Universal Curves in the Center Problem for Abel Differential Equations
UUNIVERSAL CURVES IN THE CENTER PROBLEM FOR ABELDIFFERENTIAL EQUATIONS
ALEXANDER BRUDNYI
Abstract.
We study the center problem for the class E Γ of Abel differential equa-tions dvdt = a v + a v , a , a ∈ L ∞ ([0 , T ]), such that images of Lipschitz paths˜ A := (cid:0)(cid:82) · a ( s ) ds, (cid:82) · a ( s ) ds (cid:1) : [0 , T ] → R belong to a fixed compact rectifiable curve Γ.Such a curve is called universal if whenever an equation in E Γ has center on [0 , T ], thiscenter must be universal, i.e. all iterated integrals in coefficients a , a of this equationmust vanish. We investigate some basic properties of universal curves. Our main resultsinclude an algebraic description of a universal curve in terms of a certain homomorphismof its fundamental group into the group of locally convergent invertible power serieswith product being the composition of series, explicit examples of universal curves andapproximation of Lipschitz triangulable curves by universal ones. Introduction
In the paper we study the center problem for the Abel differential equation(1.1) dvdt = a v + a v with coefficients a , a in the space L ∞ ([0 , T ]) of bounded Lebesgue measurable real func-tions on an interval [0 , T ] (cid:98) R . Recall that equation (1.1) has a center on [0 , T ] if for allsufficiently small initial values v the corresponding solution satisfies v ( T ) = v (0) := v .The center problem is to describe explicitly the set of coefficients a , a for which thecorresponding equations (1.1) have centers on [0 , T ]. The problem is closely related tothe classical Poincar´e Center-Focus problem for planar polynomial vector fields, see, e.g.,[BRY], [I] and references therein. Recently there has been an intensive study of the cen-ter problem for Abel differential equations focused on some composition conjectures forequations with piecewise smooth coefficients, see [AL, A1, A2, A3, A4, B2, B3, B4, B5,B6, B7, BlRY, BFY1, BFY2, BRY, BY, C, CGM1, CGM2, CGM3, GGL, P1, P2, PRY].It is known, see, e.g., [B1], that the set of centers of equation (1.1) consists of pairs a = ( a , a ) satisfying an infinite system of equations c i = 0, i ∈ N , where(1.2) c i := (cid:88) i + ··· + i k = i c i ,...,i k ( i ) · I i ,...,i k ( a ) , all i , . . . , i k ∈ { , } ,c i ,...,i k ( i ) := ( i − i + 1)( i − i − i + 1)( i − i − i − i + 1) · · · ,I i ,...,i k ( a ) := (cid:90) · · · (cid:90) ≤ s ≤···≤ s k ≤ T a i k ( s k ) · · · a i ( s ) ds k · · · ds . In general, qualitative analysis of this system is highly arduous because of complexity ofthe involved equations and absence of apparent recursive relations between them. The
Mathematics Subject Classification.
Primary 34C07; Secondary 37C27.
Key words and phrases.
Abel equation, center problem, universal curve, first return map, fundamentalgroup.Research supported in part by NSERC. a r X i v : . [ m a t h . C A ] M a y ALEXANDER BRUDNYI simplest and, in a certain statistical sense, the most frequently occurring centers, so-called universal centers , are defined by vanishing of all iterated integrals I i ,...,i k ( a ). Such centersadmit an equivalent characterization in terms of the tree composition condition (see [B2,Cor. 1.12, Th. 1.14]): Suppose that image Γ ˜ A of the map (1.3) ˜ A := (˜ a , ˜ a ) : [0 , T ] → R , where ˜ a i ( t ) := (cid:90) t a i ( s ) ds, i = 1 , , is Lipschitz triangulable. Then equation (1.1) has universal center on [0 , T ] if and only ifthere are a finite metric tree T and continuous maps ˆ A : [0 , T ] → T , ˆ A (0) = ˆ A ( T ) , and p : T → Γ ˜ A such that ˜ A = p ◦ ˆ A . Recall that Γ (cid:98) R is a Lipschitz triangulable curve if there exist a finite subset S ⊂ Γand Lipschitz and locally bi-Lipschitz arcs h , . . . , h k : (0 , → R such that Γ \ S = (cid:116) ≤ i ≤ k h i (0 , R admitting piecewise C parametrization or images of nonconstant analyticmaps [0 , T ] → R , see, e.g., [BY, Ex. 5.1].It was proved in [B6, Sec. 3.1] that if ˜ A is nonconstant analytic, then tree T in theabove characterization of universal centers of equation (1.1) is a closed interval in R . Inthis case we say that ˜ A satisfies a composition condition . In general, tree compositionand composition conditions are not equivalent, i.e. there exist Lipschitz maps [0 , T ] → R which factor through continuous maps into non-interval trees but do not factor throughcontinuous maps into intervals, see [BY, Sec. 3].The famous composition conjecture states that all centers of equations (1.1) with poly-nomial coefficients are universal (equivalently, for all centers of such equations the corre-sponding maps ˜ A satisfy composition conditions). This conjecture is false already for Abelequations with coefficients being trigonometric polynomials, see [BFY1] for the distinctionbetween these two cases.One associates with each connected Lipschitz triangulable curve Γ (cid:98) R containing theorigin an (uncountable) family of equations (1.1) with Γ ˜ A ⊂ Γ. Universal centers of thisfamily admit a simple topological description. Specifically, equation (1.1) with Γ ˜ A ⊂ Γhas universal center on [0 , T ] if and only if path ˜ A : [0 , T ] → Γ is closed and contractiblein Γ. This is equivalent to the above formulated tree composition condition for ˜ A , see[B2] for details. At present, there is no way to explicitly describe nonuniversal centersof Abel differential equations. However, there exist Lipschitz triangulable curves Γ (cid:98) R satisfying the property that whenever an equation (1.1) with Γ ˜ A ⊂ Γ has a center on [0 , T ],this center must be universal. We call such curves Γ universal . The basic examples ofuniversal curves are (containing 0 ∈ R ) connected rectangular curves composed of finitelymany intervals each parallel to one of the coordinate axes, see [B4], or connected Lipschitztriangulable curves whose fundamental groups are either trivial or isomorphic to Z , see[B2].In the present paper we study analytic and geometric properties of universal curves. Inparticular, we show that the set of “sufficiently smooth” universal curves is dense withrespect to the Hausdorff metric on compact subsets of R in the space of all connectedLipschitz triangulable curves in R containing the origin.The paper is organized as follows. In the next section we formulate the main resultsof the paper accompanied by some important open problems in the area. These resultsinclude an algebraic description of a universal curve in terms of a certain homomorphismof its fundamental group into the group of locally convergent invertible power series with NIVERSAL CURVES IN THE CENTER PROBLEM FOR ABEL DIFFERENTIAL EQUATIONS 3 product being the composition of series, explicit examples of universal curves and approx-imation of Lipschitz triangulable curves by universal ones. Section 3 is devoted to proofsof our results. 2.
Formulation of Main Results
A Characterization of Universal Curves.
We recall some results presented in[B2]. Let us introduce operations ∗ and − on the set A of pairs a = ( a , a ) of coefficientsof equations (1.1) by the formulas(( a , a ) ∗ ( b , b )) ( t ) := (cid:40) (2 b (2 t ) , b (2 t )) if 0 ≤ t ≤ T (2 a (2 t − T ) , a (2 t − T )) if T ≤ t ≤ T ; (cid:0) ( a , a ) − (cid:1) ( t ) := ( − a ( T − t ) , − a ( t − T )) , ≤ t ≤ T. It is well known that for a, b ∈ A iterated integrals I i ,...,i k , see (1.2), satisfy(2.1) I i ,...,i k ( a ∗ b ) = I i ,...,i k ( a ) + k − (cid:88) j =1 I i ,...,i j ( a ) · I i j +1 ,...,i k ( b ) + I i ,...,i k ( b ) . (2.2) I i ,...,i k ( a − ) = ( − k I i ,...,i k ( a ) . For a ∈ A by P ( a ) we denote the first return map of the corresponding equation (1.1)(associating with a sufficiently small initial value r the value of the corresponding solutionof (1.1) at T ). It is represented as a convergent in a neighbourhood of 0 power series inthe initial value r of the Abel equation,(2.3) P ( a )( r ) = r + ∞ (cid:88) i =1 c i r i +1 , where c i is the weighted sum of iterated integrals in (1.2).By G c [[ r ]] we denote the group of convergent in neighbourhoods of 0 power series ofform (2.3) with product ◦ defined by the composition of series. Then for a, b ∈ A we have(2.4) P ( a ∗ b ) = P ( a ) ◦ P ( b ) , P ( a − ) = P ( a ) − . An important question is (cf. [B3, Question 2, page 481]):
Problem 1.
Is it true that the first return map P : A → G c [[ r ]] is an epimorphism (i.e.each series in G c [[ r ]] is the first return map of an Abel equation (1.1) )? Next, let Γ (cid:98) R be a connected Lipschitz triangulable curve containing the origin.According to the definition, Γ is a finite connected one-dimensional CW -complex and soit is homotopically equivalent to the wedge sum of finitely many circles. In particular, thefundamental group π (Γ) of Γ with base point 0 ∈ R is isomorphic, for some m ∈ Z + , tothe free group with m generators F m . (Here F stands for the trivial group.)By A (Γ) we denote the subset of elements a ∈ A such that ˜ A ( T ) = 0 and ˜ A ([0 , T )) :=Γ ˜ A ⊂ Γ (recall that ˜ A ( t ) := (cid:82) t a ( s ) ds , t ∈ [0 , T ], see (1.3)). Since Γ admits a Lipschitztriangulation, the set A (Γ) is uncountable. (For instance, the derivative of a closedLipschitz path passing through the origin and determined as the product, in the sense ofalgebraic topology, see, e.g., [Hu], of Lipschitz arcs forming the triangulation of Γ belongsto A (Γ).) Clear A (Γ) is closed with respect to operations ∗ and − . Next, for a ∈ A (Γ)the closed Lipschitz path ˜ A represents an element [ ˜ A ] of π (Γ), and the correspondence A (Γ) (cid:51) a (cid:55)→ [ ˜ A ] ∈ π (Γ) determines an epimorphism of monoids Ψ Γ : A (Γ) → π (Γ):(2.5) Ψ Γ ( a ∗ b ) = Ψ Γ ( a ) · Ψ Γ ( b ) , Ψ Γ ( a − ) = Ψ Γ ( a ) − , a, b ∈ A (Γ) . ALEXANDER BRUDNYI
Since for each γ ∈ π (Γ) the first return map P is constant on Ψ − ( γ ), see [B2], thereexists a homomorphism ˆ P Γ : π (Γ) → G c [[ r ]] such that(2.6) P = ˆ P Γ ◦ Ψ Γ on A (Γ) . Theorem 2.1.
Curve Γ is universal if and only ˆ P Γ is a monomorphism. In particular, this implies that the group ˆ P Γ ( π (Γ)) ⊂ G c [[ r ]] is isomorphic to F m . Problem 2.
Describe all possible finitely generated subgroups of G c [[ r ]] . For instance, one can show that a “generic” finitely generated subgroup of G c [[ r ]] isfree and each finitely generated solvable subgroup of G c [[ r ]] is isomorphic to free abeliangroup Z m , where m can be any natural number. However, it is even not known whetherthe fundamental group of a compact Riemann surface of genus ≥ G c [[ r ]], see, e.g., [B4, B5] and references therein.2.2. Explicit Examples of Universal Curves.Proposition 2.2.
Lipschitz triangulable curves Γ ⊂ R containing the origin whose fun-damental groups π (Γ) are either isomorphic to F (i.e. trivial) or isomorphic to F ( ∼ = Z ) (see Figure 1 (A), (B) below) are universal. Figure 1. Examples of universal curves Γ: (A) π (Γ) is trivial; (B) π (Γ) ∼ = Z ;(C) a rectangular curve.A curve Γ (cid:98) R is called rectangular if it the union of finitely many intervals each parallelto one of the coordinate axes (see Figure 1 (C)). In [B4, Th. 2.1] it was proved that everyconnected rectangular curve Γ containing the origin is universal . The proof is heavily relyupon a deep result of [Co] on the structure of a certain subgroup of automorphisms of C .We exploit the result of [B4] in the proof of Theorem 2.7 but now we use its corollary [B5,Th. 2.4] to describe other explicit examples of universal curves.Suppose P , P , Q , Q are real polynomials without constant terms such that coeffi-cients of all of them together are algebraically independent over Q . Consider polynomialpaths A i := ( P i , Q i ) : [0 , → R , i = 1 ,
2. Next, define the set of points S := { ( m · A (1) , n · A (1)) ∈ R : ( m, n ) ∈ Z } . NIVERSAL CURVES IN THE CENTER PROBLEM FOR ABEL DIFFERENTIAL EQUATIONS 5
We join each pair of points v , v ∈ S such that v − v = A (1) by curve X v ,v := { v + A ( t ) : t ∈ [0 , } and each pair of points w , w ∈ S such that w − w = A (1)by curve Y w ,w := { w + A ( t ) : t ∈ [0 , } thus obtaining a curvilinear lattice L in R .Assume that polynomials P i , Q i were chosen so that maps A i : [0 , → R are embeddingsand all possible curves X v ,v and Y w ,w (referred to as edges of L ) are either disjoint orintersect by one of the points v , v , w , w only (see Figure 2). Then the following resultholds. Proposition 2.3.
Each connected curve Γ (cid:98) L containing the origin composed of finitelymany edges of L is universal. Figure 2. An example of a curve Γ composed of finitely many edges of a lattice L .The following problem is a particular case of [B3, Question 3, page 484]: Problem 3.
Is it true that any connected piecewise linear curve Γ (cid:98) R containing theorigin is universal? Remark 2.4.
Some piecewise linear curves are universal due to Proposition 2.3. Ingeneral, one easily show that the first return map of the Abel equation dvdt = av + bv , t ∈ [0 , a, b ∈ R , | a | + | b | (cid:54) = 0, is P a,b ( r ) := ψ − a,b (cid:0) a + ψ a,b (cid:0) r (cid:1)(cid:1) if a (cid:54) = 0 r √ − br if a = 0; (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) here ψ a,b ( x ) := − ax + b ln | ax + b | , x ∈ R . (Note that ψ a,b is invertible near ±∞ and P a,b admits a series expansion near 0.)Then Problem 3 is equivalent to the following one (cf. [B4] and the main result of [Co]): Problem 3 (cid:48) . Is it true that a composition P a ,b ◦ P a ,b ◦ · · · ◦ P a n ,b n such that all vectors ( a i , b i ) and ( a i +1 , b i +1 ) , ≤ i ≤ n − , n ∈ N , are noncollinear is never the identity map? Lipschitz Embeddings of Universal Curves. By (cid:107) · (cid:107) we denote the Euclideannorm on R . Let K (cid:98) R be a compact subset containing the origin. A continuous map F : K → R is called a Lipschitz embedding of K if there exist constants c , c > c (cid:107) x − y (cid:107) ≤ (cid:107) F ( x ) − F ( y ) (cid:107) ≤ c (cid:107) x − y (cid:107) for all x, y ∈ K. The set of Lipschitz embeddings of K sending 0 to 0 will be denoted by EL ( K ). Clearly if F ∈ EL ( K ), then F − exists and belongs to EL ( F ( K )). By I ∈ EL ( K ) we denote theidentity map. One easily shows that if F ∈ EL ( K ) satisfies (2.7), then F + λI ∈ EL ( K )for all λ ∈ ( − c , c ). ALEXANDER BRUDNYI
Theorem 2.5.
Let Γ (cid:98) R be a universal curve and F ∈ EL (Γ) satisfy (2.7) for some c , c > . Then there is an at most countable subset S of the open interval ( − c , c ) suchthat for all λ ∈ ( − c , c ) \ S the Lipschitz triangulable curves ( F + λI )(Γ) are universal. In Remark 3.1 we give an explicit description of set S .Let L ( K ) be the Banach space of Lipschitz maps F : K → R such that F (0) = 0equipped with norm (cid:107) F (cid:107) := sup x (cid:54) = y (cid:107) F ( x ) − F ( y ) (cid:107) (cid:107) x − y (cid:107) . One easily shows that if F ∈ EL ( K ) satisfies (2.7), then F + G ∈ EL ( K ) for all G ∈L ( K ) such that (cid:107) G (cid:107) < c . In particular, EL ( K ) is a nonempty open subset of L ( K ).Suppose Γ (cid:98) R is a universal curve. Let U (Γ) ⊂ EL (Γ) be the subset of Lipschitzembeddings F : Γ (cid:44) → R , F (0) = 0, such that curves F (Γ) are universal. We equip U (Γ)with topology induced from EL (Γ). The following result shows that U (Γ) is a “massive”dense subset of EL (Γ). Theorem 2.6.
There exists an at most countable family of closed nowhere dense subsets S i ⊂ EL (Γ) such that U (Γ) = EL (Γ) \ (cid:0) ∪ i S i (cid:1) . In view of this result, the following problem seems to be plausible:
Problem 4.
Let Γ (cid:98) R be a connected Lipschitz triangulable curve containing the origin.Is it true that there exists an F ∈ EL (Γ) such that curve F (Γ) is universal? Approximation of Lipschitz Triangulable Curves by Universal Ones.
Inwhat follows, H denotes the Hausdorff 1-measure on R and d H the Hausdorff metric onthe set of compact subsets of R . Let Γ (cid:98) R be a Lipschitz triangulable curve containingthe origin. We say that x ∈ Γ has order k (written ord( x ) = k ) if for all sufficiently smallopen disks D x centered at x , Γ \ { x } has k connected components in D x \ { x } . (Note thatsince Γ is triangulable, ord( x ) is correctly defined.) By S Γ ⊂ Γ we denote the (finite) setof points of order (cid:54) = 2. Our main result is
Theorem 2.7.
Suppose Γ is piecewise linear. Then there exists a sequence { Γ i } i ∈ N ofLipschitz triangulable curves containing the origin such that (a) Each Γ i is a universal curve with π (Γ i ) ∼ = π (Γ) ; (b) ord( x ) ≤ for all x ∈ Γ i , i ∈ N ; (c) Each Γ i \ S Γ i is a closed C ∞ submanifold of R \ S Γ i ; (d) lim i →∞ H (Γ i ) = H (Γ) and lim i →∞ d H (Γ i , Γ) = 0 . If (d) is valid for a sequence of curves { Γ i } i ∈ N and a curve Γ, then we say that { Γ i } i ∈ N converges to Γ with respect to H -measure and metric d H . Since each Lipschitz triangula-ble curve can be approximated with respect to H -measure and metric d H by a sequenceof piecewise linear curves (because each Lipschitz arc [0 , → R admits such approxima-tion), as an immediate corollary of Theorem 2.7 we obtain: Corollary 2.8.
Suppose Γ (cid:98) R is a Lipschitz triangulable curve containing the origin.Then there exists a sequence { Γ i } i ∈ N of universal curves converging to Γ with respect to H -measure and metric d H such that each Γ i \ S Γ i is a closed C ∞ submanifold of R \ S Γ i . The following problem is related to the composition conjecture for Abel equations withpolynomial coefficients (see the Introduction).
NIVERSAL CURVES IN THE CENTER PROBLEM FOR ABEL DIFFERENTIAL EQUATIONS 7
Problem 5.
Do there exist universal curves Γ (cid:98) R with π (Γ) ∼ = F m , m ≥ , whichare images of analytic maps [0 , T ] → R ? The same question is for images of polynomialmaps. Regarding to this problem we mention that universal centers of Abel equations withreal analytic coefficients can be easily described in terms of vanishing of finitely manymoments in the coefficients of the equations, see the algorithm in the Introduction of [B7].2.5.
Remarks on Nonuniversal Curves.
While the structure of universal centers ofequation (1.1) is well understood, nonuniversal centers of this equation are of obscurenature requiring further scrutiny. If equation (1.1) has a nonuniversal center on [0 , T ]and the image Γ ˜ A of the corresponding map ˜ A is Lipschitz triangulable, then Γ ˜ A is anonuniversal curve; in particular, homomorphism ˆ P Γ ˜ A : π (Γ ˜ A ) → G c [[ r ]] (see (2.6))has a nontrivial kernel (enclosing the minimal normal subgroup containing the element[ ˜ A ] ∈ π (Γ ˜ A ) represented by path ˜ A ). This implies that for every b = ( b , b ) ∈ A (Γ ˜ A )such that the closed path ˜ B := (cid:82) · b ( s ) ds : [0 , T ] → Γ ˜ A is homotopic to a nontrivialelement of Ker ˆ P Γ ˜ A the corresponding Abel equation dvdt = b v + b v has a nonuniversalcenter on [0 , T ]. Abel equations with nonuniversal centers can be obtained, e.g., by theapplication of the Cherkas transformation to certain elements of the Lotka-Volterra orDarboux components of the set of centers of planar polynomial vector fields of degree2, see [Bl, Sect. 2.5.3]. In these case coefficients ( a , a ) of the obtained Abel equationsare trigonometric polynomials of degrees ≤
4; hence the corresponding curves Γ ˜ A areLipschitz triangulable (and nonuniversal). We mention also the following simple property:if Γ ⊂ ˜Γ (cid:98) R is a pair of connected Lipschitz triangulable curves containing the originand Γ is nonuniversal, then ˜Γ is nonuniversal as well.3. Proofs
Proof of Theorem 2.1.
First, assume that Γ is universal. Suppose ˆ P ( g ) = 1 ∈ G c [[ r ]] forsome g ∈ π (Γ). Let a g ∈ A (Γ) be such that Ψ Γ ( a g ) = g . Then due to (2.6), P ( a g ) = 1,i.e. equation (1.1) corresponding to a g has center on [0 , T ]. Since Γ is universal, this centeris universal. Then by [B2, Th. 1.14], the closed Lipschitz path ˜ A g := (cid:82) · a g ( s ) ds : [0 , T ] → Γis contractible, that is g = Ψ Γ ( a g ) = 1 ∈ π (Γ). This shows that ˆ P Γ is a monomorphism.Conversely, suppose that ˆ P Γ is a monomorphism. Let a ∈ A (Γ) be such that thecorresponding equation (1.1) has center on [0 , T ]. Due to (2.6) and our assumption thisimplies that Ψ Γ ( a ) = 1 ∈ π (Γ). Then the closed Lipschitz path ˜ A : [0 , T ] is contractible.In turn, by the results of [B2], the corresponding center is universal. So Γ is universal byour definition. (cid:3) Proof of Proposition 2.2. If π (Γ) = F , then ˆ P Γ : F → G c [[ r ]] is obviously a monomor-phism; thus such curve Γ is universal by Theorem 2.1.Next, if π (Γ) ∼ = F but Γ is not universal, then due to Theorem 2.1 ˆ P Γ maps π (Γ)to 1 ∈ G c [[ r ]]. Let g be a generator of π (Γ) and a g = ( a g , a g ) ∈ A (Γ) be such thatΨ( a g ) = g . Then according to our hypothesis and (2.6), equation (1.1) corresponding to a g has center on [0 , T ]. In particular, from (1.2) with i = 3 (see also notation (1.3)) weobtain(3.1) 3 · (cid:90) T ˜ a g ( s ) · a g ( s ) ds + 2 · (cid:90) T ˜ a g ( s ) · a g ( s ) ds = 0 . ALEXANDER BRUDNYI
Also, since ˜ a gi ( T ) = 0, i = 1 ,
2, we have (cid:90) T ˜ a g ( s ) · a g ( s ) ds + (cid:90) T ˜ a g ( s ) · a g ( s ) ds = (cid:0) ˜ a ( t ) · ˜ a ( t ) (cid:1) | T = 0 . Thus (3.1) is equivalent to(3.2) (cid:90) T ˜ a g ( s ) · a g ( s ) ds = 0 . The latter can be rewritten as the contour integral of 1-form xdy over the closed pathˆ A g := (cid:82) · a g ( s ) ds : [0 , T ] → Γ: (cid:90) ˆ A g xdy = 0 . Since π (Γ) ∼ = Z and Γ is Lipschitz triangulable, by the Jordan theorem R \ Γ containsonly one bounded component D Γ homeomorphic to the open unit disk. In turn, by ourdefinition ˆ A g represents the generator g ∈ π (Γ) and so by the Green formula we obtain0 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ˆ A g xdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) D Γ dxdy (cid:12)(cid:12)(cid:12)(cid:12) = Area ( D ) (cid:54) = 0 , a contradiction which shows that ˆ P Γ ( g ) (cid:54) = 1, i.e. by Theorem 2.1 Γ is a universal curve. (cid:3) Proof of Proposition 2.3.
Let Γ (cid:98) L be a curve satisfying conditions of the proposition.According to Theorem 2.1 we have to prove that homomorphism ˆ P Γ : π (Γ) → G c [[ r ]] isinjective. Consider elements a i := A (cid:48) ∈ A , i = 1 , (cid:48) stands for the derivative of apath). According to [B5, Th. 2.4, Ex. 2.5] the first return maps P ( a i ) ∈ G c [[ r ]], i = 1 , F . Further, each element g ∈ π (Γ) is representedby a path A g : [0 , → Γ which is the product (in the sense of algebraic topology) offinitely many paths A i and their inverses. Assume that ˆ P Γ ( g ) = 1 ∈ G c [[ r ]]. Then dueto (2.6) P ( A (cid:48) g ) = 1 ∈ G c [[ r ]]. Thus, the Abel equation (1.1) corresponding to the pairof coefficients A (cid:48) g has center on [0 , A (cid:48) g is a finite ∗ -product ofelements a i and their inverses. Then due to [B5, Th. 2.4] this center is universal. Thisimplies, see [B2], that path A g is closed and contractible inside its image. But A g hasimage in Γ; hence A g is contractible in Γ. This is equivalent to g = 1 ∈ π (Γ). Thereforeˆ P Γ is a monomorphism and by Theorem 2.1, Γ is a universal curve. (cid:3) Proof of Theorem 2.5.
Let (cid:96) g : [0 , T ] → Γ be a closed Lipschitz path representing anelement g ∈ π (Γ). Then for each λ ∈ ( − c , c ) the countable family of paths { (cid:96) gλ } g ∈ π (Γ) , (cid:96) gλ := ( F + λI )( (cid:96) g ), represents all elements of the fundamental group π (Γ λ ) ( ∼ = π (Γ)),Γ λ := ( F + λI )(Γ). According to formula (1.2) the first return map of the Abel equationcorresponding to the pair of coefficients (cid:96) (cid:48) gλ ∈ A has the form P ( (cid:96) (cid:48) gλ )( r ) = r + ∞ (cid:88) j =1 c gj ( λ ) r j +1 , where c gj is a real polynomial in λ of degree at most j (1 ≤ j < ∞ ).Since Γ is universal, the first return map P ( (cid:96) (cid:48) g ) ⊂ G c [[ r ]] is not the identity map (seeTheorem 2.1). Thus there exists j g ∈ N such that c gj g (0) (cid:54) = 0. This implies that c gj g is anot identically zero polynomial in λ of degree at most j g . By S j g we denote the set of itszeros in ( − c , c ). Clearly, card S j g ≤ j g . We define S := (cid:91) g ∈ π (Γ) S j g . NIVERSAL CURVES IN THE CENTER PROBLEM FOR ABEL DIFFERENTIAL EQUATIONS 9
Since π (Γ) is countable, the set S ⊂ ( − c , c ) is at most countable. By definition, foreach t ∈ ( − c , c ) \ S and g ∈ π (Γ) we have P ( (cid:96) (cid:48) gλ ) (cid:54) = id. This shows that the kernel ofthe corresponding homomorphism ˆ P Γ λ : π (Γ λ ) → G c [[ r ]] (see (2.6)) is trivial. Thereforeby Theorem 2.1, Γ λ is a universal curve.The proof of the theorem is complete. (cid:3) Remark 3.1.
It is easily seen that coefficients of all polynomials c gj belong to the minimalnumerical field F ⊂ R containing all iterated integrals in (cid:96) (cid:48) g and (cid:96) (cid:48) g . Therefore to makesure that all c gj ( λ ) (cid:54) = 0 it suffices to choose λ being a transcendental number over F .Since F is a countable set, its algebraic closure cl a ( F ) is countable as well. Thus we maydefine the required set S of the theorem as S := ( − c , c ) ∩ cl a ( F ). Proof of Theorem 2.6.
We retain notation of the proof of Theorem 2.5. For each (cid:96) g ∈ [0 , T ] → Γ, g ∈ π (Γ), and G ∈ L (Γ) consider the first return map P (cid:0) ( G ◦ (cid:96) g ) (cid:48) (cid:1) = r + ∞ (cid:88) j =1 c gj ( G ) r j +1 . Due to equation (1.2) and the Rademacher theorem (on a.e. differentiability of Lipschitzmaps), the coefficients c gj are real polynomials of degrees at most j on the Banach space L (Γ). Since Γ is universal, P (cid:0) ( I ◦ (cid:96) g ) (cid:48) (cid:1) (cid:54) = id. Therefore there exists j g ∈ N such that c gj g (cid:54)≡
0. In turn, the set of zeros Z gj g of c gj g is a closed nowhere dense subset of L (Γ).We have U (Γ) = EL (Γ) \ (cid:91) g ∈ π (Γ) (cid:92) { j : c gj (cid:54)≡ } Z gj . Since π (Γ) is countable and EL (Γ) is an open subset of L (Γ), the previous identityyields the required statement. (cid:3) Proof of Theorem 2.7.
The proof consists of two parts. In the first part we approximate Γby a sequence of piecewise linear curves satisfying all the above conditions but (c). Thenwe “smooth the corners” of curves of the constructed sequence to get the required sequence { Γ i } . I. We consider Γ as a connected finite plane graph whose vertex set contains S Γ and theorigin. By definition, each vertex not in S Γ has degree 2. First, let us show how to reducethe approximation problem to certain plane graphs whose vertices have degrees ≤ ∈ R by a small angle, without loss of generality wemay assume that all edges of Γ are not parallel to the x -axis. Suppose v ∈ S Γ has degree ≥
3. Let E ( v ) be the set of edges of Γ emanating from v . Let (cid:96) ± := { ( x, y ) ∈ R : y = v ± , sgn( v ± − v y ) = ± } be two lines parallel to the x -axis with y coordinates v + , v − largerand smaller than that of v (denoted by v y ). For sufficiently small | v ± − v y | the union ofthese lines intersect all edges in E ( v ). Let I ± ⊂ (cid:96) ± be closed intervals with endpoints in E ( v ) whose union contains all points in ( (cid:96) + (cid:116) (cid:96) − ) ∩ E ( v ). Consider a piecewise linear curveΓ v obtained from Γ by removing all parts of edges of E ( v ) between (cid:96) − and (cid:96) + except forone part for each choice of sign + or − and then adding intervals I + and I − instead (cf.Figure 3). Figure 3. Example of curves Γ and Γ v .One can easily check that (for all sufficiently small | v ± − v y | ), π (Γ v ) ∼ = π (Γ). (Indeed,in this case according to the Seifert–van Kampen theorem the fundamental group of theunion U of Γ and of the minimal convex set containing I + , I − and all parts of edges of E ( v ) between (cid:96) − and (cid:96) + is isomorphic to π (Γ). Also, Γ v is the deformation retract of U ,i.e., π (Γ v ) ∼ = π ( U ) ∼ = π (Γ) as required.) Moreover, in a neighbourhood of v all verticesof Γ v have degrees ≤ v are the same as for Γ, and the family ofsuch curves Γ v converges to Γ with respect to H -measure and metric d H (cf. (d) of thetheorem) as | v ± − v y | and the above angle of rotation of Γ about the origin tend to 0.Applying subsequently a similar procedure to other vertices of Γ with degrees ≥
3, aftera finite number of steps we obtain a piecewise linear curve of the same homotopy typeas Γ with vertices of degrees ≤ v in a neighbourhoodof v . Moreover, there exists a sequence of such curves converging to Γ with respect to H -measure and metric d H .Thus, from now on without loss of generality we will assume that all vertices of Γ havedegrees ≤ , and if a vertex has degree , then two of three edges adjacent to it lie on astraight line .As a next step, we construct a rectangular piecewise linear curve Γ r by replacing edgesof Γ by unions of cathetii of right triangles with pairwise nonintersecting interiors of theirsides and with hypotenuses lying on edges of Γ (see Figure 4 below). Also, we transferto Γ r edges of Γ parallel to one of the coordinate axes. To show that the required Γ r exists, it suffices to construct the corresponding right triangles locally in neighbourhoodsof vertices of Γ. This is possible since, due to the above property of Γ, each closedquadrant with respect to the coordinate axes centered at a vertex of Γ contains in a smallopen neighbourhood of this vertex no more than 2 edges of Γ.Figure 4. Examples of curves Γ and Γ r . NIVERSAL CURVES IN THE CENTER PROBLEM FOR ABEL DIFFERENTIAL EQUATIONS 11
Since in a right triangle the union of cathetii can be isotopically deformed to the hy-potenuse, the curve Γ r can be isotopically deformed to Γ and, in particular, π (Γ r ) ∼ = π (Γ).We make use of a specific isotopy between Γ r and Γ defined in each right triangle T withhypotenuse lying on an edge of Γ as follows. Using a suitable affine transformation L we map T to the right triangle with coordinates (0 , a,
0) and (0 , b ), a, b >
0. Thehypotenuse in this triangle is given by equation y = ba ( a − x ) for x ∈ [0 , a ]. We deform theunion of cathetii K := { ( x, y ) : x ∈ [0 , a ] , y = 0 } and K := { ( x, y ) : x = 0 , y ∈ [0 , b ] } along the direction of the line y = ba x so that at time t ∈ [0 ,
1] cathetus K transfers tothe line y = tb ( a − x ) a (2 − t ) , x ∈ [ a t, a ], and K to the line x = ta ( b − y ) b (2 − t ) , y ∈ [ b t, b ], see Figure5. Then we apply the inverse affine transformation L − to place the deformed break lineback inside the original triangle T .Figure 5. Isotopy between the union of cathetii and the hypotenuse.Without loss of generality we will assume that the vertex set of Γ is a subset of thevertex set of Γ r . By Φ : Γ r × [0 , → R we denote the above described isotopy betweenΓ r and Γ which deforms cathetii of each right triangle with hypotenuse an edge of Γtoward the hypotenuse by the above formulas (modulo an affine transformation). We setΓ t := Φ( t, Γ r ).Further, each piecewise linear path γ : [0 , T ] → Γ r (i.e. a path composed of finitelymany paths moving along each edge with constant velocities) can be transformed usingΦ to a piecewise linear path γ : [0 , T ] → Γ. We set γ t := Φ( γ, t ), t ∈ (0 , γ t isa piecewise linear path in Γ t . Let γ (cid:48) t ∈ A (see Section 2) be the velocity of path γ t . Werequire the following result. Lemma 3.2.
Each iterated integral I i ,...,i k ( γ (cid:48) t ) is a real polynomial in t of degree ≤ k .Proof. By definition, γ t is product (in the sense of algebraic topology) of finitely manylinear paths with images in edges of γ t . In turn, γ (cid:48) t is ∗ -product of velocities of these linearpaths. Due to formulas for iterated integrals of ∗ -products, see (2.1), (2.2), it sufficesto prove the result for velocities of linear paths appearing as factors of γ t in the aboveproduct. Then by the definition of γ t we must consider the following cases:(1) γ t does not depend on t , this means that the image of γ belongs to a vertical or ahorizontal intervals. In this case the statement of the lemma is obvious.(2) Image of γ t belongs to one of the intervals inside of a straight triangle with hy-potenuse being an edge of Γ (cf. Figure 5). Making an affine transformation of thistriangle (which clearly does not affect the final result) without loss of generality we mayassume that we are in position of Figure 5. Suppose for definiteness that image of γ t belongs to the line y = tb ( a − x ) a (2 − t ) , x ∈ [ a t, a ] (the lower line inside the triangle in Figure 5)and the endpoints of γ t have x -coordinates a t ≤ s < s ≤ a . By the definition of isotopyΦ in this case we have γ t ( s ) := Φ( γ ( s ) , t ) = (cid:18) at − t ) γ ( s )2 , bt − btγ ( s )2 a (cid:19) , s ∈ [0 , T ] . Here γ : [0 , T ] → K ⊂ R is a linear path with image in the cathetus K and endpoints s , s ∈ K . Assuming that γ (cid:48) = v ( ∈ R ), we obtain γ (cid:48) t ( s ) := (cid:18) (2 − t ) v , − btv a (cid:19) , s ∈ [0 , T ] . Since γ (cid:48) t depends linearly on t , we have (see (1.2)) that each iterated integral I i ,...,i k ( γ (cid:48) t )is a real polynomial in t of degree at most k as required.All other choices of γ t are treated similarly. (cid:3) Using Lemma 3.2 we prove
Proposition 3.3.
There exists an at most countable subset Z ⊂ [0 , such that for each t ∈ [0 , \ Z the piecewise linear curve Γ t is universal.Proof. Since Γ r := Γ is a rectangular curve, it is universal, see [B4]. Let (cid:96) , . . . , (cid:96) m beclosed rectangular paths in Γ r containing the origin representing generators of π (Γ r ) ∼ = F m (without loss of generality we assume that m ≥ g ∈ π (Γ r ) is representedby a closed rectangular path (cid:96) g which is the finite product of paths (cid:96) i or their inverses.Further, the isotopy Φ induces isomorphisms π (Γ t ) ∼ = π (Γ r ). In particular, closed paths( (cid:96) g ) t : [0 , T ] → Γ t , g ∈ π (Γ r ), are mutually distinct and represent all elements of π (Γ t ).According to Lemma 3.2 and formula (1.2) (containing expressions for coefficients of seriesexpansions of the first return maps of Abel equations), P (( (cid:96) g ) (cid:48) t )( r ) = r + ∞ (cid:88) j =1 c gj ( t ) r j +1 , where c gj is a real polynomial in t of degree at most j (1 ≤ j < ∞ ).Since Γ r is universal, the first return map P ( (cid:96) (cid:48) g ) ⊂ G c [[ r ]] is not the identity map (seeTheorem 2.1). Thus there exists j g ∈ N such that c gj g (0) (cid:54) = 0. This implies that c gj g is anot identically zero polynomial in t of degree at most j g . By Z j g we denote the set of itszeros in [0 , Z j g ≤ j g . We define Z := (cid:91) g ∈ π (Γ r ) Z j g . Since π (Γ r ) is countable, the set Z ⊂ [0 ,
1] is at most countable. By definition, for each t ∈ [0 , \ Z and g ∈ π (Γ r ) we have P (( (cid:96) g ) (cid:48) t ) (cid:54) = id. This shows that the kernel of thecorresponding homomorphism ˆ P Γ t : π (Γ t ) → G c [[ r ]] (see (2.6)) is trivial. Therefore byTheorem 2.1, Γ t is a universal curve.The proof of the proposition is complete. (cid:3) Now, we choose a sequence { t i } i ∈ N ⊂ [0 , \ Z converging to 1. Then according toour construction and the previous proposition piecewise linear curves Γ t i are universal,isotopic to Γ and the sequence of such curves converges to Γ as t i → H -measure and metric d H (see (d) of Theorem 2.7). This finishes part I of the proof. II.
In this part given a piecewise linear universal curve Γ we approximate it (withrespect to H -measure and metric d H ) by a sequence of isotopic universal curves havingthe same sets of points of order (cid:54) = 2 and C ∞ outside of these sets.As in part I , we isotopically deform Γ to smooth its corners. Specifically, let V be the setof vertices of Γ of degree 2 whose adjacent edges do not lie on a straight line. Fix a familyof mutually disjoint open connected neighbourhoods U v of v ∈ V such that each U v doesnot contain other vertices but v . We deform simultaneously parts of Γ in all U v , v ∈ V ,preserving points of Γ outside of these neighbourhoods. To this end we map each U v by asuitable affine transformation L v onto the angle R := { ( x, y ) ∈ R : y = | x | , x ∈ [ − , } NIVERSAL CURVES IN THE CENTER PROBLEM FOR ABEL DIFFERENTIAL EQUATIONS 13 (see Figure 6). We construct an isotopy for this angle which preserves parts of the edgessituated outside the open disk of radius √ L − v attach the deformed piece of the curve to Γ.Figure 6. Smoothing the corner of the angle formed by the graph y = | x | .For this purpose consider an auxiliary C ∞ function ρ ( x ) := (cid:40) e − x if x >
00 if x ≤ . Then the isotopy φ : R × [0 , → R is given by the formula φ (( x, y ) , t ) := (cid:0) x, th ( xt ) (cid:1) ( y := | x | , x ∈ R ), where(3.3) h ( x ) := 12 + ( | x | − ) ρ (2 | x | − ρ (2 | x | −
1) + ρ (2 − | x | ) . (By the definition, h is a C ∞ function on R equals if | x | ≤ and | x | if | x | ≥ × [0 , → R we denote the (above described) isotopy given by the formulaΦ( z, t ) := L − v ( φ ( L v ( z ) , t )), ( z, t ) ∈ U v × [0 , v ∈ V . (Without loss of generalitywe assume that Φ is correctly defined, i.e., that U v were chosen so small that for each t ∈ [0 ,
1] images Φ( U v , t ) for distinct v are pairwise disjoint.) We set Γ t := Φ(Γ , t ) andfollow the lines of the proof given in step I . Specifically, for a closed piecewise linear path γ : [0 , T ] → Γ representing a nontrivial element of π (Γ) we define γ t := Φ( γ, t ) : [0 , T ] → Γ t . Then, for this choice of γ t , we require an analog of Lemma 3.2: Lemma 3.4.
Each iterated integral I i ,...,i k ( γ (cid:48) t ) is a real polynomial in t of degree ≤ k .Proof. Adding additional vertices, if necessary, without loss of generality we may assumethat each edge of Γ contains at most one vertex from V .Since iterated integrals are constant on the set of derivatives of closed paths representingthe same element of π (Γ), one can choose γ so that it is a finite product of linear pathswhose images are edges of Γ. Hence, due to formulas (2.1) and (2.2), it suffices to provethe result for a linear path γ whose image is an edge e of Γ. If e joins two vertices notin V , then γ t = γ for all t and the required result is obvious. For otherwise, making asuitable transformation L v , without loss of generality we may assume that γ is a linearmap onto a subinterval of the line y = x with endpoints (0 ,
0) and ( a, a ) for some a > γ , we see that itsuffices to prove the lemma for γ whose image is interval [(0 , , (1 , γ ( s ) := ( s, s ), s ∈ [0 , γ t ( s ) = (cid:0) s, th ( st ) (cid:1) , s ∈ [0 , γ t as product of paths γ t ( s ) := γ t ( ts ), s ∈ [0 , γ t ( s ) := γ t ( t + (1 − t ) s ), s ∈ [0 , γ it , i = 1 ,
2. We have γ (cid:48) t ( s ) := ( t, th (cid:48) ( s )) , s ∈ [0 , . Thus, see (1.2), I i ,...,i k ( γ (cid:48) t ) = t k · I i ,...,i k ( γ (cid:48) ) (= const · t k )is a polynomial in t of degree at most k .In turn, γ (cid:48) t ( s ) := (cid:18) − t, (1 − t ) · h (cid:48) (cid:18) − tt s (cid:19)(cid:19) = (1 − t, − t ) , s ∈ [0 , , since h ( x ) = x for x ≥
1. Hence, I i ,...,i k ( γ (cid:48) t ) = (1 − t ) k k !is a polynomial in t of degree at most k as well.This completes the proof of the lemma. (cid:3) Repeating word-for-word the arguments of the proof of Proposition 3.3 (now based onLemma 3.4) we get in our case
Proposition 3.5.
There exists an at most countable subset Z ⊂ [0 , such that for each t ∈ [0 , \ Z the curve Γ t (which is C ∞ outside the set S Γ ) is universal. Choosing a sequence { t i } i ∈ N ⊂ [0 , \ Z converging to 0 we obtain that curves Γ t i ,having the same sets S Γ i and C ∞ outside of these sets, are universal, isotopic to Γ and thesequence of such curves converges to Γ as t i → H -measure and metric d H .This finishes the proof of part II and hence of the theorem. (cid:3) Remark 3.6.
According to our construction, the sequence of universal curves { Γ i } i ∈ N converging to Γ has the property that for each x ∈ S Γ i of order 3 there is an open disk D x centered at x such that Γ i ∩ ( D x \ { x } ) is the union of three open intervals. Using themethod of part II of the proof of Theorem 2.7 one can additionally smooth curves Γ i toobtain new curves (cid:101) Γ i satisfying conditions (a)–(d) of the theorem and such that for each x ∈ S (cid:101) Γ i , ord( x ) = 3 , in an open disk D x centered at x , (cid:101) Γ i is the union of a connected C ∞ curve and its tangent at x having infinite order contact at x . Moreover, one can easilyshow that such a curve (cid:101) Γ i can be obtained as the image of a C ∞ map ˜ A : [0 , T ] → R which corresponds to an Abel equation (1.1) with C ∞ coefficients. We leave the detailsto the reader. References [AL] M. A. M. Alwash and N. G. Lloyd, Non-autonomous equations related to polynomial two-dimensionalsystems, Proc. R. Soc. Edinb. A (1987), 129–152.[A1] M.A.M. Alwash, On the Composition Conjectures, Electronic Journal of Differential Equations, Vol.2003 (2003), No. 69, pp. 14.[A2] M.A.M. Alwash, The composition conjecture for Abel equation, Expo. Math. (2009), 241–250.[A3] [M.A.M. Alwash, Polynomial differential equations with piecewise linear coefficients, Differ. Equ.Dyn. Syst. (2011), no. 3, 267–281.[A4] M.A.M. Alwash, Composition conditions for two-dimensional polynomial systems, Diff. Eq. and Appl.,Volume 5, Number 1, February 2013.[B1] A. Brudnyi, An explicit expression for the first return map in the center problem, J. Differ. Equations (2004), 306–314. NIVERSAL CURVES IN THE CENTER PROBLEM FOR ABEL DIFFERENTIAL EQUATIONS 15 [B2] A. Brudnyi, On the center problem for ordinary differential equations, Amer. J. Math. (2) (2006),419–451.[B3] A. Brudnyi, Formal paths, iterated integrals and the center problem for ordinary differential equations,Bull. Sci. math. (2008), 455–485.[B4] A. Brudnyi, Center problem for ODEs with coefficients generating the group of rectangular paths, C.R. Math. Acad. Sci. Soc. R. Can. (2009), no. 2, 33–44.[B5] A. Brudnyi, Free subgroups of the group of formal power series and the center problem for ODEs, C.R. Math. Acad. Sci. Soc. R. Can. (2009), no. 4, 97–106.[B6] A. Brudnyi, Composition conditions for classes of analytic functions, Nonlinearity (2012), 3197–3209.[B7] A. Brudnyi, Moments finiteness problem and characterization of universal centers of ODEs withanalytic coefficients, to appear. arXiv:1305.4303.[BlRY] M. Blinov, N. Roytvarf and Y. Yomdin, Center and moment conditions for Abel equation withrational coefficients, Funct. Differ. Equ. (2003), no. 1-2, 95–106.[BFY1] M. Briskin, J.-P. Francoise and Y. Yomdin, The Bautin ideal of the Abel Equation, Nonlinearity (1998), 41–53.[BFY2] M. Briskin, J.-P. Francoise and Y. Yomdin, Center conditions, compositions of polynomials andmoments on algebraic curves, Erg. Theory Dyn. Syst. (1999), 1201–1220.[Bl] M. Blinov, Center and composition conditions for Abel equation, Thesis. Weizmann Institute of Sci-ence, 2002.[BRY] M. Briskin, N. Roytvarf and Y. Yomdin, Center conditions at infinity for Abel differential equations,Annals of Math. (1) (2010), 437–483.[BY] A. Brudnyi and Y. Yomdin, Tree composition condition and moments vanishing, Nonlinearity (2010), 1651–1673.[C] C. Christopher, Abel equations: composition conjectures and the model problem, Bull. Lond. Math.Soc. (2000), 332-338.[CGM1] A. Cima, A. Gasull, and F. M˜anosas, Centers for trigonometric Abel equations, Qual. TheoryDyn. Syst. (2012), 19–37.[CGM2] A. Cima, A. Gasull, and F. M˜anosas, A simple solution of some composition conjectures for Abelequations, J. Math. Anal. Appl. (2013), 477-486.[CGM3] A. Cima, A. Gasull, and F. M˜anosas, An explicit bound of the number of vanishing doublemoments forcing composition, J. Diff. Equations (3) (2013), 339–350.[GGL] J. Gin´e, M. Grau and J. Llibre, Universal centers and composition conditions, Proc. London Math.Soc. (3) (2013), 481–507.[Co] S.D. Cohen, The group of translations and positive rational powers is free, Quart. J. Math. Oxford(2) (1995), 21–93.[Hu] S.-T. Hu, Homotopy Theory, New York, 1959.[I] Yu. Il’yashenko, Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc. (N.S.) (3)(2002), 301–354.[MP] M. Muzychuk and F. Pakovich, Solution of the polynomial moment problem, Proc. Lond. Math. Soc. (3) (2009), 633–657.[P1] F. Pakovich, On the polynomial moment problem, Math. Res. Lett., , no. 2-3 (2003), 401–410.[PRY] F. Pakovich, N. Roytvarf and Y. Yomdin, Cauchy type integrals of algebraic functions, Isr. J. Math.,Vol. 144 (2004), 221–291.[P2] F. Pakovich, On rational functions orthogonal to all powers of a given rational function on a curve,arXiv:0910.2105. Department of Mathematics and StatisticsUniversity of CalgaryCalgary, AlbertaT2N 1N4
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