were asked by the referee that I can’t answer, at least notfully.

– One is whether the roots are the only exceptional points for the Shiffman-Russakovskii measure. This measure is defined to be

µx = limn→∞

∑y∈f−n(x)

degy(NF )4n

δy

and it is proved in [RS], [Gu] that for all x in the complement of a pluripolar set Ethat the limit exists and is a measure µ independent of x. The set E is called theexceptional set, and in the case of a quadratic Newton map, each root is completelyinvariant, and so exceptional. The referee asks if there are any others.

Well, yes. It is fairly easy to see that the points at infinity are exceptional, andin two different ways. There are two points of indeterminacy p1 and p2 at infinity(see Lemma 1.5.6), and if x 6= p1,p2, then

µx =13µ+

13δp1 +

13δp2 .


On the other hand,

µp = µq =12δp1 +

12δp2 .

The really interesting question is: are there any exceptional points besides theroots in the finite plane? It is tempting to think not in general, but in exceptionalcases there definitely are others. In the degenerate case where the 4 roots lie ona pair of parallel lines (see Section 1.6), these lines are completely invariant, anddisjoint from the support of µ. Thus all their points are exceptional. There may beother cases where there are extra exceptional points; we don’t know.

– Another question is whether the points of indeterminacy (see Proposition 1.5.4)are in the support of µ. It is very tempting to think so in general, but we are notconfident. In the same degenerate case as above, it is definitely false: the two pointsof indeterminacy in the finite plane that remain in that case are not in the supportof µ unless Re (1/B) = 1/2, and perhaps not even then.

1

Fundamental properties of Newton maps

Given two vector spaces V and W of the same dimension, and a mapping F :V →W , the associated Newton map NF : V → V is given by the formula

NF (x) = x− [DF (x)]−1(F (x)). (1.1)A well-known theory asserts that if x0 ∈ V satisfies F (x0) = 0 and if the derivative[DF (x0)] is invertible, then x0 is an attracting fixed point of the Newton map, andthe rate of attraction is at least quadratic.

1.1. Generalities about Newton’s method

The Newton map behaves very pleasantly under linear, or even affine, changes ofvariables.

Lemma 1.1.1. If A : V → V is affine and invertible, and L : W →W is linear andinvertible, then

NL◦F◦A = A−1 ◦NF ◦A. (1.2)

Proof. This is a straightforward application of the chain rule:

NL◦F◦A(x) = x−[D(L ◦ F ◦A)(x)

]−1(L ◦ F ◦A)(x)= x−

([DA(x)]−1 ◦ [DF (A(x))]−1 ◦ L−1 ◦ L ◦ F ◦A

)(x)

= x− [DA(x)]−1 ◦ [DF (A(x)]−1(F (A(x))).

(1.3)

On the other hand, we have

(A−1 ◦NF ◦A)(x) = A−1(A(x)− [DF (A(x))]−1(F (A(x))

). (1.4)

Since A−1 is affine, we have

A−1(y + z) = A−1(y) + [DA]−1(z) (1.5)

for any y and z, and we have not indicated where the derivative is evaluated sinceit is constant. This gives

(A−1 ◦NF ◦A)(x) = A−1(A(x))− [DA(x)]−1 ◦ [DF (A(x))]−1(F (A(x))), (1.6)which agrees with equation (1.3). ¤

It is clear from the definition that even if F is defined on all of V , the associatedNewton map will not be defined where the derivative of F is not invertible; we willcall this locus the polar locus PF . It is generically of codimension 1, and is given bythe single equation

PF = {x ∈ V | det[DF (x)] = 0 } . (1.7)

11


Off the polar locus, the Newton map is differentiable; the derivative is a bitdelicate to compute. In one dimension, the derivative of the Newton map is givenby

(NF )′(x) =F (x)F ′′(x)F ′(x)2

. (1.8)

We want to extend this computation to several dimensions. This requires a bit ofnotation to decide what the second derivative is. We will write the Taylor expansionof F in a neighborhood of x as

F (x + u) = F (x) + [DF (x)](u) +12

[D2F (x)](u,u) + . . . , (1.9)

so that1k!

[DkF (x)] : V × · · · × V →W (1.10)

denotes the symmetric k-linear mapping whose restriction to the diagonal is theterms of degree k of the expansion. A kth derivative with fewer than k argumentsis an operator, a function of the missing arguments; this will occur below where asecond derivative with a single argument is a linear operator of the missing argu-ment.

Lemma 1.1.2. The derivative of the Newton map is given by the formula

[DNF (x)](u) = [DF (x)]−1[D2F (x)](u, [DF (x)]−1F (x)

). (1.11)

This formula reduces to the simple one above in dimension 1 (Equation (1.8)).

Proof. Write the Taylor expansion:

x + u− [DF (x + u)]−1F (x + u)

= x + u−(

[DF (x)] + [D2F (x)](u) + o(|u|))−1(

F (x) + [DF (x)]u + o(|u|))

= x + u−(

[DF (x)](I + [DF (x)]−1[D2F (x)](u) + o(|u|)

))−1(F (x) + [DF (x)]u + o(|u|)

)= x + u−

(I − [DF (x)]−1[D2F (x)](u) + o(|u|)

)[DF (x)]−1

(F (x) + [DF (x)]u + o(|u|)

)= x + u− [DF (x)]−1F (x)− u + [DF (x)]−1[D2F (x)](u, [DF (x)]−1F (x)) + o(|u|).

(1.12)

¤

Of course, since roots of F are superattractive for Newton’s method, the deriv-ative vanishes there, as is clear from Lemma 1.1.2. It is often useful to know thequadratic terms also.

Lemma 1.1.3. If F (x0) = 0 and [DF (x0)] is invertible, then we have

NF (x0 + u) = x0 +(

12

[DF (x0)]−1[D2F (x0)](u,u))

+ o(|u|2). (1.13)

Proof. If F (x0) = 0, then the second factor of [DF (x0 + u)]−1F (x0 + u) has afactor of u, so the quadratic terms of the first factor contribute only to cubic terms

Newton in C2 13

of the Newton map. Thus

x0 + u− [DF (x0 + u)]−1F (x0 + u)= x0+u−(

I −[DF (x0)]−1[D2F (x0)](u))

[DF (x0)]−1(

[DF (x0)]u +1

2[D2F (x0)](u,u)

)+o(|u|2)

= x0 +

(1− 1

2

)[DF (x0)]

−1[D2F (x0)](u,u) + o(|u|2). ¤(1.14)

The case where [DF (x0)] is not invertible, i.e., where the root x0 is multiple, ismore complicated, it is the second degenerate case discussed in Section 1.6.

1.2. The intersection of graphs

There is a geometric description of the Newton map to find the roots of F1(x) =0, . . . , Fn(x) = 0, where Fi : Cn → C are analytic functions, which directly general-izes the standard picture for a function of one variable.

x0 x1 x2

Figure 7. The graphical representation of the Newton map in dimension 1.

Consider the hypersurfaces Yi ⊂ Cn+1 of equation z = Fi(x). At a point a ∈ Cn,we can consider the affine subspaces Ki,a ⊂ Cn+1 tangent to Yi at the points(a, Fi(a)), and their intersections Hi with the subspace z = 0. Then

NF (a) =n⋂i=1

Hi. (1.15)

In particular, if for some point a these subspaces intersect in more than 1 point,then NF is not defined there. Such points are called points of indeterminacy, andthe behavior of NF near such points presents essential difficulties, and is one of themain themes of this paper. If these subspaces do not intersect at all, at least indimension 2 we can define NF at such a point a by extending NF to projectivespace, and setting NF (a) to be an appropriate point at infinity; we will say that abelongs to the polar locus.


Intersecting graphs

We have not found this description to give much insight for general functions Fi,but when they are of the form

Fi(x) = xi − fi(x1, . . . , x̂i, . . . , xn), (1.16)

where the hat indicates that the variable is omitted, the geometric description isextremely illuminating.

We will refer to the system of equations

F1(x) = 0, . . . , Fn(x) = 0, i.e., x1 = f1(x2, . . . , xn), . . . , xn = fn(x1, . . . , xn−1)(1.17)

as the intersection of graphs. This paper is mainly about intersections of conics,but as we will see in Equation (1.63), the intersection of conics can be put in sucha form, so everything we say in this section will apply to that case.

In that case, the hypersurfaces Yi are cylinders, made up of lines parallel to theline of equation z = xi, xj = 0 if j 6= i through the manifold Xi ⊂ Cn of equationxi = fi(x1, . . . , x̂i, . . . , xn). In particular the planes Ki,a are tangent to Yi alongthe entire generator through a, and Hi,a is the affine subspace of Cn tangent to themanifold Xi at the point

(a1, . . . , ai−1, fi(a1, . . . , âi, . . . , an), ai+1, . . . , an). (1.18)

Intersecting graphs in dimension 2

For the rest of this section, we will study the intersection of graphs when n = 2,i.e., we will study the Newton map NF associated to the equations

y = f(x), x = g(y), i.e., F(xy

)=(y − f(x)x− g(y)

).

The formula for NF is then

NF

(xy

)=(xy

)−[f ′(x) −1−1 g′(y)

]−1 (f(x)− yg(y)− x

), (1.19)

but we will hardly ever use the formula; instead we will use the following geometricdescription.

At a point (a, b), draw the vertical line to the point (a, f(a)) and the horizontalline to the point (g(b), b). At these points, draw the tangents to the respectivecurves; the intersection of these tangent lines is NF (a, b). This is illustrated inFigure 8.

Note that the Newton map NF takes the vertical line x = a to the line tangentto X1 at (a, f(a)), and also takes horizontal lines to lines tangent to X2.

Newton in C2 15

(a,b)

NF(a,b)

(g(b),b)

(a,f(a))

y=f(x)

x=g(y)X2

X1

(a,b)(g(b),b)

(a,f(a))

y=f(x)

x=g(y)X2

X1

(c,d)(g(d),d)

(c,f(c))

Figure 8. The graphical representation of the intersection of graphs in dimension 2. On

the left we see the image NF

(ab

), and on the right we see a point

(ab

)of indeterminacy,

and a point

(cd

)on the polar locus.

The polar locus and points of indeterminacy

If the tangent line to the curve y = f(x) through (x, f(x)) and the tangent linethrough (g(y), y) to the curve of equation x = g(y) are parallel, then NF (x, y) isnot a point of C2. However, it is easy to see that if we take the range of NF to beP2, then such a point maps to a point of the line at infinity, quite specifically thepoint “at the end” of the pair of parallel lines.

Thus such points belong to the polar locus of NF . This polar locus is the curveP of equation

f ′(x)g′(y) = 1. (1.20)

Suppose now that f is a polynomial of degree p and g is a polynomial of degree q.Since for any m 6= 0,∞ there are p − 1 points (counted with multiplicity) of thegraph of f where the tangent has slope m, and q − 1 points where the graph of ghas slope 1/m, we see that P covers the line at infinity with degree (p− 1)(q − 1).

The points of P where the tangent lines above are not only parallel but coincideare special: they are points of indeterminacy of NF . They correspond exactly tothe pairs of points x1 ∈ X1 and x2 ∈ X2 such that the tangent line to X1 at x1coincides with the tangent line to X2 at x2.

The equations describing the points of indeterminacy in C2 are

y = f(x) + f ′(x)(g(y)− x), x = g(y) + g′(y)(f(x)− y), f ′(x)g′(y) = 1. (1.21)

These equations are not independent: the first and third imply the second, and thesecond and third imply the first, but the first two define the union of the set ofindeterminacy and of the roots, i.e., X1 ∩X2.

We will be interested throughout this paper in performing blow-ups on surfacesto remove points of indeterminacy. If X is a surface, and x ∈ X is a point, we willdenote the blow-up of X at x by X̃x.


inProposition 1.2.1. Let (a, b) be a point of indeterminacy of N , and supposethat the common tangent

f(a) + f ′(a)(x− a)− y = g(b) + g′(b)(y − b)− x = 0 (1.22)is not an inflectional tangent at either of the points of tangency (a, f(a)) and(g(b), b), i.e., where f ′′(a) 6= 0, g′′(b) 6= 0. Assume also that the points (a, f(a))and (g(b), b) are distinct. Blow up (a, b) once, with exceptional divisor E. ThenNF is well-defined on a neighborhood of E in C̃2(a,b).

Proof. There is no loss in assuming that (a, b) = (0, 0); in that case let us compute

NF

(xy

)to first order in x and y. The tangent lines to be intersected have equations

Y = f(x) + f ′(x)(X − x), X = g(y) + g′(y)(Y − y), (1.23)and these have solutions which are easy to write down:X(1− f ′(x)g′(y)) = g(y)− g′(y)f(x)− xf ′(x)g′(y)− yg′(y)

= g(0) + yg′(0) +(g′(0) + yg′′(0)

)(f(0) + xf ′(0)

)− xf ′(0)g′(0)− yg′(0) +O(x2 + y2)

= yf(0)g′′(0) +O(x2 + y2),

Y (1− f ′(x)g′(y)) = f(x)− f ′(x)g(y)− yf ′(x)g′(y)− xf ′(x)

= f(0) + xf ′(0) +(f ′(0) + xf ′′(0)

)(g(0) + yg′(0)

)− yg′(0)f ′(0)− xf ′(0) +O(x2 + y2)

= xg(0)f ′′(0) +O(x2 + y2),(1.24)

where we have used the identities f(0) + f ′(0)g(0) = g(0) + g′(0)f(0) = 0.Further we have (1− f ′(x)g′(y)) = −xg′(0)f ′′(0)− yf ′(0)g′′(0) +O(x2 + y2), so

altogether this gives

X = − yf(0)g′′(0)

xg′(0)f ′′(0) + yf ′(0)g′′(0)+O(x2 + y2),

Y = − xf′′(0)g(0)

xg′(0)f ′′(0) + yf ′(0)g′′(0)+O(x2 + y2).

(1.25)

In particular, we see that if (x, y) approaches the origin on the line y = mx, thenNF (x, y) approaches(

− mf(0)g′′(0)

g′(0)f ′′(0) +mf ′(0)g′′(0),− f

′′(0)g(0)g′(0)f ′′(0) +mf ′(0)g′′(0)

). (1.26)

This formula provides the extension of NF to E. Note that if f ′′(0) = 0 or g′′(0) = 0,the exceptional divisor is collapsed to a single point by NF , except for the directionof the polar curve which is a point of indeterminacy, and similarly if f(0) = 0 (whichimplies g(0) = 0). ¤

Inverse images and the topological degree of the Newton mapIt is quite easy to visualize the set of inverse images N−1F (a, b). From (a, b), drawall the tangent lines to X1 and X2, and from each point of tangency with X1 drawa vertical line, and from each point of tangency with X2 draw a horizontal line.Then the intersections of the family of horizontal lines with the family of verticallines form the inverse image of (a, b).

Newton in C2 17

The line x = c is such a vertical line if b = f(c) + f ′(c)(a− c); similarly, the liney = d is such a horizontal line if a = g(d) + g′(d)(b− d).

Suppose now that f(x) = xp+ . . . is a polynomial of degree p and g(y) = yq+ . . .is a polynomial of degree q. Then the equation satisfied by c is

f(c) + f ′(c)(a− c)− b = cp +(pcp−1 + . . .

)(a− c) = (1− p)cp + · · · = 0. (1.27)

This is always a polynomial of degree exactly p, which has p roots counted withmultiplicity. Similarly, the values of d satisfy a polynomial of degree q.

The algebraic degree of NF is easy to compute from formula (1.19). The highestdegree terms are among those of xf ′(x)g′(y), yf ′(x)g′(y), f(x)f ′(x), g(y)g′(y), withdegrees respectively p+ q− 1, p+ q− 1, 2p− 1, 2q− 1. Even if p = q, the monomialsare distinct, so they cannot cancel, and any one of highest degree determines thealgebraic degree. Moreover, p+ q − 1 is between 2p− 1 and 2q − 1. This gives thefollowing result.

inTheorem 1.2.2. The Newton map associated to the system of equations y −f(x) = 0, x− g(y) = 0, where f and g are polynomials of degree p and q respec-tively, has topological degree degNF = pq. The algebraic degree is max{2p −1, 2q − 1}.

In particular the inverse image of any point forms a finite set, and if the pointsof these sets are counted with multiplicities these sets all have the same number ofelements.

The critical locus

We will now describe the critical locus of the mapping NF .

inProposition 1.2.3. The critical values of the mapping NF as above are pre-cisely the union of the curves X1, X2 and any inflectional tangents to either ofthe curves. The critical curve C1 that maps to X1 has equation x = g(y) +g′(y)(f(x) − y), and N(x, y) = x on C1. The curve C2 which maps to X2 hasequation y = f(x) + f ′(x)(g(y)− x), and N(x, y) = y on C2.

Proof. First let us see that if (a, b) is a point of a line l tangent to a curve C at(x, y), if (x, y) is not a point of inflection of C, and if (a, b) 6= (x, y), there existsan neighborhood U of (a, b) and an analytic function φ : U → C such that the linejoining (c, d) ∈ U to φ(c, d) is tangent to C at φ(c, d) and φ(a, b) = (x, y). Thisfollows from the implicit function theorem: if C has equation y = f(x), then thecoordinates (u, v) = φ(c, d) satisfy the equations

Φ((

cd

),(uv

))=(f ′(u)(u− c)− (v − d)

v − f(u)

)= 0, (1.28)

so (u, v) are analytic functions of (c, d) in a neighborhood of (a, b) if[∂G1/∂u ∂G1/∂v∂G2/∂u ∂G2/∂v

]=[f ′(u) + (u− c)f ′′(u) −1

−f ′(u) 1

]


is invertible at((

ab

),(xy

)), which is evidently the case when u 6= c and f ′′(x) 6= 0.

Thus a point (a, b) ∈ C2 is not a critical value of NF if (a, b) /∈ X1∪X2 and (a, b)is not on an inflectional tangent to either X1 or X2. We leave the converse to thereader. ¤

The extension to the line at infinity

The Newton mapping N associated to the system of equations y = f(x), x = g(y)can be extended to the line at infinity (with some points of indeterminacy) if f andg are rational functions. We will work this out when f and g are polynomials ofdegree p and q respectively. We will use the formula (1.19).

Set(xy

)=(tmt

), so that m is parametrizing the line at infinity, and calculate

the leading terms in t as t → ∞. Writing only the leading terms, which are ofdegree p+ q − 1 in the numerator and p+ q − 2 in the denominator, we find

N(tmt

)=(

(1− 1/p)tm(1− 1/q)t

)+O(1) (1.29)

when m 6= 0 and t→∞. Thus N extends to the line at infinity, by the linear map

m 7→ p(q − 1)q(p− 1)m. (1.30)

The two points at infinity on the axes, corresponding to m = 0 and m = ∞, arepoints of indeterminacy of N . We will show that if we blow them up once, the mapN extends to the exceptional divisor, and maps the exceptional divisor to the lineat infinity with degrees p− 1 and q − 1 respectively.

Indeed, the line t 7→(ta

)tends to the point of the exceptional divisor at [1 : 0 : 0]

as t→∞. The horizontal line maps to the line tangent to X2 at the point (g(a), a),and as t→∞, the image of

(ta

)goes to infinity on that line. Since there are q− 1

lines tangent to X2 with given slope, this means that the map from the exceptionaldivisor to the line at infinity is of degree q− 1. Similarly, the exceptional divisor atthe point [0 : 1 : 0] maps to the line at infinity with degree p− 1.

Note that by putting together the inverse images of points at infinity in thepolar curve, the two exceptional divisors and the line at infinity itself, we havefound (p− 1)(q− 1) + (p− 1) + (q− 1) + 1 = pq points, and since N has degree pq,we have them all.

1.3. The Russakovskii-Shiffman measure

This section is essentially due to Adrien Douady, who found a mistake in an earlierversion of the paper, and proceeded to correct it with a much simpler approach,which is the one presented here.

Russakovskii and Shiffman [RS] have constructed a measure µ on P2 which de-scribes the distribution of inverse images of points in the finite plane C2. Morespecifically, let f : P2 ∼∼> P2 be a rational mapping. It has a topological degree dt:the generic number of inverse images of a point, and an algebraic degree da which isthe degree of the algebraic curve given as the inverse image of a generic line. They

Newton in C2 19

prove that if dt > da, then there is a pluripolar exceptional set E ⊂ P2 such that ifx ∈ P2 − E, then the sequence of probability measures

µn =1dnt

∑y∈f−n(x)

δy, (1.31)

where δy is the unit mass at y, converges to a limit measure µf independent of x.The mappings NF considered above, to solve y = f(x) and x = g(y) when f is a

polynomial of degree p and g is a polynomial of degree q, satisfy their hypothesis:they have topological degree pq, by Theorem 1.2.2, and algebraic degree max{2p−1, 2q−1}. In fact, they consider the case of intersecting conics, more specifically thesystem of equations (1.64) when a = b = 0 in their paper. In this case the pluripolarexceptional set contains the roots and the line at infinity. Russakovskii and Shiffmanask whether µf charges the points of indeterminacy on the line at infinity; theircomputer experiments suggest that it does not.3 We will prove that this is indeedthe case, and more generally that for the Newton map to solve y = f(x), x = g(y)where f and g are polynomials, the Russakovskii-Shiffman measure never chargesthe line at infinity.

In the case of intersecting conics, we will see that the roots are their only in-verse images, so they belong to the exceptional set. The points at infinity arealso exceptional: their inverse images, as appropriately weighted, give 1/3 of theRussakovskii-Shiffman measure plus a certain measure carried at infinity, in factgiving mass 1/3 at each of the points of indeterminacy at infinity. That part ofthe mass can be further analyzed: in the infinite blow-up described in Chapters 5and 6, this measure is carried by the ends of the infinite tree. There are no otherexceptional points.

inTheorem 1.3.1. Let N be the Newton mapping to solve the equations y =f(x), x = g(y), where f and g are polynomials of degree p and q respectively.Then the corresponding Russakovskii-Shiffman measure does not charge the lineat infinity.

Proof. By Lemma 1.1.1, we may take f and g monic and centered, so that

f(x) = xp + αp−2xp−2 + · · ·+ α0, g(y) = yq + βp−2yq−2 + · · ·+ β0. (1.32)

Given a point (a, b) ∈ C2, the points (x, y) ∈ N−1F (a, b) satisfy the equations

f(x) + f ′(x)(a− x) = b, g(y) + g′(y)(b− y) = a. (1.33)

The first is the equation of degree p

(1− p)xp + apxp−1 +(αp−2 − (p− 2)αp−2

)xp−2 + · · ·+ α0 + α1a− b = 0. (1.34)

Lemma 1.3.2. There exist constants K1,K2 such that at least one root x1 satisfies

|x1| ≤ K1 (sup{|a|, |b|, 1})1/p (1.35)

3Added in proof: Guedj [Guedj] has found a new proof of the existence of the Russa-kovskii-Shiffman measure, which shows far more, and in particular shows that it nevercharges the points of indeterminacy.


and all roots satisfy

|xi| ≤ K2 sup{|a|, |b|, 1}. (1.36)

Proof of Lemma 1.3.2. We see from the constant term that the smallest rootin absolute value must be

≤∣∣∣∣α0 + α1a− bp− 1

∣∣∣∣1/p . (1.37)This is clearly at most K1(sup{|a|, |b|, 1})1/p for an appropriate K1.

The other inequality comes from the usual bound on roots:

|f(x) + f ′(x)(a− x)− b| ≥ (p− 1)|x|p −(p−1∑i=0

|γi|)|x|p−1 (1.38)

where γi are the coefficients of (1.34), and |x| > 1. But then|f(x) + f ′(x)(a− x)− b| > 0 (1.39)

when |x| > 4p(sup |αi|)(sup{|a|, |b|, 1}). ¤ Lemma 1.3.2

A probabilistic lemma

Lemma 1.3.2 can be understood as follows. Every point (a, b) has m = pq inverseimages; think of them as its children. More precisely, think we have a population, onthe plane, say with a cultural home at the origin. Every member of this populationhas m children. Of these, all but one might be adventurous and settle at a distanceK2D from the origin, where D is the distance of their parent’s home from the origin.But at least one is homebound and settles a distance at most a distance K1D1/p

from the origin. Now suppose the parent divides his fortune equally among hischildren. Can the adventurous ones make the fortune drift further and further fromthe origin, or do the homebound ones of the succeeding generations manage to keepthe fortune from wandering too far. The object of the following lemma is to saythat the second alternative holds: the pth root wins over the multiplication by K2,however large it might be.

Let (X,µ) be a probability space, L : X → R a function, and φ1, . . . , φk : X → Xbe maps and λ1, . . . , λk,m ∈ R be numbers such that

L ◦ φi(x) ≤{L(x) + λi if L(x) ≥ mm+ λi if L(x) < m.

(1.40)

Let us suppose that∑ki=1 λi < 0. Note that in that case there exists α > 0 such

that

1k

k∑i=1

eαλi ≤ 1, (1.41)

because the derivative of the function t 7→∑ki=1 e

tλi is negative at t = 0. Define

µ̃ =1k

k∑i=1

(φi)∗µ, (1.42)

Newton in C2 21

and set h(r) = µ{x| L(x) ≥ r} and h̃(r) = µ̃{x| L(x) ≥ r}.

Lemma 1.3.3. If h(r) ≤ e−α(r−m), then h̃(r) ≤ e−α(r−m).

Proof. Define µi = (φi)∗µ, and hi(r) = µi{x| L(x) ≥ r}.If r > m+ λi and L(φi(x)) > r, then L(x) > m. Indeed, otherwise we have

r < L(φi(x)) ≤ m+ λi, (1.43)a contradiction. So in that case

hi(r) = µi{x| L(x) ≥ r} = µ{x| L(φi(x)) ≥ r}≤ µ{x| L(x) ≥ r − λi} = h(r − λi) ≤ e−α(r−m−λi).

(1.44)

If r ≤ m+ λi, then the same inequality holds more easily:hi(r) ≤ 1 ≤ e−α(r−m−λi). (1.45)

So

h̃(r) =1k

k∑i=1

hi(r) ≤1k

k∑i=1

e−α(r−m−λi)

= e−α(r−m)(

1k

k∑i=1

eαλi

)≤ e−α(r−m).

(1.46)

¤ Lemma 1.3.3

Proof of the Theorem

We will apply this where X = C2, L(x) = log+ ‖x‖ using the sup-norm, and wherethe φi are the branches of the inverse image under Newton’s method; the fact thatthese cannot be taken to be continuous is irrelevant since we are doing measuretheory. The inequalities of Lemma 1.3.2 become: we can choose φ1 so that

L(φ1(x)) ≤1dL(x) + logK1, (1.47)

and all the φi satisfy

L(φi(x)) ≤ L(x) + logK2, (1.48)We will apply Lemma 1.3.3 setting λ2 = · · · = λk = logK2; clearly there exists λ1such that

∑ki=1 λi < 0 and m such that if x ≥ m, then

L(φ1(x)) ≤ L(x) + λi. (1.49)Now take any probability measure µ0 on C2, and define µn by

µn+1 =1pqN∗Fµn. (1.50)

This formula needs a bit of justification: how do we pull back measures? Byformula (1.33), every point has exactly pq inverse images counted with multiplicity,so the formula

(NF )∗g(x) =∑

y∈N−1F (x)

g(y) (1.51)


is well defined as an operator from the space of continuous functions g with compactsupport on C2 to itself (in the sum, y is counted with multiplicity; recall that everypoint has pq inverse images counted with multiplicity). The pull-back of measuresis the transpose of (NF )∗. Russakovskii-Shiffman prove that the limit of the µnexists and is independent of µ0 if µ0 has a density; we will denote it by σ.

Thus we may suppose that µ0 has support in L(x) ≤ m. Then h0(r) = 0 ≤e−α(r−m) when r ≥ m. Therefore hn(r) ≤ e−α(r−m) when r ≥ m for all n.

1.4. Invariant currents

In addition to the Russakovskii-Shiffman measure, there are a number of otherinvariant currents.

One (or rather, several, one for each root) was explored at considerable lengthin [HP]. For each root, there is an invariant 1-1 current with support in the basinof that root.

More precisely, let x0 be a non-degenerate solution of F (x) = 0. Then the limit

Gx0(x) = limn→∞12n

log |NnF (x)− x0|

exists and is a continuous plurisubharmonic function on the basin of x0. Considerthe 1-1 current ddcGx0 which is a current on the basin; as shown in [HP], it issupported on a sort of non-linear cone over the Julia set of the rational functionassociated to the root (see Section 1.7, and Figure 9). Since this set is invariant,the support of the Russakovskii measure must be contained in the intersection ofthe closures of all these “cones”.

Figure 9. This figure represents Newton’s method for the map

F

(xy

)=

(y − x2 − ax− y2 − b

), a = −1− 2i, b = −3 + i.

The drawings are in the parabola y = x2+a, parametrized by x, with −2 ≤ Rex, Imx ≤ 2.As we will see in Proposition 1.5.7, this parabola is one component of the critical value

locus, hence relevant to the dynamics. On the left we see the four basins, colored in

different colors. On the right we see (in black) an approximation of the support of the

Laplacian, and in red an approximation of the boundaries of the basins.

Newton in C2 23

There is another invariant current, defined as follows. Lift NF : P2 ∼∼> P2 to ahomogeneous cubic mapping ÑF : C3 → C3, and define

G̃(z) = limn→∞

13n

log |ÑnF (z)|.

This limit was originally considered in [HP] for endomorphisms of Pk; Sibony [Si]showed that the result is true in greater generality.

Let h : Pk ∼∼> Pk is a dominating rational map of algebraic degree d, andH : Ck+1 → Ck+1 a homogeneous polynomial map of degree d that lifts h. ThenSibony defines h to be algebraically stable if for all n ≥ 0, Hn does not vanish on ahypersurface. The way this can fail in dimension 2 is if h collapses some curve to apoint, which after some iterates lands on a point of indeterminacy.

Sibony proves that if h is algebraically stable, then the limit

Gh(z) = limn→∞

1dn

log |Hn(z)|

defines a plurisubharmonic function on Ck+1; of course this function is −∞ on thelines above the points of indeterminacy of h, and all of their inverse images. Hefurther shows that ddcGh is the pull-back of a positive 1-1 current on Pk, whichdoes not charge pluripolar sets.

All of this applies to the map NF , which is algebraically stable since it doesn’tcollapse anything, so we do obtain an invariant 1-1 current S on P2.

We do not understand S at all well: it is a lot easier to say what it isn’t thanto say what it is. It’s not the current in the basins described above; in fact it’s noteven related: because of the 3 rather than 2 in the denominator, the Sibony currentmust be supported on the boundaries of the basins.

It is also not true that S ∧ S is the Russakovskii-Shiffman measure. The cor-responding statement is true for endomorphisms of P2 as proved in [HP], but toobtain an invariant measure one must pull-back and divide by the topological de-gree. For endomorphisms, this is d2, but not for rational maps, and S ∧ S if itexisted would be obtained by pulling back and dividing by d2. Presumably S ∧ Sdoes not exist at all; we only know how to take wedges of positive currents withcontinuous potentials.

When you look at pictures of plane sections of Newton maps, the main featuresthat stand out are the “smooth curves” separating the basins. In Section 2.4, wewill show that under some circumstances there actually are topological 3-manifoldsseparating the basins, that intersect plane sections in quasi-circles. These manifoldsnaturally carry a 1-1 current, but as they are only 2 inverse images of themselves,this current is also not related to S.

We conducted a few experiments to investigate S numerically. These experimentsindicate that S is supported on the entire separator of the basins, but that most ofthe support at “manifold points” comes from “bubble sequences” (see Section 2.5)accumulating at such points.

1.5. The intersection of conics

One particularly simple example of Newton’s map is the one used to find the inter-section of two quadratic curves; in fact this might be the simplest non-degenerate


case, since the intersection of a line and a curve of any degree gives a singularNewton map.

Two such curves usually intersect in four points, and we will need to understandthe space of quadratic functions vanishing at four distinct points.

Lemma 1.5.1. (a) The space of quadratic functions vanishing at four distinctpoints is 2-dimensional, and GL2 acts transitively (by composition on the left) onpairs of such functions that are linearly independent.

(b) Among these functions exactly two define parabolas (possibly degenerate),and exactly three define pairs of lines.

From this and Lemma 1.1.1, we immediately see:

inCorollary 1.5.2. Newton’s method to find the intersection of two conics dependsonly on the intersection points and not on the choice of curves.

The family of curves given by a 2-dimensional vector space of functions is a1-dimensional family of curves, called a pencil of curves. Lemma 1.5.1 says thatthe conics passing through four distinct points form a pencil. More generally, thecurves of degree d passing through (d2 + 3d − 2)/2 points form a pencil. Whend = 3, we have (d2 + 3d − 2)/2 = 8, and given any eight points in C2 in generalposition, we can consider the Newton map defined by any two cubics passing bythese points. This map is independent of the choice of cubics in the pencil. Awell-known theorem asserts that if three cubics are concurrent in eight points, thenthey are also concurrent in a ninth; the argument above gives another proof of thisfact.

Recall that the Grassmanian of k-subspaces in Cn has dimension k(n−k). Usingthis formula, we see that the Newton maps to find the intersections of two curves ofdegree d form, after normalization, a family of dimension d2 + 3d− 8. Indeed, thisis the dimension of the Grassmanian of planes in the vector space of polynomialfunctions in 2 variables of degree d, normalized to vanish at

(00

),(

10

)and

(01

).

Moreover, any two bases of a plane determine the same Newton map, and any twodistinct planes determine distinct Newton maps.

Proof of Lemma 1.5.1. By an affine change of variables in the domain, the fourpoints can be moved, for instance to

(00

),(

10

),(

01

),(αβ

); so up to conjugation

by an affine mapping the corresponding Newton maps depend on the two parametersα, β. We will find it convenient to set

A =1− αβ

and B =1− βα

(1.56)

as these quantities arise frequently in computation.The general quadratic function vanishing at

(00

),(

10

),(

01

),(αβ

)is

P (x2 − x) +Qxy +R(y2 − y), P,Q,R ∈ C (1.57)where Q = PA+ RB. We can take as our equations any two linearly independentfunctions in this family. For instance we can choose P = 1, R = 0, and P = 0, R = 1,to find

F(xy

)=(x2 +Axy − xy2 +Bxy − y

)(1.58)

Newton in C2 25

and finally the Newton map

NF

(xy

)=(xy

)−[

2x+Ay − 1 AxBy 2y +Bx− 1

]−1 (x2 +Axy − xy2 +Bxy − y

)=

1∆

(x(Bx2 + 2xy +Ay2 − x−Ay)y(Bx2 + 2xy +Ay2 −Bx− y)

) (1.59)where

∆ = 2Bx2 + 4xy + 2Ay2 − (2 +B)x− (2 +A)y + 1. ¤ (1.60)

This form of the Newton map to intersect two conics will be useful in Section1.6, but for our present purposes, a different one is better.

The parabolic normalization

We could have required that Px2 +Qxy+Ry2 be a perfect square: this will happenwhen

Q2 = (PA+RB)2 = 4PR. (1.61)

Up to multiples, there are two such equations in general, defining parabolas.By an affine change of coordinates in the domain we can impose that the axes of

the two parabolas in a given pencil should be the coordinate axes, and the remainingfreedom to scale the coordinates allows us to write these parabolas as

y = x2 + a and x = y2 + b. (1.62)

This corresponds to setting

F(xy

)=(x2 − y + ay2 − x+ b

). (1.63)

Thus the intersection of two generic conics can be reduced to the equations (1.62),and in particular, everything mentioned in Sections 1.2 and 1.3 applies to this case.

In this form, the Newton mapping becomes

NF

(xy

)=(xy

)− 1

4xy − 1

[2y 11 2x

](x2 − y + ay2 − x+ b

)=

14xy − 1

(2x2y + y2 − 2ay − b2xy2 + x2 − 2xb− a

).

(1.64)

The following Proposition will be one of our central tools.

inProposition 1.5.3. The lines joining the roots are invariant under Newton’smethod, and in these lines Newton’s method induces the 1-dimensional Newton’smethod to find the roots of a quadratic polynomial.

More generally, if a line contains d intersections of two curves of degree p,then that line is invariant by Newton’s method, and the restriction to the line isthe one-dimensional Newton’s method to solve the polynomial of degree p withthose p roots.

Of course, when p > 2, this situation is not generic; usually no line contains morethan two intersection points of two curves of degree p.


Proof. In the case of conics, we can use formula (1.59). Any invariant line can beturned into the x-axis by a change of variables, by putting the two roots on it at(

00

)and

(10

)and a third root at

(01

), so it is enough to show that the x-axis is

invariant, and this is clear because of the factor y in the second coordinate.In the general case, let l be a line containing p intersection points of two curves

C1 and C2 of degree p. Then N−1(l) is an algebraic curve of degree 2p − 1 (seeTheorem 1.2.2). Since l passes through the roots, its inverse image does too. Infact at each root N−1(l) has a double point. We will study this in detail in Section1.7, where we show that if we blow up the root, the Newton map extends to theblow-up, and maps the exceptional divisor to itself by a rational map of degree 2(independently of p). So the proper transform of N−1(l) intersects the exceptionaldivisor in two points, and projected back down to P2, the curve N−1(l) has a doublepoint. Thus each root counts as 2 intersections of l and N−1(l), giving 2p in all.But a curve of degree 2p− 1 can intersect a line in 2p points only if it is reducible,and one of its components is the line. This proves that l is invariant.

To see that N |l is conjugate to the Newton map NP where P is the polynomialwith roots the roots of F in l, observe that it is a rational map of degree p with psuper-attractive points. It also fixes infinity, with derivative there d/(d− 1), as youcan check from formula (1.29), or you can use Fatou’s theorem (Equation (1.75)) tosee that this is the only possible value. The difference of our map and the Newtonmap for the polynomial P is a rational function of degree ≤ 2p with 2p + 1 roots,so it is 0. ¤

We will refer to the lines joining pairs of roots as the invariant lines of the Newtonmap NF . Their intersections (other than the roots) also play an important role:they are points of indeterminacy of NF . If the Newton map were defined there, theorbit would need to be in both lines, so the point of intersection would have to befixed, and it isn’t.

inProposition 1.5.4. The points of indeterminacy of NF in the finite plane C2 arethe three intersections of the invariant lines other than the roots. If the parabolasof equation y = x2 + a and x = y2 + b are not tangent, then the mapping NFextends to the blow-up of C2 at these points, mapping each exceptional divisorto a line tangent to both parabolas.

Proof. This follows from combining the observation that these points must bepoints of indeterminacy with Proposition 1.2.1. There are exactly 3 lines tangentto both parabolas, which tells us that these are all the points of indeterminacy inthe finite plane, and that a single blow-up resolves the indeterminacy. ¤

Remark. Proposition 1.5.3 explains some features of Figure 1. For instance, itexplains why the lines joining two roots are colored in only two colors. On theselines, there are points where the other basins touch; the points where two linesjoining pairs of roots intersect stand out, for instance, the point on the x-axis atx = −1. This point is a point of indeterminacy, and the other points on the x-axis where the blue basin touches this line are the inverse images of the point of

Newton in C2 27

indeterminacy. They accumulate at infinity, and at the point(

1/20

)which is the

inverse image of infinity. For such a real picture of Newton’s method, these points ofindeterminacy and their inverse images are the most striking features of the picture.We will investigate what these “fingers” of blue look like in the complex in Section2.5. 4

We also obtain the following statement of elementary geometry:

Let Q be the quadrilateral whose vertices {A,B,C,D} are the intersections oftwo parabolas. Through the point E where the line through A,B intersects the linethrough C,D, draw the parallels to the axes of the parabolas; these intersect theirparabola at G and H. Then the line through G,H is tangent to both parabolas.

We will now specialize the results of section 1.2 to the case of conics.

The polar locus for intersection of conics

Lemma 1.5.5. For the mapping F of (1.63), the polar locus PF is the hyperbolaof equation 4xy = 1.

From the point(xy

)of the polar locus we can draw a vertical line to the point(

xx2 + a

)and a horizontal line to the point

(y2 + by

); at these points the tangents

to the parabolas are parallel, and the Newton map extends to the polar locus (with

the points of indeterminacy removed by mapping(xy

)to the point at infinity in

the direction of these parallels.Notice that the polar locus goes through all the points of indeterminacy (the

three at finite distance and the two on the line at infinity), and also through thecenters of the 6 segments joining pairs of roots.

Lemma 1.5.6. The map NF extends to the line at infinity, except at the pointsp1,p2 at infinity on the axes of the parabolas (1.62). Moreover, the extension isthe identity on the line at infinity (except at these points), and the eigenvalues of

the derivative are 1 and 2. The eigenvalue 1 corresponds to the direction of the lineat infinity.

Remark. Lemma 1.5.6 says that the line at infinity is repelling, and this appearsto contradict Theorem 1.3.1. This apparent contradiction comes from the fact thatonly one inverse image of a point near infinity is still near the same point at infinity;the other inverse images carry the mass away from the line at infinity. 4

The critical locus for intersection of conics

Proposition 1.2.3 gives a complete description of the critical locus and critical valuelocus of NF .


inProposition 1.5.7. The critical value locus of the mapping NF is the union ofthe two parabolas y = x2 + a and x = y2 + b.

The critical locus is the union of the cubics of equation

2xy2 − x2 + 2xb− y + a = 0 and 2x2y − y2 + 2ay − x+ b = 0. (1.65)

The parabolas and critical cubics are illustrated in Figure 10, together with theway in which the cubics map to the parabolas (as double covers, ramified at 4points, as one might expect).

Figure 10. Left: The two critical cubics and the critical value parabolas, when a = −4and b = −3. We have also drawn in the polar curve of equation 4xy = 1. The horizontalparabola and its inverse image are drawn dark; the vertical parabola and its inverse image

are drawn light. Right: One of the cubics, and its image parabola: the inverse image of

the horizontal parabola maps horizontally to its image.

The topological degree of the Newton mapping

Theorem 1.2.2 immediately tells us the topological degree of the mapping NF .

inProposition 1.5.8. The Newton mapping has topological degree 4.

In fact, the proof of Theorem 1.2.2 tells us a lot more: the inverse images of apoint form a rectangle, centered at that point.

Newton in C2 29

Figure 11. The inverse images of the point marked with a heavy dot are the vertices ofthe indicated rectangle.

Proposition 1.5.8 has the following remarkable consequence.

inTheorem 1.5.9. For the Newton map NF , the basin of each root is connected.

Proof. The mapping is locally four-to-one near the roots. So we can choose aconnected neighborhood U0 of a root such that U1 = N−1F (U0) is connected. DefineUk = N−1F (Uk−1); we must prove that all the Uk are connected. Suppose Uk isthe first disconnected one, choose x ∈ Uk, and choose a path γ in Uk−1 connectingNF (x) to a point of U0. By a small perturbation, we may assume that γ does notintersect the critical value locus Γ1 ∪ Γ2, or the three double tangents L1, L2, L3.Then the inverse image of γ consists of four arcs, all ending at points of U1. Onearc must lead to x in Uk. ¤

1.6. Degenerate cases

There are two special cases of intersections of conics which are not well treatedby our mapping NF ; the case where the roots lie on a pair of parallel lines, and thecase where the two parabolas are tangent. In this section we will start describingthese cases, but there is much more to say than we will say here.

The intersections of a conic with two parallel lines

In the case where two conics intersect in four points which lie on two parallel lines,this pair of lines is one of the parabolas in the pencil of conics passing through thefour points, and our parabolic normalization of the Newton map does not apply.The form (1.58) is better adapted to the present degenerate case.

The four points(

00

),(

10

),(

01

),(αβ

)lie on two parallel lines when α = 1,

or β = 1, or α = −β. These are equivalent under an affine change of variables, sowe can restrict ourselves to the case α = 1, i.e., A = 0. A bit of computation showsthat in this case, the first coordinate of NF is simply x2/(2x − 1), which is theone-variable Newton map for solving x2 − x = 0. Thus all points with Rex < 1/2are attracted to the line x = 0, and all points with Rex > 1/2 are attracted to


the line x = 1. Within each of these lines we are again faced with a one-variableNewton map, so the dynamics are simple. Thus all the really interesting dynamicsoccurs in the real 3-dimensional manifold Rex = 1/2.

In this 3-manifold, we have what might be called a fibered family of rationalfunctions, i.e., a map of the form

G : R/Z× P1 → R/Z× P1,(θy

)7→(

2θgθ(y)

)(1.66)

where gθ is the rational function

y 7→ y(yz2 +Bz − y)

(z + 1)((2y − 1)(z − 1) +Bz) , z = e2πiθ. (1.67)

There has been some work on dynamical systems of this sort, mainly when gθis a polynomial [Se]. Computer investigation indicates that the system is quiteinteresting: we intend to write another paper on the subject.

In the case where three roots are aligned, there does not appear to be any wayof making sense of the Newton map.

The case where the parabolas are tangent

The intersection of parabolas also contains a degenerate subcase: those where thetwo parabolas are tangent. The locus in the (a, b)-plane where this occurs is easy

to parametrize by the slope m at the point of tangency(xy

): the relations

2x = m, 2my = 1 lead to m 7→(ab

)=(y − x2x− y2

)=(

1/(2m)−m2/4m/2− 1/(4m2)

).

(1.68)The equation of this curve is

28(a2b2 + a3 + b3) + 2532ab = 33; (1.69)

it is a quartic of genus 0, with three cusps; the real locus of this quartic is shownin Figure 12.

Figure 12. The locus of pairs (a, b), −4 ≤ a, b ≤ 4 for which the parabolas y = x2 + aand x = y2 + b are tangent.

Newton in C2 31

In one dimension, a multiple root is an attracting fixed point of the Newtonmap, but not superattractive. In two dimensions, the situation is a great deal morecomplicated. It is easier to study if we go back to a variant of the form (1.59),

and consider the pencil of conics that both pass through(

00

),(

10

),(

01

)and have

slope m at the origin. The space of quadratic functions defining such conics is2-dimensional, and we may take as basis the two functions

xy, mx2 − y2 −mx+ y. (1.70)(The curve xy = 0 is a degenerate such conic.) The Newton map associated to theequations xy = 0, mx2 − y2 −mx+ y = 0 is(

xy

)7→ 1

2mx2 −mx+ 2y2 − y

(x(mx2 + y2 − y)y(mx2 + y2 −mx)

). (1.71)

In this form, it is fairly easy to see that the origin is a point of indeterminacy,in particular, it is not an attractive fixed point. But it almost is: in the x-axis andin the y-axis it is superattracting. In the line of tangency y = mx it is linearlyattracting: the Newton map restricts to x 7→ x/2.

As a point of indeterminacy, the origin is a bit more complicated than the pointsexamined so far: after one blow-up, the exceptional divisor is collapsed to the pointcorresponding to slope m, except for the point corresponding to slope −m, whichmust be blown up again to make the Newton map well defined.

Computer investigation seems to show that this case is not much simpler thanthe general case, as Figure 13 indicates.

Figure 13. The picture represents a slice of C2, for the dynamical system (1.71). In redis the basin of the double root, in green the basin of

(10

), in blue the basin of

(01

).

Comparison of the two ways of writing the Newton map

Except for the degenerate cases, every map of the form (1.59) is conjugate to amap of the form (1.64), and vice-versa. Let us spell this out. Consider the spaceN of conjugacy classes of Newton maps used to find the points of intersection oftwo conics that intersect in four distinct points, not on parallel lines. Then the(α, β)-plane, with the six lines α = 0, 1, β = 0, 1, α+β = 0, 1 removed, is a 24-fold


cover of N , with the fiber above a point corresponding to the 24 ways of labelingthe roots.

The (a, b)-plane, with the quartic curve corresponding to tangency of the parabo-las y = x2 + a and x = y2 + b removed, is a double cover of N , with the fiber abovea point corresponding to labeling the parabolas passing through the roots.

There is no covering mapping between these two covering spaces, and the fiberedproduct over N of the two covering spaces is a 48-fold cover, corresponding tolabeling both the roots and the parabolas. Despite this, we will treat both forms asthe “same family”, and label NF the Newton maps associated to both.

The relation between the families does not respect reality: in the case wheretwo real conics intersect in a real non-convex quadrilateral, the Newton map in thefirst form has real coefficients, but not in the second form: the change of variablesbringing this quadrilateral to the second form is not real, since the parabolas in thepencil of conics passing by those points are not real.

1.7. The one-variable rational functions associated to the roots

At each root a of F , i.e., at each fixed point of NF , the quadratic terms of NFinduce a rational function Ga on the projective line Pa associated to the tangentspace to C2 at a. This projective line can also be understood as the exceptionaldivisor one obtains if the root is blown up.

Our two normalizations of Newton’s method give this rational function in twodifferent forms. In the representation (1.59), it is easy to see the three fixed points ofthe rational function; they are the points of Pa corresponding to the three invariantlines through a. In particular, for the root at the origin they are the points of Pacorresponding to the axes, and to the line of equation αy = βx.

In the representation (1.64), on the other hand, the points of Pa correspondingto vertical and horizontal lines are the critical points of ga. Thus we will naturallyfind ga in the form

z 7→ z2 + cz2 + d

, (1.72)

with critical points at 0 and ∞.Comparing these two normalizations, we see that in the normalization (1.64), the

rational function ga is never a polynomial, or conjugate to a polynomial. Indeed,polynomials have a fixed critical point, and the vertical or horizontal lines are neverinvariant lines of NF in the form (1.64). In the form (1.59), polynomials do arise,precisely when the roots are on pairs of parallel lines.

Since we know where the roots are, it is easiest to deal with the form (1.59). Thequadratic terms at the origin give the rational function

z2 +BzAz + 1

(1.73)

with fixed points

0, with multiplier B;

∞, with multiplier A;β

α, with multiplier α+ β.

(1.74)

Newton in C2 33

Note that these multipliers do satisfy the Fatou relation [Mi2]∑ 11−mi

= 1 , (1.75)

where the sum is over the fixed points, and the mi are the multipliers of the fixedpoints. In the case where one of the multipliers is 1, a more elaborate relation isrequired, but a look at equations (1.74) shows that this case does not arise: it wouldrequire α = 0, or β = 0, or α + β = 1, i.e., that three roots be on a line, and wehave seen that in that case, there is no Newton map, even degenerate.

From this we can easily read off the multipliers of the other roots. The affinemap that permutes the roots as follows:(

αβ

)7→(

10

)7→(

00

)7→(

01

)7→(α′β′)

(1.76)

is (xy

)7→(

y/β−x−Ay + 1

)(1.77)

and in particular(α′β′)

=(

1/β(α+ β − 1)/β

).

Thus the multipliers of the fixed points previously at(

10

), now at

(00

), are

A′ =β − 1

α+ β − 1 , B′ = 1− α , and α′ + β′ = α+ β

β. (1.78)

One can of course continue this way for the other roots.Figure 14 illustrates the multipliers.

B BB-1

A α+β

α+βα+β−1 α−1

α

β−1 β

1 - β

AA - 1

α+β β

α+β α

10 ((

00 ((

01 ((

αβ ((

1−α

Figure 14. Each invariant line corresponds to a fixed point of a rational function at eachroot it passes through; this picture represents the corresponding multipliers.

There is a way to condense this information into a cleaner algebraic form (whichstill gives no relation between the dynamics of the various rational maps). Notethat one way of parametrizing rational functions with a distinguished fixed point isby the multiplier m of that fixed point, and the product µ of the multipliers at the


other two fixed points. We can then summarize the information in Figure 14 by thefollowing statement.

inProposition 1.7.1. Let L be an invariant line of NF , and suppose m,µ andm′, µ′ are the coordinates of the two rational functions corresponding to rootson L, with m and m′ being the multipliers of of the fixed points correspondingto L. Then

m′ =m

m− 1 and µ = µ′. (1.79)

2

Invariant 3-manifolds associated to invari-ant circles

The most striking feature of computer drawings for Newton’s method in thecomplex is the apparent existence of smooth 3-manifolds in the dynamical plane,which make up most of the apparent boundary of the basins. We will prove inTheorem 2.4.1 that under appropriate circumstances, there actually are invariant3-manifolds in P2, which do belong to the boundaries of the basins.

In computer pictures, these manifolds look remarkably smooth. The picturesare misleading; sections of these manifolds by complex lines are quasi-circles, or atleast quasi-arcs, which are non-differentiable on a dense subset (though they maybe differentiable almost everywhere).

These manifolds are some sort of “stable manifolds” of invariant circles. Buttheir existence does not follow from any variant of the stable manifold theorem, asthere are always parts of them that are not attracted to the circles. In the finalanalysis, we control the topological structure of these manifolds using a completelydifferent tool, the λ-lemma of [MSS] concerning holomorphic motions.

We also prove, in Theorem 2.3.1, that every point at infinity except the pointsof indeterminacy has an unstable manifold. This is much less delicate than the pre-vious result, and explains most of the self-similarity one sees in computer pictures:the points that appear to be centers of self-similarity are simply inverse images ofpoints at infinity.

In Sections 2.1 and 2.2 we will write our Newton map in the form (1.59).

2.1. The circles in the invariant lines

In each invariant line, the bissector of the roots is invariant under NF , and if we addthe point at infinity, this bissector becomes a circle, in which NF is angle-doubling.More specifically, consider the x-axis. The bissecting line is parametrized by

θ 7→ 12− i

2cot θ, and NF

(12− i

2cot θ

)=

12− i

2cot 2θ. (2.1)

Of course, NF is expanding along that line; we will compute how it behaves inthe normal direction. Since the x-direction is invariant, the derivative is triangular,and the derivative in the normal direction is given by the linear terms in y of thesecond component of NF , i.e., by

Bxy −Bx2y−2Bx2 +Bx+ 2x− 1 =

Bx(1− x)(2x− 1)(1−Bx)y. (2.2)

Set x = (1 + it)/2; we want to compute the average of this second component withrespect to the invariant measure

1π|dθ| = 1

π

|dt|(1 + t2)

, (2.3)

35


we see that the logarithm of the average is given by

1π

∫ ∞−∞

log∣∣∣∣B((1 + it)/2)((1− it)/2)it(1−B(1 + it)/2)

∣∣∣∣ dt1 + t2 . (2.4)inTheorem 2.1.1. The Lyapunov numbers of the circle Rex = 1/2, y = 0 are

λ1 = 2, corresponding to the circle itself, and

λ2 =

{ |B| if the point of indeterminacy 1/B is in the basin of 1∣∣∣ BB−1 ∣∣∣ if the point of indeterminacy 1/B is in the basin of 0. (2.5)The rational functions g(

00

) and g(10

) corresponding to the roots on the x-axis each have a fixed point corresponding to the x-axis itself, with multipliersrespectively B and B/(B − 1). Moreover, one of these roots attracts the point ofindeterminacy q1 =

(1/B

0

), unless Re (1/B) = 1/2.

So we see that Theorem 2.1.1 can be restated without reference to the x-axis, soit applies to the invariant circles in any invariant line.

inTheorem 2.1.1’. The second Lyapunov exponent λ2 of the invariant circlein an invariant line l is the absolute value of the multiplier of the fixed pointcorresponding to l of the rational function at the root in l that does not attractthe point of indeterminacy in l.

The second Lyapunov exponent λ2 above always satisfies |λ2 − 1| ≤ 1, so we dohave 0 ≤ λ2 ≤ λ1.

If the point of indeterminacy is on the circle, then |B| = |B/(1−B)|, and in thatcase this number is λ2.

Proof. By the ergodic theorem, the second Lyapunov exponent exists almosteverywhere, and is an invariant function on the circle. Angle doubling is an ergodicmap of the circle to itself, hence this function is constant (almost everywhere), andequal to its space average. In other words, we have

λ2 = exp(

1π

∫ ∞−∞

log∣∣∣∣B((1 + it)/2)((1− it)/2)it(1−B(1 + it)/2)

∣∣∣∣ dt1 + t2)

= exp(

1π

∫ ∞−∞

log∣∣∣∣ B(1 + t2)2it(2−B(1 + it))

∣∣∣∣ dt1 + t2).

(2.6)

Evaluating this integral is an entertaining exercise in the calculus of residues.Write it as the sum of

1π

∫ ∞−∞

(log |B|− log 2 + log |t+ i|+ log |t− i|− log |t|

) dt1 + t2

= log |B| − log 2 + log 2 + log 2 + 0(2.7)

and1π

∫ ∞−∞

log∣∣∣∣ 12−B(1 + it)

∣∣∣∣ dt1 + t2 ={ − log |1−B| − log 2 if Re (1/B) < 1/2− log 2 if Re (1/B) > 1/2.

(2.8)

Newton in C2 37

The only terms requiring care in the first integral are the ones involving log |t±i|.To evaluate them we observe that the function log(z + i) has an analytic branchin the upper half-plane, so we can close the interval [−R,R] with a half-circle inthe upper half-plane, to form a closed curve ΓR bounding a half-disc containing thepole of 1 + z2 at i. Applying the residue formula, we find

limR→∞

∫ΓR

log(z + i)dzz2 + 1

= 2πilog 2i

2i= π log 2i. (2.9)

In the standard way, let R→∞, so that the contribution from the half-circle tendsto 0. Finally, we are only interested in the real part, giving π log 2.

The third integral is evaluated by residues also; we need to avoid the zero of1−B(1 + iz)/2, which is in the upper half-plane or the lower half-plane dependingon the sign of Re (1/B) − 1/2, so we integrate around a semi-disc in the otherhalf-plane.

In the final analysis, the average normal expansion on the circle is{ ∣∣∣ B1−B ∣∣∣ if Re (1/B) < 1/2|B| if Re (1/B) > 1/2.

(2.10)

This makes sense: one of the two roots on that line is distinguished, the onethat attracts the point of indeterminacy 1/B. What we are finding depends onthe non-distinguished point, and is the multiplier of the fixed point of the rationalfunction associated to the invariant line itself.

Note that the multiplier is always a number in [0, 2]; the value 0 occurs whenB = 0, i.e., β = 1, which means that the roots are on the two parallel lines y = 0 andy = 1. This is the first degenerate case of section 1.6, where we see that the parallellines are superattracting, so the circles on them are superattracting attracting inthe normal direction.

The value 2 is realized exactly if B = 2, i.e., when (α, β) is on the line of equation2α+β = 1. This means that the average of two roots, i.e., the point halfway betweenthem, is on the line connecting the other two roots.

This happens for instance for the case α = β = 1/3. This particular case (notthe general one where a circle has normal derivative 2) is an especially symmetricsituation, when one root is at the center of gravity of the other three. Up toconjugacy, there is only one such Newton map, which also arises when solving thetwo equations

x2 = y, y2 = x, (2.11)

when the roots are (00

),(

11

),(ωω2),(ω2

ω

)with ω a primitive cube root of 1; in this case again

(00

)is at the center of gravity

of the other three roots.

Pesin theory and stable manifolds

Pesin theory, together with Theorem 2.1.1, implies that if λ2 < 1, then the invariantcircle is an “ergodic attractor” [PS]. In particular, almost every point has a stable


manifold, which is a Riemann surface in C2 made up of points attracted to thecircle, and which is part of the separator of the basins. This gives some sort ofa “measure theoretic” 3-dimensional real manifold in the separator for each circlewith λ2 < 1. But we have no precise idea what the topological structure of such anobject is. Later, we will show that under appropriate circumstances its closure is agenuine topological manifold.

2.2. Periodic cycles on invariant circles

The map φ : C→ P2 given by

φ(u) 7→(u/(u− 1)

0

)(2.12)

conjugates u 7→ u2 to Newton’s method. In particular, it maps the unit circle to theinvariant circle in the x-axis. Set ζm,l = e2πil/(2

m−1), so that ζ2m

m,l = ζm,l. Suppose

that ζ2m′

m,l 6= ζm,l for all m′ with 1 < m′ < m. Then the points

φ(ζm,l), φ(ζ2m,l), . . . , φ(ζ2m−1

m,l ) (2.13)

are precisely the cycles of length m on the invariant circle in the x-axis. Thederivative of NF at φ(ζ) has eigenvalues

2ζ andBζ

(ζ + 1)(1 + ζ(B − 1)

) . (2.14)Set ζ = ζm,l. The second eigenvalue of the corresponding cycle is

Mm,l =m−1∏k=0

Bζ2k

(ζ2k + 1)(1 + ζ2k(B − 1)

) = Bm∏m−1k=0

(1 + ζ2k(B − 1)

) . (2.15)Let us check this simplification when m = 3 so ζ7 = 1. In the deniminator, we find(1+ζ)(1

table of contentspi.math.cornell.edu/~hubbard/newtonseveral.pdf · 4.7. the action of mappings fˆ...

Documents