Branner, Bodil

In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias

Online Research Database In Technology

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 18, 2017Polynomial Vector Fields in One Complex VariableBranner, BodilPublication date:2008Link back to DTU OrbitCitation (APA):Branner, B. (2008). Polynomial Vector Fields in One Complex Variable. Paper presented at Congress in memoryof Adrien Douady, Institut Henri Poincaré, .POLYNOMIAL VECTOR FIELDS of ONE COMPLEX VARIABLEBodil BrannerTechnical University of Denmark The talk is based on Adrien's joint work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias.IN THE MEMORY of ADRIEN DOUADYIntroductionBasic propertiesCombinatorial invariantsAnalytic invariantsqc-equivalenceThe structure theoremCONTENTNOTATIONConsider the set of monic centered polynomials of degreePd d ≥ 2P (z) = zd + ad−2zd−2 + · · · + a0 a = (a0, . . . , ad−2) ∈ Cd−1parametrized byThe associated vector field in C isξP (z) = P (z)ddz zP (z)We study the maximal trajectories t !→ γ(t, z0) ξP t ∈ Rγ′(t, z0) = P (γ(t, z0)) and γ(0, z0) = z0of with i.e. maximal solutions toThe space of polynomial vector fields of fixed degree dΞd = {ξP }P∈Pd ! Cd−1is•EQUILIBRIUM POINTS of ξP – ROOTS of PPROPOSITION If  with mulitiplier thenis a simple root ofis holomorphically conjugate in a neighborhood of to the linear vector fieldζζP λ = P ′(ζ)ξPλzddz.THREE CASES:!λ < 0 !λ = 0 !λ > 0!λ > 0!λ < 0sink center sourcePROPOSITION If ζ is a multiple root of P of multiplicity m > 1 ξPthenthenhasm− 1 attracting directions and m− 1 repelling directions.THREE QUADRATIC EXAMPLESP (z) = z2 − 1 P (z) = z2 + 1 P (z) = z2ζ = 1ζ = −1 ζ = ± i ζ = 0Two simple equilibrium points Two simple equilibrium points One double equilibrium pointcentersa sourcea sinkTHREE QUADRATIC EXAMPLESP (z) = z2 + 1 P (z) = z2ζ = ± i ζ = 0Two simple equilibrium points Two simple equilibrium points One double equilibrium pointcentersa sourcea sinksink and sourcewith non-real multiplierstypical:a singletondecomposed into disjoint classes: P (z) = z2 + aTHE SPACE of QUADRATIC VECTOR FIELDS Ξ2 ! Ca = 0a > 0real-analytic one-dim submanifoldcomplex  submanifolda ∈ C \ [0,+∞[GOALsuch thatThe goal is to decompose into disjoint classeseachhave the same qualitative dynamicsallis connectedA classis maximalA class is either or part of theWithinis characterized by ais associated two integersTo each classa vector field is uniquely determined by anCCCCξP ∈ Cstructurally stable bifurcation locuscombinatorial invariant I(C)analytic invariantC .C ξPΞdAP = (A1P , . . . , AsP , T 1P , . . . , ThP ) ∈ Hs × Rh+.•••s = s(C) ≥ 0 and h = h(C) ≥ 0 satisfying s+ 12h ≤ d− 1THE STRUCTURE THEOREMGiven a combinatorial data set with associated integers and and a tupleThen there exists a unique such that the vector field has combinatorial invariant and analytic invariant Each isomorphic tohence of real-dimensionAny pair of vector fields are dynamically equivalent:mapping trajectories of mapping trajectories of trajectories of preserving orientation but not necessarily the parametrization by time.MOREOVERd ≥ 2 , I s = s(I)h = h(I)A = (A1, . . . , As, T 1, . . . , Th) ∈ Hs × Rh+P ∈ Pd ξPI(C) = I AP = A,..C Cd−1 Hs × Rh+2s+ h .ξ1, ξ2 inthere exists a quasi-conformal mapping is a real-analytic submanifold ofCΨ : Ĉ→ Ĉ ξ1ξ2 ,,ontoDUAL DESCRIPTIONMeromorphic vector fields Meromorphic abelian differentialsξf (z) = f(z)ddzωf =1f(z)dzobey similar transformation laws: If ϕ : U → V is a holomorphic coordinate change and w = ϕ(z) thenϕ∗(ξf ) = ξg and ϕ∗(ωg) = ωf whereg(ϕ(z)) = ϕ′(z)f(z)The singularities of ξf and ωf are the zeros and the poles of fRECTIFYING COORDINATESThe two descriptions complement each other. The advantage of the differentials are that they can be integrated.In any simply connected domain avoiding zeros of f  the differential ωf has an antiderivative, unique up to addition by a constant,.Note thatNote thatwhereThe coordinates are called  rectifying coordinatesφ∗(ξf ) = ξg g(φ(z)) = φ′(z)f(z) =1f(z)f(z) = 1w = φ(z) of ξf ..φ(z) =∫1f(z)dzξg(w) = g(w)ddwωg =1g(w)dwOne sink and one source Two centers One double equilibrium pointRECTIFYING COORDINATES for THE QUADRATIC EXAMPLESa horizontal strip a vertical stripφφ φS0S0S1 S1S10a planeC \ {S0 ∪ S1} C \ { i ]−∞,−1] ∪ i[+1,+∞[ } Ĉ \ { 0}THE SINGULARITY at ∞(C∗, ξP ) is holomorphically conjugate to (C∗, f(z) ddz)by z !→ 1zwheref(z) = − 1zd−2(1 + ad−2z2 + · · · + a0zd).Hence, ξP has a pole of order d− 2 at ∞ .IN PARTICULARξ0(z) = zdddzis holomorphically conjugate to f0(z)ddz= − 1zd−2ddzφ0 : Ĉ \ {0}→ C, φ0(z) = − 1d− 11zd−1is a branched covering, mapping∞ to 0 .RECTIFYING COORDINATES for ξ0(z) = zdddzEXAMPLE d = 3φ0φ0HH-S0S2S1S3THE SPECIAL ROLE of ∞ – SEPARATRICESPROPOSITION Every ξP is holomorphically conjugate to ξ0 in neighborhoods of infinity, by a conjugating map tangent to the identity at ∞ .CONSEQUENCESThere areNote, that for any pointincoming trajectories to and outgoing trajectories fromTheir asymptotes are the half lines in directions where d− 1 ∞ ∞∞d− 1 .δ! = exp(2pii#2(d− 1))! ∈ L = { 0, 1, . . . , 2d− 3 } or ! ∈ Z/2(d− 1) .on such a trajectory it takes a finite amount of time to get toz0if incoming and to come from  if outgoing.∞SEPARTRICESThe separatrices are the maximal trajectories of ξP incoming to and outgoing from∞ .A separatrix is homoclinic  if both outgoing from and incoming to ∞ .∞ is a saddle point for ξP .LIMITING BEHAVIOUR of trajectories γ(t, z0), t ∈] t−, t+ [PROPOSITION Each sink or source is the landing point of at least one separatrix.Each multiple equilibrium point is the landing point of at least one separatrixany of the attracting or repelling directions.tangent to limt→t+γ(t, z0) =limt→t−γ(t, z0) = sink or multiple equilibrium point if source or multiple equilibrium point ift+ = +∞t− = −∞∞∞t+ ∈ R+t− ∈ R−LABELING the SEPARATRICES and THE SEPARATRIX GRAPHA separatrix is labeled according to its asymptote,! even corresponds to a separatrix incoming to! odd corresponds to a separatrix outgoing from∞∞THE SEPARATRIX GRAPH ΓP is the closure in Ĉ of the separatrices. HenceS! ! ∈ {0.1, . . . , 2d− 3}....ΓP =⋃!=1,...,2d−3S! ∪⋃sink, source, mult.ζ ∪ {∞}SYMBOLICALLYd = 2 d = 3 d = 4Mark δ! ∈ S1 .THE COMBINATORIAL INVARIANT of ξP ∈ ΞdDEFINITION of the COMBINATORIAL INVARIANT IP of ξP :IP  is an equivalence relation with a specified subsetsatisfying: on∼P L = {0, 1, . . . , 2d− 3}HP ⊂ LHP consists of the labels for which  is a homoclinic separatrix.For !1, !2 ∈ HP :!1 ∼P !2 ⇐⇒ S!1 = S!2HP is saturated by ∼PL \HP consists of the labels for which lands at , a sink, a source, or a multiple equilibrium point. !1, !2 ∈ L \HPFor :!1 ∼P !2 ⇐⇒ S!1 and S!2 lands in C at the samethat there are three possible kinds of equivalence classes inNOTE L \HP :For mixed the multiplicity of is iff    the ordered cyclic sequence of labels inchanges parity times.!!..S!S![!]oddevenmixedcorresponding tosinksourcemultiple equilibrium point[!]ζζ .ζζ being am [!]2(m− 1)COMBINATORIAL INVARIANTS for the QUADRATIC EXAMPLESS0 S0S1 S1S10EQUIVALENCE CLASSES of ∼P[0] even[1] odd[0, 1] mixedHP = {0, 1}[0, 1]A CUBIC EXAMPLE with HP = ∅ and its COMBINATORIAL INVARIANTP (z) = (z + 1)2(z − 2)ζ = −1ζ = 2a double equilibrium pointa sourceAttracting petal of ζ = −1 := S0S2S1S3C \ { }Repelling petal of ζ = −1 :=Half plane left ofS0 ∪ S2S1 ∪ S3a sepala sepalEQUIVALENCE CLASSES of ∼P[0] odd[1, 2, 3] mixedTwo sepals := the intersection ofthe attracting and the repelling petal.an interpetalCUBIC EXAMPLES with HP != ∅ and their COMBINATORIAL INVARIANTone center,one double equilibrium three centersone center,one sink, one  source S12S12 S12S0S0S3S3 S30HP = {1, 2} = [1, 2] HP = {1, 2} = [1, 2]HP = {0, 1, 2, 3}[0] odd[3] even[3, 0] mixed[1, 2] , [3, 0]SUBDIVISION of LABELS if HP != ∅:polynomial-like ofpolynomial-like ofdegree d1degree d2d2 = 2d1 = 1TYPES of ZONESA zone Z is a connected component of Ĉ \ ΓP . There are three types, classified by the type of the holomorphic conjugacyφ : (Z, ξP )→(• , ddz) is isomorphic to a HORIZONTAL STRIP. is isomorphic to an UPPER or LOWER HALF PLANE. is isomorphic to a horizontal strip.two distinct equilibrium points: Ana source or a multiple equilibrium, a sink or a multiple equilibrium, such thatconsists of one or two incoming separatrices and one or two outgoing separatrices,αω-zone ∃∀γ(t, z0)and possibly some homoclinics.A sepal-zone∃ a multiple equilibrium such that ∀z0 ∈ Zz0 ∈ ZAconsists of one incoming separatrices and one outgoing separatrices, and possiblysome homoclinics.is isomorphic to a HALF UPPER orcenter-zone∀ z0 ∈ Z\ γ(t, z0) is periodic of period T = 2pi|P ′(ζ)|∂Z∂Z∂Z consists of one or several homoclinics.•••ζ the and the is ζcontains one center ζ and ζZ\ζLOWER CYLINDER. α-limitα-limitω-limitthe of isω-limitζωζωζαζα..COMBINATORIAL CLASSESA combinatorial class C consists of all ξP IP = I(C)with . The integerss = s(C) and h = h(C)are the numbers of αω-zones (numbers of strips) and the number of homoclinics  (half the number of labels in       ) respectively.HPA COMBINATORIAL DATA SETDEFINITIONGiven , an equivalence relation on L = {0, 1, . . . , 2d− 3} , and a specified subsetH ⊂ L consisting of labels,if it satisfies:2h ≥ 0 odd and evenh h(∼, H) is a combinatorial data setis non-crossing.is saturated by , and each equivalence class in and anconsists of anlabel.Zones in the disc model are of the three types: αω, sepal, center∼∼H H oddeven•••∼d ≥ 2..ANALYTIC INVARIANTS for a given ξPEach homoclinic separatrix for is assigned the positive real time it takes to travel along Each is assigned the complex ''time'' it takes to travel along the transversal in the zone closest to theIn each case the invariant can be expressed asthe oriented trajectory from to∫loop1P= 2pii∑ζ left of loopRes(1P, ζ)= −2pii∑ζ right of loopRes(1P, ζ)where the loop is either the homoclinic or the transversal and the summation is over all equilibrium points left of the loop (or all right of the loop).ζα ζαξPξPζTPAPαω-zone ofζαα-limit point tofrom∞ ∞∞ ∞..We choose AP ∈ H .QC DYNAMICAL EQUIVALENCEAMONG αω-zones , represented in rectifying coordinates:affine map, mapping the base {1, A1} to the base {1, A2}piecewise affine map; on the triangle, mapping the base {T1, A1} to the base {T2, A2}..Suppose ξP1 , ξP2 belong to the same combinatorial class. Then they have the same qualitative dynamics. In rectifying coordinates the equivalence is given through piecewise affine mappings.QC DYNAMICAL EQUIVALENCEAMONG center-zones : affine or piecewise affine mappings,mapping the base to the base{T1, i} {T2, i} ,AMONG sepal-zones with homoclinics : piecewise affine mappings.Suppose ξP1 , ξP2 belong to the same combinatorial class.or the base {T j1 ,−i} to the base {T j2 ,−i} .THE STRUCTURE THEOREMGiven d ≥ 2 , a combinatorial data set is the number of equivalence classes inPROOFSTRUCTURALLY STABLEBIFURCATION SETwhereandis the number of andforto a class, which is isomorphic to. There exists a unique realizing the above, i.e. andA = (A1, . . . , As, T 1, . . . , Th) ∈ Hs × Rh+s h(∼P , HP ) = (∼, H)AP = A by surgery. From the rectified building blocks we construct a Riemann surfacewith a vector field and prove that  is isomorphic to and that there exists a uniquesuch thatthen the number ofis holomorphically conjugate toIfis in the bifurcation set.has only sinks and sources, no homoclinic separatrices and no multiple equilibrium points, takes its maximal value . It follows that belongs  . Hence is structurally stable.If has a homoclinic separatrix or a multiple equilibrium point then and. Hence(∼, H)(∼, H)αω-zonesαω-zonesMM(M, ξM)H ξP ∈ Ξd.ξMξPξPξPξPP ∈ Pd (Ĉ, ξP )Ĉ.s = d− 1Hd−1 ξPs < d− 12s+ h < 2(d− 1)ILLUSTRATING the SURGERY in a FAMILIAR case d = 3ILLUSTRATING the SURGERY in a FAMILIAR case d = 3ILLUSTRATING the SURGERY in a FAMILIAR case d = 3

Polynomial Vector Fields in One Complex Variable

http://orbit.dtu.dk/ws/files/5530757/ADmemorialBB.pdf

Polynomial Vector Fields in One Complex Variable

Abstract

Similar works

Full text

Available Versions

Online Research Database In Technology