Karima, Mohaby

Tamae, Sayaka

Taniguchi, Masahiko

Kyoto University Research Information Repository

TitleTopological representation of the branched covering structureinduced from a real rational function (Extensions of thehistorical calculus transforms in the geometric function theory)Author(s)Karima, Mohaby; Tamae, Sayaka; Taniguchi, MasahikoCitation数理解析研究所講究録 (2010), 1717: 29-37Issue Date2010-10URL http://hdl.handle.net/2433/170329RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityTopological representationof the branched covering structureinduced from a real rational functionM.Karima, S.Tamae, and M..TaniguchiGraduate School of Humanities and Sciences,Nara Women’s University1 IntroductionLet $f$ : $\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$ be a rational function of degree $n$ . We say that $f$ is a generic(complex) rational function if the preimage $f^{-1}(z)$ of any point $z\in\hat{\mathbb{C}}$ consistsof either $n$ or $n-1$ points; the points of the latter type are called simpleramification points. The set of all simple ramification points of $f$ is denoted$\Sigma(f)$ and consists of $2n-2$ points.Two rational functions $f_{i}$ : $\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$ are called covering-equivalent if thereexists a M\"obius transformation $\phi:\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$ such that $f_{1}=f_{2}o\phi$ . Let $CH_{0,n}$be the set of all equivalence classes of complex generic rational functionsof degree $n$ . The correspondence $f\mapsto\Sigma(f)$ generates a covering $C\Phi_{n}$ :$CH_{0,n}arrow CQ_{0,n}$ , where $CQ_{0,n}$ is the configuration space consisting of all n-tuples of unordered distinct points on $\hat{\mathbb{C}}$ . We assume that $CH_{0,n}$ is providedwith the weakest topology for which the map $C\Phi_{n}$ is continuous.Remark 1. The degree $h_{n}$ of the covering $C\Phi_{n}$ , and its analogs for arbitrarymeromorphic functions, are called the Hurwitz numbers. These numbers arisein many situations in mathematical physics. The Hurwitz numbers $h_{n}$ canbe calculated.$h_{n}= \frac{n^{n-3}(2n-2)!}{n!}$ ;in fact, this result was apparently known already to Hurwitz himself. Alsosee [2] and [3].1717 2010 29-37 29Next, let $\tau$ be an anti-holomorphic involution. A real mtional functionis a complex rational function $f$ : $\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$ such that $\overline{f(\tau p)}=f(p)$ for any$p\in\hat{\mathbb{C}}$ . A real rational function $(\tau, f)$ is said to be generic if $f$ is generic.Clearly, $\overline{\Sigma(f)}=\Sigma(f)$ for any real rational function $f$ .Definition 1. Two real rational functions $(\tau_{i}, f_{i})(i=1,2)$ are called equiv-alent if there exists a M\"obius transformation $\phi$ such that$f_{1}=f_{2}o\phi$, $\phi 0\tau_{1}=\tau_{2}0\phi$ .Let $RH_{0,n}$ denote the space of all equivalence classes of generic real ra-tional functions of degree $n$ .The topology of $CH_{0,n}$ generates a topology on $RH_{0,n}$ ; in this topology$RH_{0,n}$ is not connected.The points of $RH_{0,n}$ are called the equivalence classes of generic realmtional functions. Since there might be no confusions, we write $(\tau, f)$ simplyas $f$ and $f\in RH_{0,n}$ means that the equivalence class of $f$ belongs to $RH_{0,n}$ .Now the main result of Natanzon, Shapiro, and Vainshtein is as follows.Theorem 1 ([2]). The set of all connected components of the space $RH_{0,n}$ isin $a$ 1-1-correspondence with the set of the equivalence classes of all gardensof weight $n$ .Here, the object, called a garden, consists of a weighted labeled directedplanar chord diagram and of a set of weighted rooted trees each of whichcorresponds to a face of the diagram.Definition 2. By a planar chord diagmm (of order $2l$ ) we mean a circledrawn on the plane together with $2l$ points on this circle partitioned into $l$pairs in such a way that, for any two pairs, the chords joining the points fromthe same pair do not intersect. The above $2l$ points are called the vertices ofthe chord diagram; the chords joining the vertices from the same pair and thearcs of the circle joining adjacent vertices are called the edges. The notionof its faces is defined in a usual way (except for the outer face of the graph,which is not a face of the diagram).We say that a planar chord diagram is directed if its edges are directedin such a way that the boundary of each face becomes a directed cycle.Fixing the standard orientation of the plane, we call a face of a directed30planar chord diagram positive if the face lies to the left when we traverse itsboundary according to the chosen direction, and negative otherwise.A planar chord diagram is said to be weighted if each edge is equippedwith a nonnegative integer (weight), and labeled if there exists a bijection $\beta$(labeling) that takes the vertex set of the diagram to the set $\{$ 1, 2, $\ldots,$ $2l\}$ .Two labelings $\beta_{1}$ and $\beta_{2}$ are said to be cyclically equivalent if $\beta_{1}(v)-\beta_{2}(v)$$mod 2l$ is a constant not depending on the choice of a vertex $v$ .Definition 3. A rooted tree is, by definition, a tree with one distinguishedvertex called the mot; all the other vertices of the tree are said to be inner.We say that a rooted tree is weighted if each its vertex is equipped with apositive integer (weight).Definition 4. A garden is a weighted labeled directed planar chord diagramwith a weighted rooted tree (possibly consistingjust of its root) correspond-ing to each face of the diagram. The weights of the inner vertices of thetrees are arbitrary positive integers, and the weight of the root of the treecorresponding to the face $j$ equals $t_{J}$ defined below. The (totat) weight ofthe garden equals twice the sum of the weights of all the inner vertices of alltrees plus the sum of the weights of all roots.Here, for any face $j$ of a labeled directed planar chord diagram, we de-note by $d_{j}$ the number of descents in the sequence of vertex labels orderedcyclically along the boundary of the face, and by $t_{j}$ the sum of $d_{j}$ and theweights of all the edges along the boundary of the face $j$ .Two gardens are said to be equivalent if there exists a bijection of thevertex sets of the corresponding chord diagrams which preserves chords, theirorientation, labels (up to the cyclic equivalence), rooted trees, and weights.2 The case of degree 2 or 3Every real rational function is equivalent to another real rational functionwith $\tau=J$ , where $J$ is the complex conjugation. And we can see that thelatter is a rational function with real coefficients.(Indeed, if such a funciton $R(z)$ is $P(z)/Q(z)$ with polynomials $P(z),$ $Q(z)$ ,we may assume that the leading coefficient of $Q(z)$ is 1. Then $hom$ the as-sumtion, $P(z)/Q(z)=\overline{P}(z)/\overline{Q}(z)$ , where $\overline{P}(z),\overline{Q}(z)$ are the polynomialsobtained from $P(z),$ $Q(z)$ by replacing the coefficients with the complex con-jugates of them. Since $Q(z)$ and $\overline{Q}(z)$ are monic, and have the same zeros,31we conclude that $Q(z)=\overline{Q}(z)$ . And hence, we can conclude similarly that$P(z)=\overline{P}(z).)$Here, we also recall how to get a garden from a real rational functionDefinition 5. Take an $f\in RH_{0,n}$ , and represent $\Sigma=\Sigma(f)$ as $\Sigma=\Sigma_{R}\cup\Sigma_{I}$ ,where $\Sigma_{R}$ is the set of real critical values of $f$ and $\Sigma_{I}$ is the set of its non-realcritical values. The number of elements in $\Sigma_{R}$ is denoted by $2l(\Sigma)$ .Let $S(f)$ be the preimage of the real line $\overline{\mathbb{R}}=\mathbb{R}\cup\infty$ under $f$ . Forevery element in $\Sigma_{R},$ $S(f)$ contains exactly 4 arcs incident to it. These arcstogether with $\overline{\mathbb{R}}\subset S(f)$ define a 2-dimensional cell complex on $\hat{\mathbb{C}}$ . The 2-cellsof this complex are called the faces of $S(f)$ . Here, $S(f)$ may contain simpleclosed curves called ovals as the connected components.To construct $G(f)$ we start from a planar chord diagram of order $2l(\Sigma)$ .The vertices of the diagram correspond to the critical points with real criticalvalues, and the chords correspond to the arcs of $S(f)$ lying inside the circle.Thus, the faces of the diagram correspond to the faces of $S(f)$ lying insidethe circle. The orientation of the edges is induced by the orientation of $\overline{\mathbb{R}}$ inthe image. To define the labeling of the chord diagram, consider the naturalorder $<$ on $\Sigma_{R}$ (if $\infty$ belongs to $\Sigma_{R}$ , we assume that it is the biggest criticalvalue). The label of a critical point equals the number of the correspondingcritical value under this order. To define the weights, consider an arbitrarypoint $x\in\overline{\mathbb{R}}-\Sigma_{R}$ , and for any given arc (or oval) let $w(x)$ be the number ofpreimages of $x$ lying on this arc (or oval). The weight of the arc (or oval) isthen defined as the minimum of $w(x)$ over all $x\in\overline{\mathbb{R}}-\Sigma_{R}$ .The root of the tree corresponds to the boundary of the face, and theinner vertices correspond to the ovals contained in the face. The weight ofan inner vertex is equal to the weight of the corresponding oval.Now noting that covering-equivalence should be taken by a real M\"obiustransformation, we see the following$Th\infty rem2$ . Every real mtional function of degree 2 is covering-equivalentto an element of the families$\{c+\frac{az+b}{z^{2}+1}\}$ , $\{z+c+\frac{b}{z}\}$ , $\{\pm z^{2}+a\}$ ,where $a,$ $b,$ $c\in \mathbb{R}$.Next, gardens of the total weight 2 are321. a garden of order $0$ with one mot only,2. a garden of order 2 with two roots.The first garden represents any generic element in the subfamily$\{z+c+\frac{b}{z}$ $b<0\}$ ,and the second garden represents any generic element in the subfamilies$\{c+\frac{az+b}{z^{2}+1}\}$ , $\{z+c+\frac{b}{z}$ $b>0\}$ , $\{\pm z^{2}+a\}$ .Proof. According as $\infty$ is a non-critical point or a critical one, we can showthat the given quadratic rational function is covering-equivalent to an elementof the first two families or of the third one, respectively. Here, every elementof the first family has poles $\pm i$ , and every one of the second family has twopoles on $\overline{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$ . (See the remark below.)Next, it is easy to see from the definition, that there are two kinds ofgardens.Now, for the first family, both of the critical points are real and finite if$a\neq 0$ and $0,$ $\infty$ if $a=0$ . For the second family, it is clear that $b\neq 0$ . If $b>0$ ,the family $z+c+(b/z)$ has two real critical points. For the third family,critical points are $0,$ $\infty$ . Thus every generic functions in these families arerepresented by the second garden.While, if $b<0$ in the second family, a generic element has no real criticalpoints, and is represented by the first garden. $\square$Remark 2. In the first family, by rotating around $\pm i$ , we may assume that$\infty$ is always a critical point. Hence we can replace the first family by asimpler one:$\{c+\frac{b}{z^{2}+1}\}$ .Lemma 1. Every real mtional functoin of degree 3 is covering-equivalent toan element of the families$\{z+c+\frac{az+b}{z^{2}+d}\}$ $(a, b, c, d\in \mathbb{R})$$\{\pm z^{2}+bz+c+\frac{a}{z}\}$ $(a, b, c\in \mathbb{R})$$\{z^{3}+bz+c\}$ $(b, c\in \mathbb{R})$33Pmof. Write the given real rational function as $P(z)/Q(z)$ with suitablepolynomials $P(z)$ and $Q(z)$ . If the degree of $Q(z)$ is 3, then the equation$Q(z)=0$ has a real solution $x_{0}$ . Sending $x_{0}$ to $\infty$ , we may assume that thedegree of $Q(z)$ is less than 3.The rest of the proof is similar to that in case of degree 2. $\square$Remark 3. The third family contains no generic real rational functions.The next lemma is easy to see from the definition.Lemma 2. Gardens of the weight 3 are1. a garden of order $0$ with one root only,2. anotgher garden of order $0$ with one rooted tree with one inner vertex,3. a garden of order 2 with two roots,4. a garden of order 4 with three roots.Now, if $d<0$ in the first family in Lemma 1, then by setting $d=-A^{2}$with a positive $A$ and using new real parameters $B$ and $D$ , we can rewritethe function as$f(z)=z+c+ \frac{B}{z-A}+\frac{D}{z+A}$ .If $B>0,$ $D>0$ , then there are two cases:1. $f(z)$ has 4 real critical points, or2. $f(z)$ has two real critical points and two non-real ones.Here, the phase transition occurs at the locus defined by$\Phi=(4A^{2}-B-D)^{3}-108BDA^{2}=0$ .So we devide the first family into the followings1. $E_{2}=\{(A, B, D)|\Phi>0, BD<0, or \Phi<0, B>0, D>0\}$2. $E_{4}=\{(A, B, D)|\Phi<0, BD<0, or \Phi>0, B>0, D>0\}$3. $E’=\{(A, B, D) I B<0, D<0\}$ .Here the locus defined by $\Phi=0$ is as in Figure 1.34Figure 1: The horizontal plane is $\{B+D=0\}$ ; the vertical one is $\{A=0\}$ .Theorem 3. 1. In the first family, if $d<0$ , then generic elements in thesubfamily $E_{2}$ or $E_{4}$ , or $E’$ are represented by the third garden, the $4$ -thone, or the first one, respectively.2. If $d>0$ in the first family, then we can write elements as$f(z)=z+c+ \frac{B}{z-iA}+\frac{\overline{B}}{z+iA}$ $(d=A^{2})$with real $A,$ $c$ and a complex $B=a+ib$. And if$\Phi’=-(4d+2a)^{3}+108(a^{2}+b^{2})d>0$ ,35then generic elements are represented by the third garden, and if $\Phi’<0$ ,generic ones are represented by the second one.3. For the second family, set$\Psi=a(a-b^{3}/27)$ .Then generic elements are represented by the third garden or the $4$ -thone, respectively, according as $\Psi>0$ or $\Psi<0$ .Proof. First, we consider the first family, and note that, if $d=0$ , then thefunction has a critical value $\infty$ , and is equivalent to an element of the secondfamily. Hence we may assume that $d\neq 0$ .The case 1). Assume that $d<0$ , and set$F(z)=(z^{2}-A^{2})^{2}-B(z+A)^{2}-D(z-A)^{2}$ .Elements in the first subfamily $E_{2}$ has 2 real critical points, and hence isrepresented by the third kind of gardens. Elements in $E_{4}$ has 4 real criticalpoints.These were shown in [1] when $B>0,$ $D>0$ . Actually, $\Phi=0$ gives thelocus where the number of critical points changes by 2. Here for instance,if we put $B=D=A^{2}/4$ , then $F(O)>0$ and $F(\pm A)<0$ , and hence theequation $F=0$ has 4 real solutions. On the other hand, $\Phi>0$ , we have theassertion in this case.And if $BD<0$ , assume for instance $B+D=0$ . Then$F(z)=(z^{2}-A^{2})^{2}-4ABz$and hence $F=0$ has exactly 2 real solutions, since$F(-A)F(A)<0$ .Here it is clear that $\Phi>0$ .Finally, we also know (cf. [1]) that, if $B<0,$ $D<0$ , then the element hasno real critical points. Since all poles belongs to $\overline{\mathbb{R}}$ , we have the assertion for$E’$ .The case 2). If $d>0$ in the first family, then the above $\Phi$‘ is the sameas $\Phi$ , and again $\Phi^{f}=0$ gives the locus where the number of critical pointschanges by 2. Here for instance, if $B$ is sufficiently near to $0$ , then $f(z)$ as36in Theorem 3.2) is strictly increasing, and hence a diffeomorphism of $\mathbb{R}$ ontoitself. Since $iA$ is a pole, such a function has a single inner vertex.On the other hand, if $B$ is sufficiently near to $0$ , then $\Phi’$ is negative, andwe have the assertion 2).The case 3). Generic elements in the second family always have a criticalpoint at $\infty$ , and we can conclude the assertion by direct calculations. $\square$References[1] M. Karima, Bell domains and critical points parameters for triply con-nected planar domains, peprint.[2] S. Natanzon, B. Shapiro and A. Vainshtein, Topological classificationof generic real rational functions, J. Knot Theory Ramffications, 11(2002), 1063-1075.[3] B. Shapiro and A. Vainshtein, Counting real rational functions with allreal critical values, Mosc. Math. J., 3 (2003), 647-659.37

Topological representation of the branched covering structure induced from a real rational function (Extensions of the historical calculus transforms in the geometric function theory)

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Topological representation of the branched covering structure induced from a real rational function (Extensions of the historical calculus transforms in the geometric function theory)

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