Dias, Kealey

A vector field in the space of degree d monic, centered single variable complex polynomial vector fields has a combinatorial structure which can be fully described by a combinatorial data set consisting of an equivalence relation and a marked subset on the integers mod 2d-2, satisfying certain properties, corresponding to an equivalence relation on the 2d-2 separatrices. This data set can be equivalently represented in a combinatorial disk model, by labeling the 2d-2 roots of unity on the unit circle by the asymptotic directions of the separatrices and joining the points in the same equivalence class by geodesics in the unit disk with respect to the Poincaré metric. The goal is to utilize this fact in order to convert the problem of counting the combinatorial classes into a so-called bracketing problem: a combinatorial problem involving pairings of parentheses placed in a string of elements in a valid way. We first enumerate all combinatorial classes with respect to degree d, and then we enumerate the combinatorial classes having a specific dimension q in parameter space. In both cases, a recursion equation and implicit expressions for the algebraic generating functions are calculated, and asymptotic growth questions are considered

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Downloaded from orbit.dtu.dk on: Dec 19, 2017Enumeration of Combinatorial Classes of Single Variable Complex Polynomial VectorFieldsDias, KealeyPublication date:2008Document VersionEarly version, also known as pre-printLink back to DTU OrbitCitation (APA):Dias, K. (2008). Enumeration of Combinatorial Classes of Single Variable Complex Polynomial Vector Fields.Poster session presented at Bobfest: Dynamics and Topology on the occasion of  Bob Devaney's 60th birthday,Tossa de Mar, Spain, .References[C ] Louis Comtet. Advanced Combinatorics. D. Reidel Publishing Company, Dordrecht-Holland,1974.[D ] Tom Davis. Catalan Numbers., http://mathcircle.berkeley.edu/BMC6/ps/catalan.pdf, 2006.[DES ] A. Douady, F. Estrada, and P. Sentenac. Champs de vecteurs polynomiaux sur C. Unpub-lished manuscript.Conclusions/Future workA recursion equation is successfully computed for the number of combinatorial classes for vec-tor fields in Ξd when the separatrices are labeled. An estimate for the asymptotic growth of thesequence is given.I have also computed a recursion equation for the number of combinatorial classes of a givendimension within Cd−1, however, a two-varirable generating function has not successfully com-puted as of date.Generating Function and Asymptotic GrowthThe generating function for c2s can be computed (after simplification) to beG(t) =∑s≥0c2st2s =12[11− (2Gt2 − t) +11− (2Gt2 + t)]. (5)Now G(t) is algebraic since equation (5) gives 4t4G3 − 4t2G2 + (1 + t2)G − 1 = 0. A simplecalculation and application of the implicit function theorem shows that G(t) has positive radiusof convergence at t = 0, where the radius can be determined by solving the system of equations4t4G3 − 4t2G2 + (1 + t2)G − 1 = 0∂∂G[4t4G3 − 4t2G2 + (1 + t2)G − 1]= 12t4G2 − 8t2G + 1 + t2 = 0, (6)giving R = 12√−22 + 10√5 ≈ .3002830998. Now1R= lim sups→∞2s√|c2s| = lims→∞2s√c2s (7)since c2s is a strictly positive and increasing sequence. By the definition of limit, one can con-clude that for large enough s, (1R− )2s< c2s <(1R+ )2s, (8)that is, c2s ≈(1R)2sfor s large enough. Numerical evidence is given below.2 4 6 8 10 12 141234567891011log(c2s) vs. s+11 2 3 4 5 6 7 8 9 10 11−3−2−101234log(c2(k+1))−log(c2k) vs. kThe Complete SolutionIf we allow for non-structurally stable vector fields, then an involution on the ends is not enoughto describe the combinatorial structure. We also need to include pairings of separatrices, cor-responding to homoclinic separtrices for the vector field. This is represented by a bracketingproblem where we distinguish between pairings of separatrices and pairings of ends. This willbe represented by placing round brackets () and square brackets [] in the string of the 2(d − 1)elements s0s1s2...s2s−1 in the valid way. An example is given below to demonstrate.e2s3s4e5s5e0e4e1e3s2 s1s0 s0e2s3s4e5s5e0e4e1e3s2 s1s0e2s3s4e5s5e0e4e1e3s2 s1s0e2s3s4e5s5e0e4e1e3s2 s1s0e2s3s4e5s5e0e4e1e3s2 s1[s0(s1s2)s3][s4s5]e0 e3 e4 e5s0s1s2[s3s4]s5e3 e4s0s1s2(s3s4)s5(s0(s1s2)s3)(s4s5)(s0[s1(s2s3)s4]s5)e1 e4Figure 2: Disk models for some non-structurally stable vector fields of degree d = 4. Theinvolution on the ends is marked by the dashed curves. The curves representing the separatrices,as well as the dashed curves joining the ends may not cross. The representation of the involutionas a bracketing problem is displayed below each figure. A round paring (s`1...s`2) represents ahomoclinic separatrix s`1 = s`2, and a square pairing [s`1...s`2] represents a pairing of the endse`1 and e`2.The problem reduces to counting the number of ways c2s one can place pairs of parentheses(some round, some square) in the valid way in a string of length 2s.We consider first the situation when no distinction is made between square and round brackets.There are l terms defined by the outermost brackets: l1 terms consist of one element, and theother terms must consist of an even number of elements of which there are li each, i = 2, 4, ..., 2s.Example: (s0(s1s2)s3)︸ ︷︷ ︸ s4︸︷︷︸ s5︸︷︷︸ has 3 terms: one of length 4 and 2 of length 1. Note that l1 isalways even and the li must furthermore satisfyl1 + l2 + l4 + ... + l2s = ll1 + 2l2 + 4l4 + ... + 2sl2s = 2s. (1)There are l!/(l1!l2!l4!...l2s!) ways to arrange these l terms. Within each pair of parentheses havingi ≥ 4 elements, there might be more pairs of parentheses nested inside these. Since the 2 out-ermost elements contained in this pair of pars is already paired off, there are i − 2 elements thatmight contain parentheses. Hence, each term of length i contributes ci−2.Now if we distinguish between the square and round brackets, and there are k2j = 0, ..., l2jsquare parentheses in the outermost bracketing of length 2j, j = 1, ..., s, then the number of waysto arrange the terms becomesl!(l1!l2!l4!...l2s!)s∏j=1(l2jk2j). (2)Putting these elements together, one can deduce that the number of combinatorial configurationsfor a sequence having 2s elements isc2s =2s∑l=1∑(l1,l2,l4,...,l2s)l2∑k2=0l4∑k4=0· · ·l2s∑k2s=0s∏j=1(l2jk2j)l!(l1!l2!l4!...l2s!)cl20 cl42 ...cl2s2s−2, s ≥ 1, c0 = 1(3)where the (11, l2, l4, ..., l2s) satisfy (1). Sincel2j∑k2j=0( l2jk2j)= 2l2j, equation (4) reduces toc2s =2s∑l=1∑(l1,l2,l4,...,l2s)l!(l1!l2!l4!...l2s!)(2c0)l2(2c2)l4...(2c2s−2)l2s, s ≥ 1, c0 = 1. (4)Valid BracketingsThe bracketing must satisfy certain rules so that they are in accordance with what can happen fora vector field. Pairs of parentheses placed in a string of elements is called a valid bracketing if:1. there are an equal number of right and left parentheses2. the number of left parentheses must be greater than or equal to the number of right, readingfrom left to right Example: ())()( not valid3. there must be at least one element between successive left (resp. right) parentheses Example:((e0e1)e2e3) not valid4. There must be an even number of elements in each pair of parentheses Example: (e0e1e2)e3not validStructurally Stable Vector FieldsFor structurally stable vector fields, this disk model is dual to a non-crossing involution on the2d − 2 ends e`. An involution on the ends can be seen as one way that 2d − 2 people can shakehands without any handshake crossing another. You might recognize then that the number ofcombinatorial classes (with the labelling of the separatrices) for the structurally stable vectorfields is just the Catalan number C(d − 1) [DES]. Note that this is equivalent to the number ofways to make “valid” pairings of d − 1 pairs of parentheses [D]. An example is given below todemonstrate.s3e1s1e2s2e3e4e5s5s0 s0e1s1e2e3s3s4e0s0e2s2s4e5s3e3e4s5e0e1s1e0s5e5s4e4s2e1s1e2s2e3e4s3 s0e2e3e4e5e0s3s5s0s4e5s5e0e1s1s2s4(e0(e1e2)e3)(e4e5) (e0(e1(e2e3)e4)e5) (e0(e1e2)(e3e4)e5) (e0e1)(e2e3)(e4e5)(e0e1)(e2(e3e4)e5)Figure 1: Disk models for the structurally stable vector fields of degree d = 4. The involutionon the ends is marked by the dashed curves. The representation of the involution as a bracketingproblem is displayed below each figure.SummaryA vector field ξP in the space Ξd of degree d monic, centered complex polynomial vector fieldshas a combinatorial structure which can be fully described by an equivalence relation on the 2d−2separatrices. This equivalence relation can be equivalently represented in a combinatorial diskmodel, by labelling the points exp( 2pii`2d−2), ` = 0, ..., 2d − 3 on D¯ by s` and joining the points inthe same equivalence class by geodesics in D¯ with respect to the Poincare´ metric. Here, the com-binatorics can be fully described by non-crossing invoutions between even and odd separatricess` corresponding to homoclinic separatrices and between the even and odd ends e`, the principalpoints defined by the 2d − 2 accesses to infinity between the separatrices s`−1 and s`, whichcorrespond to αω-zones. These disk models can be completely described by bracketing prob-lems: combinatorial problems involving pairings of parentheses placed in a string of elements(e0e1...e2d−3 and s0s1...s2d−3 in our case). This specific problem is similar to the generalizedbracketing problem of Schro¨der [C], and a similar method is used to calculate a recursion equa-tion for the number of combinatorial classes c2(d−1) for vector fields in Ξd. Furthermore, animplicit expression for the generating function for c2(d−1) is computed, and asymptotic growthquestions are considered.Kealey Dias1, Bodil Branner11:Department of Mathematics, Technical University of DenmarkCounting Combinatorial Classes of Complex Polynomial Vector Fields

Enumeration of Combinatorial Classes of Single Variable Complex Polynomial Vector Fields

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Enumeration of Combinatorial Classes of Single Variable Complex Polynomial Vector Fields

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