DDYNAMICS OF GENERALIZED NEVANLINNAFUNCTIONS
TAO CHEN AND LINDA KEEN
Abstract.
In the early 1980’s, computers made it possible to ob-serve that in complex dynamics, one often sees dynamical behaviorreflected in parameter space and vice versa. This duality was firstexploited by Douady, Hubbard and their students in early work onrational maps. See [DH, BH] for example.Here, we continue to study these ideas in the realm of transcen-dental functions.In [KK1], it was shown that for the tangent family, λ tan z ,the way the hyperbolic components meet at a point where the as-ymptotic value eventually lands on infinity reflects the dynamicbehavior of the functions at infinity. In the first part of this paperwe show that this duality extends to a much more general classof transcendental meromorphic functions that we call generalizedNevanlinna functions with the additional property that infinity isnot an asymptotic value. In particular, we show that in “dynami-cally natural” one dimensional slices of parameter space, there are“hyperbolic-like” components with a unique distinguished bound-ary point whose dynamics reflect the behavior inside an asymptotictract at infinity. Our main result is that every parameter point insuch a slice for which the asymptotic value eventually lands on apole is such a distinguished boundary point.In the second part of the paper, we apply this result to the fam-ilies λ tan p z q , p, q ∈ Z + , to prove that all hyperbolic componentsof period greater than 1 are bounded. Introduction
In the early 1980’s, computers made it possible to observe that incomplex dynamics, one often sees dynamical behavior reflected in pa-rameter space and vice versa. This duality was first exploited byDouady, Hubbard and their students in early work on rational maps.See [DH, BH] for example. Here, we continue to study these ideas inthe realm of transcendental functions.In [KK1], it was shown that for the tangent family, λ tan z , the waythe hyperbolic components meet at a point where the asymptotic value Mathematics Subject Classification. a r X i v : . [ m a t h . D S ] M a y TAO CHEN AND LINDA KEEN eventually lands on infinity, reflects the dynamic behavior of the func-tions at infinity. This is very different from what one sees for the expo-nential family exp z + c (see [DFJ, RG, Sch]) where all the hyperboliccomponents are unbounded. A crucial difference between these fami-lies is that for the exponential family infinity is an asymptotic valuewhereas for the tangent family it is not.In the first part of this paper we define a much more general class oftranscendental meromorphic functions that we call generalized Nevan-linna functions . We show that families of this class with the additionalproperty that infinity is NOT an asymptotic value exhibit this dual-ity; that is, in a one dimensional slice of the parameter space, theway the hyperbolic components meet at a point where the asymptoticvalue eventually lands on infinity reflects the dynamical behavior of thefunctions at infinity.To describe these functions and state our theorems, we need somebackground. The dynamical plane of a meromorphic map is dividedinto two sets: the Fatou or stable set on which the iterates are welldefined and form a normal family, and the
Julia set , its complement.The Julia set can be characterized as the closure of the set of repellingperiodic points, or equivalently, the closure of the set of pre-poles,points that map to infinity after finite iteration. A good introductionto meromorphic dynamics can be found in [Ber].The points over which a meromorphic function is not a regular cov-ering map are called singular values. There are two types of singu-lar values: critical values (images of zeroes of f (cid:48) , the critical points )and asymptotic values (points v = lim t →∞ f ( γ ( t )) where γ ( t ) → ∞ as t → ∞ ). If an asymptotic value is isolated, the local inverse there is thelogarithm. We denote the class of meromorphic functions with finitelymany singular values by F T . This class is particularly tractable: fam-ilies in
F T have parameter spaces with natural embeddings into C n ,where n is a simple function of the number of singular values; all as-ymptotic values are isolated; their Fatou domains have a simple clas-sification because there are no wandering domains (see Section 1 fordefinitions).The focus in this paper is on the subclass M ∞ of functions in F T for which infinity is not an asymptotic value. These necessarily haveinfinitely many poles and this behavior at infinity has consequences forthe dynamical and parameter spaces (see e.g. [BK]). In particular, it isvery different from that for entire functions or transcendental functionswith finitely many poles.
YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 3
A general principle in dynamics is that each stable dynamical phe-nomenon is “controlled” by a singular point. For example, each attract-ing or parabolic periodic cycle always attracts a singular value. Usingthis principle, one can define one dimensional slices of the parameterspace of a family in M ∞ that are “compatible with the dynamics” bykeeping all but one of the dynamical phenomena fixed and letting theremaining one, controlled by the “free singular value v ”, vary. Lookingfor regions in the slice where the free singular value is attracted to anattracting cycle gives a picture of how the “hyperbolic-like” compo-nents of the slice fit together around the bifurcation locus. Dynamically natural slices were defined in [FK] for families of func-tions in M ∞ . It was shown there that for those slices, as for slices ofparameter spaces of rational maps, the bifurcation locus contains pa-rameters distinguished by functional relations. There are Misiurewiczpoints where v lands on a repelling periodic cycle and centers where v lands on a super-attracting cycle. Another type of distinguished pa-rameter, not seen for rational maps, is a virtual cycle parameter , where v lands on a pole.Dynamically natural slices contain two different kinds of “hyperbolic-like” domains in which the functions are all quasiconformally conjugateon their Julia sets: capture components , where the free asymptoticvalue is attracted to one of the fixed phenomena — an attracting orsuper-attracting cycle, and shell components , where the free asymptoticvalue is the only singular value attracted to an attracting cycle. In ashell component, the period of the attracting cycle is constant andthe multiplier map is a well-defined universal covering map onto thepunctured disk. Properties of shell components for general families in M ∞ were studied in detail in [FK]. In particular, it was proved thatthe boundary of every shell component contains a special point, the virtual center where the limit of the multiplier map is zero. One of themain results proved there is Theorem A.
For families in M ∞ , a virtual center on the boundaryof a shell component in a dynamically natural slice is a virtual cycleparameter and any virtual cycle parameter on the boundary is a virtualcenter. In this paper, we concern ourselves with a special subclass in M ∞ .We start with the Nevanlinna functions , the family N r ⊂ F T of func-tions with r > M p,q,r by pre- and post-composing by polynomialsof degrees q and p respectively. We then look at M p,q,r ⊂ M ∞ , thesubset of functions in ˜ M p,q,r all of whose asymptotic values are finite. TAO CHEN AND LINDA KEEN
We first define the embedding of M p,q,r into C p + q + r +3 and then studyshell components of dynamically natural slices for this embedding. Weinclude the proof, given originally in [FK], that Proposition A. If h is topologically conjugate to a meromorphic func-tion f = P ◦ g ◦ Q in ˜ M p,q,r , and if h is meromorphic, then it is alsoin ˜ M p,q,r ; that is there is a function ˜ g ∈ N r and polynomials ˜ P , ˜ Q ofdegrees p, q respectively such that h = ˜ P ◦ ˜ g ◦ ˜ Q . A corollary is
Corollary A.
There is a natural embedding of ˜ M p,q,r into C p + q + r +4 and hence an embedding of M p,q,r into C p + q + r +3 . Theorem A does not preclude the existence of virtual cycle parame-ters that are not on the boundary of a shell component. For example,such a parameter might be buried in the bifurcation locus. Our firstnew theorem says this cannot happen.
Theorem B.
In a dynamically natural slice in M p,q,r , every virtualcycle parameter lies on the boundary of a shell component. Corollary B, which follows directly from Theorems A and B, saysthat in M p,q,r the notions of virtual center and virtual cycle parameterare equivalent. Corollary B.
In a dynamically natural slice of M p,q,r , every virtualcycle parameter is a virtual center and vice versa. In [FK] it was proved that for families in M ∞ , shell componentsof period 1 in dynamically natural slices are always unbounded and itwas conjectured that, in contrast to families of entire functions, those ofperiod greater than 1 are always bounded. The conjecture was provedtrue in the tangent family in [KK1]. In this paper we prove it forthe generalization of the tangent family, F = λ tan p ( z q ). This is asubfamily of M p,q,r . Theorem C.
Every shell component of period greater than in the λ plane of the family F is bounded. The paper is organized as follows. Part 1 is a discussion of the generalfamily M p,q,r and dynamically natural slices of its parameter space. InSection 1 we give a brief overview of the basic theory, set our nota-tion and discuss the theorem of Nevanlinna, Theorem 1.1, which weuse to define the class N r of Nevanlinna functions. Next, in Section 2,we define the classes ˜ M p,q,r and M p,q,r , and prove Proposition A and YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 5
Corollary A. In Section 2.1 we recall the properties of Fatou compo-nents that we need and define dynamically natural slices of parameterspaces and the shell components in these slices. In Section 3, we de-fine virtual cycle parameters and virtual centers, state Theorem A andprove Theorem B. Part 2 is a discussion of the special symmetric sub-family
F ⊂ M p,q, . In Section 4 we classify the shell components byperiod and discuss the special properties of the components of periods1 and 2. Finally, in Section 5 we prove Theorem C. Part The family M p,q,r Basics and Tools
Meromorphic functions.
In this paper, unless we specificallysay otherwise, we always assume that an entire or meromorphic mapis transcendental and so has infinite degree. If we mean a map of finitedegree we call it polynomial or rational. We also always assume infinityis an essential singularity. We need the following definitions:
Definition 1.
A point v ∈ C is called a singular value of f if, forsome small neighborhood of v , some branch of f − is not well defined.If c is a zero of f (cid:48) , it is a critical point and its image v = f ( c ) is a critical value . The branch with f − ( v ) = c is not well defined so acritical value is singular. A point is also singular if there is a path γ ( t ) such that lim t →∞ γ ( t ) = ∞ and lim t →∞ f ( γ ( t )) = v . The limit v issingular and called an asymptotic value of f . Singular values may becritical, asymptotic or accumulations of such points. We denote the setof singular points of f by S f . If an asymptotic value v is isolated, and γ is an asymptotic pathfor v , we can find a nested sequence of neighborhoods of v , D r,v with r →
0, and a particular branch g of f − such that V r = g ( D r,v ) is anested sequence of neighborhoods containing γ and ∩ g ( D r,v ) = ∅ . Then r can be chosen small enough so that the map f : V r → D r,v \ { v } isa universal covering map. In this case V r is called an asymptotic tract for the asymptotic value v and v is called a logarithmic singular value .The number of distinct asymptotic tracts of a given asymptotic valueis called its multiplicity . Definition 2.
A ray β approaching infinity is called a Julia ray or Julia direction for the meromorphic function f if f assumes all (but atmost two) values infinitely often in any sector containing β . In the literature these are sometimes called singularites of f − . Note this notion is not the same as multiplicity of a critical point.
TAO CHEN AND LINDA KEEN
An example to keep in mind is e z with asymptotic values at 0 and ∞ . The asymptotic tracts are the left and right half planes and theJulia directions are parallel to the positive and negative imaginary axes.Another example is tan z with asymptotic values ± i , asymptotic tractsthe upper and lower half planes and Julia directions parallel to thepositive and negative real axes.In this paper, whenever we talk about the number of critical pointsand/or asymptotic values, we tacitly assume that we count with mul-tiplicity.An entire function f always has an asymptotic value at infinity.We will only be interested in meromorphic functions with S f < ∞ .These are called functions of finite type . Note that all the asymptoticvalues of a finite type function are isolated and so are logarithmic.1.1.1. Nevanlinna Functions.
Nevanlinna, in [Nev, Nev1], Chap X1,characterized families of meromorphic functions with finitely many as-ymptotic values, finitely many critical points and a single essentialsingularity at infinity. (See [DK, KK1, EreGab] for further discussion.)Recall that the Schwarzian derivative of a function g is defined by(1) S ( g ) = ( g (cid:48)(cid:48) /g (cid:48) ) (cid:48) −
12 ( g (cid:48)(cid:48) /g (cid:48) ) . Because Schwarzian derivatives satisfy the cocycle relation S ( f ◦ g )( z ) = S ( f )( g (cid:48) ( z )) + S ( g )( z )and the Schwarzian derivative of a M¨obius transformation is zero, so-lutions to equation (1) are determined only up to post-composition bya M¨obius transformation.Nevanlinna’s theorem says Theorem 1.1 (Nevanlinna) . Every meromorphic function g with p < ∞ asymptotic values and q < ∞ critical points has the property that itsSchwarzian derivative is a rational function of degree p + q − . If q = 0 ,the Schwarzian derivative is a polynomial P ( z ) . In the opposite direc-tion, for every polynomial function P ( z ) of degree p − , the solutionto the Schwarzian differential equation S ( g ) = P ( z ) is a meromorphicfunction with exactly p asymptotic values and no critical points. Remark 1.1.
The proof of the first part of this theorem involves theconstruction of the function as a limit of holomorphic functions whoseSchwarzians are rational of bounded degree. The proof of the secondpart of the theorem involves understanding the asymptotic properties ofsolutions to the equation S ( g ) = P ( z ) . In particular, there are exactly p YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 7 “truncated solutions” g , . . . , g p − that, for any (cid:15) > , have asymptoticdevelopments of the form log g k ( z ) ∼ ( − k +1 z p/ defined in the sector | arg z − πk/p | < π/p − (cid:15) . Each g k is entireand tends to zero as z tends to infinity along each ray of the sector | arg z − πk/p | < π/p and tends to infinity in the adjacent sectors. Therays separating the sectors are the Julia rays for g ( z ) . See [Hille, Nev] for details. Definition 3.
We denote the family of meromorphic functions with p < ∞ asymptotic values and no critical values by N p and call thefunctions Nevanlinna functions . One immediate corollary is that Nevanlinna functions cannot haveexactly one asymptotic value. Moreover, since polynomials of degree p − p − Corollary 1.2.
Nevanlinna functions with p < ∞ asymptotic valueshave a natural embedding into C p +2 . We are interested in the dynamical systems generated by these Nevan-linna functions and some generalizations of them. Infinity plays a spe-cial role so we separate the cases where infinity is an asymptotic valuefrom those where it is not. For these functions, the dynamics are un-changed by conjugation by an affine transformation. Thus suitablynormalized solutions have a natural embedding into C p − In his proof of the above theorem, (see also [DK], Section 1), Nevan-linna shows that for f ∈ N p , a neighborhood of infinity is divided intoexactly p disjoint sectors, W , . . . , W p − , each with angle 2 π/p . Eachof these is an asymptotic tract for one asymptotic value. The sectorsare separated by the rays that bound them, and these are Julia rays.Although two tracts may map to the same asymptotic value, tracts inadjacent sectors must map to different asymptotic values.Another immediate corollary of Nevanlinna’s theorem is that thefamily N p is topologically closed. That is, Corollary 1.3. If f is topologically conjugate to a meromorphic func-tion g in N p , and if f is meromorphic, then it is also in N p . The family M p,q,r A family that is more general than N p is the family of functions˜ M p,q,r = { f = P ◦ g ◦ Q | g ∈ N r , P, Q polynomials of degrees p, q } . TAO CHEN AND LINDA KEEN
This family is also topologically closed. We have Proposition A. If h is topologically conjugate to a meromorphic func-tion f = P ◦ g ◦ Q in ˜ M p,q,r , and if h is meromorphic, (with essentialsingularity at infinity), then it is also in ˜ M p,q,r ; that is there is a func-tion ˜ g ∈ N r and polynomials ˜ P , ˜ Q of degrees p, q respectively such that h = ˜ P ◦ ˜ g ◦ ˜ Q . The proof uses quasiconformal mappings and the Measurable Rie-mann mapping theorem. We refer the reader to the standard literature([A, LV] and for a more dynamical discussion, [BF].
Proof.
We prove the theorem for h quasiconformally conjugate to f . Itthen follows from Theorem 3.3 of [KK1] that it is true for h topologi-cally conjugate to f .Let φ µ be a quasiconformal homeomorphism with Beltrami coeffi-cient µ such that h = φ µ ◦ f ◦ ( φ µ ) − = φ µ ◦ P ◦ g ◦ Q ◦ ( φ µ ) − is meromorphic. We can use P to pull back the complex structuredefined by µ = ¯ ∂φ µ /∂φ µ to obtain a complex structure ν = P ∗ µ suchthat the map ˜ P = φ µ ◦ P ◦ ( φ ν ) − is holomorphic. Note that this is not a conjugacy since it involves twodifferent homeomorphisms. Similarly, we can use g to pull back thecomplex structure defined by ν to obtain a complex structure η = g ∗ ν such that the map ˜ g = φ ν ◦ g ◦ ( φ η ) − is meromorphic. Again this is not a conjugacy. The function˜ P ◦ ˜ g = φ µ ◦ P ◦ ( φ ν ) − ◦ φ ν ◦ g ◦ ( φ η ) − is a composition of meromorphic functions and is meromorphic. Nowset ˜ Q = φ η ◦ Q ◦ ( φ µ ) − . Although ˜ Q is not conjugate to Q , we were given that h = φ µ ◦ P ◦ ( φ ν ) − ◦ φ ν ◦ g ◦ ( φ η ) − ◦ φ η ◦ Q ◦ ( φ µ ) − = ˜ P ◦ ˜ g ◦ ˜ Q is meromorphic so that ˜ Q is also meromorphic.The main point here is that although ˜ g is not a conjugate of g , sincethe quasiconformal maps φ µ and φ ν are homeomorphisms, the map ˜ g is a meromorphic map with the same topology as g ; that is, it has r This result is proved in [FK]. Since the proof is short and the ideas used arerelevant we include it here.
YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 9 asymptotic values and no critical values so that by Corollary 1.3 ˜ g be-longs to N r . Similarly, although ˜ Q and ˜ P are not respective conjugatesof Q and P , because the quasiconformal maps φ µ , φ ν and φ η are home-omorphisms, the maps ˜ Q and ˜ P are holomorphic maps with the sametopology as Q and P ; that is, they are respectively a degree q and adegree p branched covering of the Riemann sphere with the same num-ber of critical points and the same branching as Q and P . The maps φ µ , φ ν and φ η are defined up to normalization. If we want to keep theessential singularity of h at infinity, we normalize so that ˜ Q and ˜ P arepolynomials of degrees q and p respectively, (that is, infinity is a fixedcritical point with respective multiplicities q − p − (cid:3) A corollary to this theorem is
Corollary A.
The space of functions ˜ M p,q,r has a natural embeddinginto C p + q + r +4 and the subspace M p,q,r has dimension C p + q + r +3 .Proof. A polynomial of degree d is determined by its d + 1 coefficients.If g ∈ N r , set S g = S ( g ). It is determined by its r − w , w are linearly independentsolutions of the equation w (cid:48)(cid:48) + 12 S g w = 0then S ( w /w ) = S g . Since the space of solutions to this linear differ-ential equation has dimension 2, and the solution g of the Schwarzianequation is the quotient of two such solutions, g is determined by r + 2parameters. Requiring that infinity cannot be an asymptotic value re-stricts one parameter. Therefore ˜ M p,q,r has a natural embedding into C p + q + r +4 and the subspace M p,q,r has dimension C p + q + r +3 . (cid:3) Dynamics of meromorphic functions.
Let f : C → (cid:98) C be atranscendental meromorphic function and let f n denote the n th iterateof f , that is f n ( z ) = f ( f n − )( z ) for n ≥
1. Then f n is well-defined, ex-cept at the poles of f, f , · · · , f n − , which form a countable set. Thesepoints have finite orbits that end at infinity.The basic objects studied are the Fatou set and
Julia set of thefunction f . The Fatou set F ( f ) of f is defined by F ( f ) = { z ∈ C | f n is defined and normal in a neighborhood of z } , and the Julia set by J ( f ) = (cid:98) C \ F ( f ) . Note that the point at infinity is always in the Julia set. If f is ameromorphic function with more than one pole, then the set of pre-poles, P = ∪ n ≥ f − n ( ∞ ) is infinite. By the Picard theorem, f n isnormal on (cid:98) C \ P . Since it is not normal on P , J ( f ) = P , (see also[BKL1]).A point z is called a periodic point of p ≥
1, if f p ( z ) = z and f k ( z ) (cid:54) = z for any k < p . The multiplier of the cycle is defined to be µ = ( f p ) (cid:48) ( z ) . The periodic point is attracting if 0 < | µ | < super-attracting if µ = 0, parabolic if µ = e πiθ , where θ is rational number,and neutral if θ is not rational. It is repelling if | µ | > D is a component of the Fatou set, then f ( D ) is either a componentof the Fatou set or a component missing one point. For the orbit of D under f , there are only two cases: • there exist integers m (cid:54) = n ≥ f m ( D ) ⊂ f n ( D ), and D is called eventually periodic ; • for all m (cid:54) = n , f n ( D ) ∩ f m ( D ) = ∅ , and D is called a wanderingdomain .Suppose that { D , · · · , D p − } is a periodic cycle of Fatou compo-nents, then either:(a) The cycle is (super)attracting: each D i contains a point of aperiodic cycle with multiplier | µ | < D i are attracted to this cycle. If µ = 0, the criticalpoint itself belongs to the periodic cycle and the domain iscalled super-attracting.(b) The cycle is parabolic: the boundary of each D i contains a pointof a periodic cycle with multiplier µ = e πiq/m , ( q, m ) = 1, m adivisor of p , and all points in each domain D i are attracted tothis cycle.(c) The components of the cycle are Siegel disks: that is, each D i contains a point of a periodic cycle with multiplier µ = e πiθ ,where θ is irrational and there is a holomorphic homeomorphismmapping each D i to the unit disk ∆, and conjugating the firstreturn map f p on D i to an irrational rotation of ∆. The preim-ages under this conjugacy of the circles | ξ | = r, r <
1, foliatethe disks D i with f p forward invariant leaves on which f p isinjective.(d) The components of the cycle are Herman rings: each D i is holo-morphically homeomorphic to a standard annulus and the firstreturn map is conjugate to an irrational rotation of the annulusby a holomorphic homeomorphism. The preimages under this YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 11 conjugacy of the circles | ξ | = r, < r < R , foliate the diskswith f p forward invariant leaves on which f p is injective.(e) D i is an essentially parabolic (Baker) domain: the boundary ofeach D i contains a point z i (possibly ∞ ) and f np ( z ) → z i for all z ∈ D i , but f p is not holomorphic at z i . If p = 1, then z = ∞ . Definition 4.
Define the post-singular set of f as the closure of theorbits of the singular values; that is, P S f = ∪ ∞ n =0 f n ( S f ) . For notational simplicity, if a pre-pole s is a singular value, ∪ ∞ n =0 f n ( s ) is a finite set and includes infinity. Each non-repelling periodic cycle is associated to some singular point.In particular we have, see e.g. [M2], chap 8-11 or [Ber], Sect.4.3,
Theorem 2.1. If { D , · · · , D p − } is an attracting, superatttracting,parabolic or Baker periodic cycle of Fatou components, then for some i = 0 , , . . . , p − , D i contains a singular value. If { D , · · · , D p − } is acycle of rotation domains (Siegel disks or Herman rings) the boundaryof each D i contains the accumulation set of some singular value. We use the following definition of a hyperbolic function in ˜ M p,q,r Definition 5. An f ∈ ˜ M p,q,r is called hyperbolic if P S f ∩ J ( f ) = ∅ . Note that if all the singular values of f are attracted to attractingor super-attracting cycles then f is hyperbolic.For a discussion of hyperbolicity for more general meromorphic func-tions, see the discussion in [Z] and the references therein.Singularly finite maps may have Baker domains but Theorem 2.2. [BKL4] If S f is finite, then there are no wanderingdomains in the Fatou set. Holomorphic families.
In this paper, we are interested in thefamily M p,q,r ⊂ ˜ M p,q,r of meromorphic functions in ˜ M p,q,r for whichinfinity is not an asymptotic value. For each choice of triples ( p, q, r )this is an example of a holomorphic family. Below we state the generaldefinitions and results we need. This definition is equivalent to standard definitions of hyperbolic functions be-cause for these function S f is finite. Definition 6 (Holomorphic family) . A holomorphic family of mero-morphic maps over a complex manifold X is a map F : X × C → (cid:98) C ,such that F ( x, z ) =: f x ( z ) is meromorphic for all x ∈ X and x (cid:55)→ f x ( z ) is holomorphic for all z ∈ C . Definition 7 (Holomorphic motion) . A holomorphic motion of a set V ⊂ (cid:98) C over a connected complex manifold with basepoint ( X, x ) is amap φ : X × V → (cid:98) C given by ( x, v ) (cid:55)→ φ x ( v ) such that (a) for each v ∈ V , φ x ( v ) is holomorphic in x , (b) for each x ∈ X , φ x ( v ) is an injective function of v ∈ V , and, (c) at x , φ x = Id .A holomorphic motion of a set V respects the dynamics of the holo-morphic family F if φ x ( f x ( v )) = f x ( φ x ( v )) whenever both v and f x ( v ) belong to V . The following equivalencies are proved for rational maps in [McM]and extended to the transcendental setting in [KK1].
Theorem 2.3.
Let F be a holomorphic family of meromorphic mapswith finitely many singularities, over a complex manifold X , with basepoint x . Then the following are equivalent. (a) The number of attracting cycles of f x is locally constant in aneighborhood of x . (b) There is a holomorphic motion of the Julia set of f x over aneighborhood of x which respects the dynamics of F . (c) If in addition, for i = 1 , . . . , N , s i ( x ) is are holomorphic mapsparameterizing the singular values of f x , then the functions x (cid:55)→ f nx ( s i ( x )) form a normal family on a neighborhood of x . Definition 8 ( J − stability) . A parameter x ∈ X is a J -stable param-eter for the family F if it satisfies any of the above conditions. The set of non J -stable parameters is precisely the set where bifurca-tions occur, and it is often called the bifurcation locus of the family F ,and denoted by B X . In families of maps with more than one singularvalue, however, it makes sense to consider subsets of the bifurcation lo-cus where only some of the singular values are bifurcating, in the sensethat the families { g in ( x ) := f nx ( s i ( x )) } are normal in a neighborhood of x for some values of i , but not for all. We define B X ( s i ) = { x ∈ X | { g in ( x ) } is not normal in any neighborhood of x . } YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 13
In this paper we investigate dynamically natural one dimensionalslices of the holomorphic family M p,q,r . In these slices, roughly speak-ing, all the dynamic phenomena but one are fixed and the last is deter-mined by a “free asymptotic value”. We will study the components ofthe complement of the bifurcation locus in these slices. The parametersin them are J -stable.Precisely, (see [FK]) Definition 9.
A one dimensional subset Λ ⊂ X is a dynamicallynatural slice with respect to F if the following conditions are satisfied. (a) Λ is holomorphically isomorphic to the complex plane puncturedat points where the function is not in the family; for example,points where the number of singular values is reduced. The re-moved points are called parameter singularities . By abuse ofnotation we denote the image of Λ in C by Λ again, and denotethe variable in Λ by λ . (b) The singular values are given by distinct holomorphic functions s i ( λ ) , i = 1 , . . . , N − , and an asymptotic value v λ that is anaffine function of λ . We call v λ the free asymptotic value ; werequire that B Λ ( v λ ) (cid:54) = ∅ . (c) The poles (if any) are given by distinct holomorphic functions p i ( λ ) , λ ∈ Λ , i ∈ Z . (d) The critical values and some of the asymptotic values are at-tracted to an attracting or parabolic cycle whose period andmultiplier are constant for all λ ∈ Λ . We call these dynamicallyfixed singular values . The remaining singular values include v λ ;if there is only one remaining value, it may have arbitrary be-havior. If there are several, they satisfy a relation that persiststhroughout Λ so that the remaining dynamical behavior is con-trolled by the behavior of v λ . Examples of how this may workare discussed in Section 4. (e) Suppose v λ is attracted to A λ , the basin of attraction of anattracting cycle that does not attract any of the dynamicallyfixed singular values. Then the slice Λ contains, up to affineconjugacy, all meromorphic maps g : C → (cid:98) C that are quasicon-formally conjugate to f λ in C and conformally conjugate to f λ on C \ A λ . This is only for convenience of exposition and can be arranged by a holomorphicchange of coordinates in X . If the attracting cycle is parabolic, the functions in the slice are not hyperbolicbut the results hold. See [FK] for further discussion (f) Λ is maximal in the sense that if Λ (cid:48) = Λ ∪ { λ } where λ is aparameter singularity, then Λ (cid:48) does not satisfy at least one ofthe conditions above. In the components of the complement of the bifurcation locus inthese slices v λ is attracted to an attracting cycle a of fixed period; thisis the period of the component . We distinguish two cases:(i) a does not attract one of the dynamically fixed singular val-ues. We call these Shell components and denote the individualcomponents by Ω and the collection by S .(ii) a attracts one of the dynamically fixed singular values in addi-tion to attracting v λ . We call these Capture components . Theirproperties are very different from those of the shell componentsand we leave a discussion of them to future work.3.
Shell components
The properties of shell components are described in detail in [FK].We summarize them here. We assume F λ is the restriction of a holo-morphic family in M p,q,r to a dynamically natural slice Λ and denotea function in F λ by f λ . We need the following definitions Definition 10.
Suppose that for some λ , { a , a , . . . , a k − , a } satisfies f iλ ( a i ) = a i +1 mod k , i = 1 , . . . , k − where a = v λ and a = ∞ ; thatis, v λ is a prepole of order k − . Then λ is called a virtual cycleparameter of order k . This is justified by that fact that if γ ( t ) is anasymptotic path for v λ = a and if h is the branch of the inverse of f k taking ∞ to the prepole a then lim t →∞ f k ( h ( γ ( t )) = v λ = a . We can think of the points as forming a virtual cycle . Definition 11.
Let Ω be a shell component in Λ and let a λ = { a , a , . . . , a k − , a k − } be the attracting cycle of period k that attracts v λ . Suppose that as n →∞ , λ n → λ ∗ ∈ ∂ Ω and the multiplier µ λ n = µ ( a λ n ) = Π k − f (cid:48) ( a ni ) → . Then λ ∗ is called a virtual center of Ω . Since the attracting basin of the cycle a λ must contain v λ , we willassume throughout that the points in the cycle are labeled so that v λ and a are in the same component of the immediate basin.The next theorem collects the main the results in [FK] about shellcomponents in a dynamically natural slice Λ for a holomorphic family YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 15 of transcendental functions with finite singular set, none of whose as-ymptotic values is at infinity. Note that Theorem A of the introductionis part (c) of the theorem.
Theorem 3.1.
Let Ω be a shell component in Λ . Then if D ∗ = { z :0 < | z | < } ( a) The map µ λ : Ω → D ∗ is a universal covering map. It extendscontinuously to ∂ Ω and ∂ Ω is piecewise analytic; either Ω issimply connected and µ λ is infinite to one or Ω is isomorphic apunctured disk and the puncture is a parameter singularity.( b) There is a unique virtual center on ∂ Ω . If the period of thecomponent is , the virtual center is at infinity.( c) Every (finite) virtual center of a shell component is a virtual cy-cle parameter and any virtual cycle parameter on the boundaryof a shell component is a virtual center.
Here we prove a stronger theorem for slices of the family M p,q,r ;recall that all their asymptotic values are finite. Note that because thefunctions are of the form f ( z ) = Q ◦ g ◦ P ( z ), if v is an asymptoticvalue of g , then Q ( v ) is an asymptotic value of f . There are p distinctasymptotic tracts corresponding to each asymptotic tract of v so thatthere are pr distinct asymptotic tracts at infinity separated by the Juliadirections. The asymptotic values, tracts and Julia directions dependholomorphically on the parameters. At each pole of f of order k thereare kqr pre-asymptotic tracts and the pull-backs of the rays in the Juliadirections separate them. Theorem B.
Let Λ be a dynamically natural slice for the meromorphicfamily M p,q,r consisting of meromorphic functions of the form f λ = Q ◦ g ◦ P all of whose asymptotic values are finite, and let λ ∗ be avirtual center parameter of order k . Then λ ∗ is on the boundary of ashell component of order k in Λ . That is, in any neighborhood of λ ∗ there exists λ ∈ Λ such that f λ has an attracting cycle of period k . As an immediate corollary to Theorem 3.1 and Theorem B is
Corollary B.
In a dynamically natural slice of M p,q,r , every virtualcycle parameter is a virtual center and vice versa. Remark 3.1.
The essence of the theorem is that the dynamic pictureat the poles is reflected in the parameter picture at the virtual centerparameters. If infinity is an asymptotic value, both the dynamics andparameter pictures are different. This is analogous to the situation for Misiurewicz points in the parameter planefor quadratic polynomials.
Proof.
Since λ ∗ is a virtual center parameter of order k , f λ ∗ has a virtualcycle a ∗ = { a ∗ = v λ ∗ , . . . , a ∗ k − , a ∗ = ∞} . For each j = 2 , . . . , k −
1, the cycle uniquely determines a branch of the inverse of f λ ∗ , f − λ ∗ ,j such that f − λ ∗ ,j ( a ∗ j ) = a ∗ j − . By abuse of notation, for readability, wedrop the j and denote all of these branches by f − λ ∗ . Because we arein a dynamically natural slice of a holomorphic family, the analyticcontinuations of the f − λ ∗ , denoted by f − λ are well defined.Note however, that in a neighborhood of the asymptotic value a ∗ ,the inverse branch of f λ ∗ is not uniquely determined. Since v λ ∗ is partof a virtual cycle, a neighborhood U of v λ ∗ has at least one pre-imagethat is in an asymptotic tract. If the multiplicity of v λ ∗ is one, thenby definition there is a unique asymptotic tract that is determinedby the virtual cycle of f λ ∗ ; we denote it by A λ ∗ and take as f − λ ∗ thebranch that maps U to A λ ∗ . Then taking the analytic continuationof this f − λ ∗ we obtain A λ as the analytic continuation of A λ ∗ . If themultiplicity of v λ ∗ is greater than one there will be a choice amongthe tracts corresponding to v λ ∗ (and hence the inverse branch f − λ ∗ ).Similarly, if there is more than one asymptotic value that varies with λ (and satisfies a functional relation with λ ), we choose the tract (or oneof them if there is more than one) corresponding to the free asymptoticvalue . In the argument below, we take A λ ∗ , or choose one of the tractsas A λ ∗ , along with the corresponding branch f λ ∗ . Since, in general,the asymptotic value is not omitted, there may also be infinitely manyinverse branches of the neighborhood that are bounded. We don’t needto concern ourselves with them here.Because we are in a dynamically natural slice we have the followingholomorphic functions:(1) v ( λ ) = v λ is the free asymptotic value of f λ .(2) If p ∗ = a ∗ k − , then p ( λ ) = p λ is the holomorphic function defin-ing the pole of f λ such that p ( λ ∗ ) = p ∗ .(3) Note that for λ in a neighborhood V ⊂ Λ of λ ∗ , the affinefunction λ (cid:55)→ v λ determines a corresponding neighborhood U of v ∗ λ in the dynamic plane of f λ ∗ and vice versa. Define themap h : V → C by h ( λ ) = f k − λ ( v λ ) = h λ . Then if V is smallenough, both p λ and h λ are in (cid:98) U ⊂ U , a small neighborhood of p ∗ in the dynamic plane of f λ ∗ .(4) In the dynamic plane of f λ , set u ( λ ) = f − ( k − λ ( p λ ) = u λ . Then u λ is the preimage of the pole p λ in a neighborhood of theasymptotic value v λ . See the examples in Part 2.
YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 17 (5) Each u λ has infinitely many inverses in the asymptotic tract A λ ; denote these by w λ,j , j ∈ Z .The main ideas for the proof are first to use the relation between λ and the free asymptotic value v λ to carefully choose and fix a λ closeto λ ∗ and then to construct a domain T ⊂ A λ such that f kλ ( T ) ⊂ T . Itwill then follow from Schwarz’s lemma that f kλ has an attracting fixedpoint in T . S Figure 1.
The map g λ on parameter space. S is a sectorinside all the asymptotic tracts A λ , λ ∈ V . Note that g ( λ ∗ ) = ∞ . Choosing λ : (See figure 1.) The asymptotic tract A λ ∗ lies betweentwo Julia rays r ∗ , r ∗ and these span an angle θ pr = 2 π/pr . We choosea small enough neighborhood V of λ ∗ such that for each f λ , λ ∈ V , theJulia rays r ( λ ) , r ( λ ) of f λ , lie within δ = δ ( V ) of r ∗ , r ∗ respectively.Each f λ has an asymptotic tract A λ between these rays and we canfind a set S ⊂ ∩ λ ∈ V A λ which lies inside the sector in (cid:98) C bounded bythe rays r ∗ and r ∗ such that S is a sector in (cid:98) C with vertex at infinityand angle θ pr − δ .As above, let U be a neighborhood of v λ ∗ and let V be the corre-sponding neighborhood in parameter space. For each λ ∈ V , v λ ∈ U and f k − λ ( v λ ) is defined. Define g λ : V → C by g ( λ ) = f k − λ ( v λ ).Since V contains λ ∗ and g λ is holomorphic, we can find R = R V suchthat g ( λ ) = | f k − λ ( v λ ) | > R. Set D R = { z ∈ C | | z | > R } and let (cid:98) S V = g − λ ( S ∩ D R ) be the “triangular” subset of V with vertex at λ ∗ .Note that the number of such triangular sets is equal to the order ofthe pole. If it is one, the set is unique. If it is greater than one, wechoose one arbitrarily. Figure 2.
The dynamic plane for f λ . The region f nλ ( T ) iscontained inside T . Constructing T : (See figure 2.) Now we work with a fixed λ ∈ (cid:98) S V and we set U = f λ ( A λ ). Since S ⊂ A λ we have f λ ( S ) ⊆ U . In U wehave v λ and u λ = f − ( k − λ ( p λ ). Let C be a circle with center v λ andradius | v λ − u λ | = η λ . Taking λ closer to λ ∗ if necessary, we may assume C lies in a compact subset of U ; that is the disk { z | | z − v λ | ≤ η λ } ⊂ U .Now (cid:98) C = f − λ ( C ) ⊂ A λ and f λ : (cid:98) C → C is an infinite to one cover so (cid:98) C contains preimages w λ,j , j ∈ Z , of u λ .We want to approximate the distance from a point on (cid:98) C to ∂A λ . Let φ : A λ → H l and ψ : U \ { v λ } → D ∗ be a conformal homeomorphismsfrom the left half plane to A λ and the punctured disk to U \ { v λ } respectively chosen such that f λ = ψ − ◦ exp ◦ φ .Because ψ is a homeomorphism, | ψ (cid:48) ( z ) | is bounded above and be-low for z in a closed disk containing C . Applying Koebe’s distortion YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 19 theorem, there are positive constants K , K (cid:48) such that for z ∈ C , K η λ (1 + η λ ) < | ψ − ( z ) | < K (cid:48) η λ (1 − η λ ) . We are assuming η λ is small so we may assume it is less than 1 / D ∗ and U containing ψ ( C ) and C respectively.We then have, for appropriate positive constants, K , K (cid:48) ,(2) K + | log η λ | < | Re log ψ − ( z ) | < K (cid:48) + | log η λ | . This says that the lift log ψ − ( C ) of the circle C lies in a vertical strip W of bounded horizontal width in H l . Let (cid:99) W denote φ − ( W ) ⊂ A λ ; itis the lift of the annulus in U to A λ .The covering group Z for the exponential map acting on H l pullsback to an infinite cyclic group Γ generated by a map γ : z (cid:55)→ γ ( z )that is the covering group acting on A λ under the map f λ so that φ ( γ ( z )) = w + 2 πi. Thus φ (cid:48) ( z ) dz = dw and φ (cid:48) ( γ ( z )) γ (cid:48) ( z ) dz = dw. This says that dwdz = φ (cid:48) ( z ) = φ (cid:48) ( γ ( z )) γ (cid:48) ( z ) , and because | φ (cid:48) ( z ) | is bounded in the closure of a fundamental domainfor Γ intersected with (cid:99) W , it is bounded for all w in (cid:99) W . Now because φ is a conformal homeomorphism it preserves the hyperbolic density sothat ρ A λ ( γ ( z )) | dγ ( z ) | = | dw || Re w | ρ A λ ( γ ( z )) | γ (cid:48) ( z ) dz | = | dw + 2 πi || Re( w + 2 πi | ) = | dw || Re w | . Therefore the hyperbolic density in A λ is invariant under the actionof γ . Equation (2), says that | Re w | is comparable to | log η λ | whichgives us the estimate we want; that is, there are positive constants K , K (cid:48) such that min ζ ∈ ∂A λ ,y ∈ (cid:98) C | y − ζ | ∼ K + K (cid:48) | log η λ | . Note that because we have a group action on A λ , there will be a ζ ∈ ∂A λ and a y ∈ (cid:98) C in each fundamental region. Now let ˜ T be a triangularregion in U with one vertex at u λ , two sides spanning an angle of θ < θ pr /m , where m is the order of the pole p λ , joining u λ to C , and third side an arc of the circle C ; the sides are chosen so that ˜ T contains v λ in its interior. Next set T = f − λ ( ˜ T ). First note that T ⊂ A λ and f − λ ( ∂ ˜ T ) is a doubly infinite curve in A λ that stays a bounded distancefrom (cid:98) C , where the bound depends on η λ and θ . The inverse images ofsides of the triangle form scallops “above” (cid:98) C (further inside A λ ).Now look at f k − λ ( ˜ T ). This is a triangular shaped region contained in f k − ( U ) with a boundary point p λ ; it contains f k − λ ( v λ ). Choose y ∈ (cid:98) C realizing the minimum above so that f − λ ( y ) is contained in f k − ( U ).Then, as p λ is a pole of order m , we have(3) | f − λ ( y ) − p λ | ∼ K | log η λ | m . Since the derivative of f λ ∗ along the orbit of v λ ∗ from v λ ∗ to p λ ∗ isbounded and varies holomorphically with λ , the derivative of f λ alongthe orbits of v λ from v λ to f k − λ ( v λ ) and u λ to p λ are also bounded for λ ∈ (cid:98) S V . Because v λ ∈ ˜ T and η λ is small, when we map by f k − λ theimages of v λ and points on C ∩ ˜ T are comparably close to the pole p λ .Specifically, for some positive constant K , we have | f k − λ ( v λ ) − p λ | ∼ K η λ << K | log η λ | m . Comparing this with the estimate (3) it follows that f k − λ ( C ∩ ˜ T ) ⊂ T and f k − λ ( v λ ) ∈ f kλ ( T ) . Therefore because v λ is inside ˜ T , f kλ ( T ) ⊂ T so that by the Schwarz lemma f kλ has a fixed point in T . (cid:3) Part The Extended Tangent Family λ tan p z q . In this part of the paper we use the results above to prove thatfor family F = { λ tan p z q } , every shell component of period n > The shell components of F The family F = f λ = λ tan p z q is a subfamily of M p,q,r = { P ◦ g ◦ Q } with P ( z ) = z p and Q ( z ) = z q and g in the one dimensional slice of N consisting of functions that fix 0 and have symmetric asymptotic values.The functions in F have one fixed critical point at 0 and have either YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 21 one asymptotic value with mulitplicity 2 q or two asymptotic valueswith multiplicity q that are opposite in sign. Specifically, the maptan z has two distinct asymptotic tracts and two asymptotic values, ± i . Each of these tracts has q pre-images under Q − . If p is even, all2 q of the asymptotic tracts map onto punctured neighborhoods of thesingle point i p λ = ( − p/ λ which is the free asymptotic value v λ . If p is odd, q of the asymptotic tracts map onto punctured neighborhoodsof i p λ and the other q tracts map onto punctured neighborhoods of( − i ) p λ . In this case we choose v λ = i p λ as the free asymptotic value.The other asymptotic value satisfies the relation v (cid:48) λ = − v λ .The punctured plane λ (cid:54) = 0 is thus a dynamically natural slice in M p,q, and the dynamics are determined by the forward orbit of v λ .The full set of shell components in this slice is denoted by S = { Ω } and we divide it into subsets depending on the period of the cycle asfollows: Definition 12. If pq is even S n = { Ω n | f λ has an attracting cycle of period n } , otherwise S n = { Ω n | f λ has one attracting cycle of period n or Ω (cid:48) n | f λ has two attracting cycles of period n } . For readability we only include the subscript on Ω if the period isnot obvious from the context.Figure 3 shows the parameter plane for the family f λ ( z ) = λ tan z .The period 1 shell components are yellow, the period 2 shell compo-nents are cyan blue. The capture components are green.4.1. Symmetries.
Note that for any λ , f ¯ λ (¯ z ) = f λ ( z ). In addition, if ω k , k = 0 , . . . , q −
1, are the q th roots of unity, f ω k λ ( ω k z ) = ω k λ tan p ( ω k z ) q = ω k f λ ( z ) . It follows that if Ω ∈ S n then Ω , ω k Ω ∈ S n .Suppose pq is even so that f λ has a single attracting cycle { z , . . . , z n } of period n with multiplier µ ( λ ). Then {− z , . . . , − z n } is a cycle for f − λ and µ ( − λ ) = µ ( λ ).If pq is odd then f λ ( − z ) = − f λ ( z ) = f − λ ( z ) and f ¯ λ ( z ) = f λ (¯ z ) . Assume that f λ has two cycles of period n . They must be symmetric:that is they are { z , . . . , z n } and {− z , . . . , − z n } and they have thesame multiplier. Figure 3.
The parameter plane for λ tan z . Now f − λ ( z ) = − z , f − λ ( − z ) = z , . . . , f m − λ ( z ) = ( − m z m +1 , sothat if n is even, f n − λ ( z ) = z and f − λ also has two cycles of period n .The set of periodic points of f − λ is the same as that for f λ but theydivide into different cycles for λ and − λ ; again, µ ( − λ ) = µ ( λ ). If,however, n is odd, f n − λ ( z ) = − z and f − λ has a single cycle of period2 n ; it has multiplier µ ( λ ). We summarize this discussion as follows. Proposition 4.1. If Ω is a hyperbolic component of the λ plane, then − Ω , ¯Ω and ω k Ω , k = 0 , q − are all hyperbolic components. That is,the parameter plane is symmetric with respect to reflection in the realand imaginary axes, rotation by π and rotation by q th roots of unity.If pq even, µ ( λ ) = µ ( − λ ) for λ ∈ Ω , while if pq is odd and thereare two cycles for λ ∈ Ω , then each has multiplier µ ( λ ) and there is asingle cycle of double the period for − λ ∈ − Ω such that for this cycle, µ ( λ ) = µ ( − λ ) . Components of S . In this section we will show that S consistsof exactly 2 q unbounded components arranged symmetrically aroundthe origin. Let η k = η + k , k = 0 , . . . , q −
1, be the roots of η qk = i and let η − k , k = 0 , . . . , q −
1, be the roots of ( η − k ) q = − i . If q is odd these arelabeled so that η − k = − η + k whereas if q is even they are labeled so that η − k = η + k . YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 23
Theorem 4.2.
The set S consists of q unbounded components, Ω ± k , k = 0 , . . . , q − , such that each is symmetric about a ray (cid:96) ± k in thedirection i − p η ± k ; that is, if λ ∈ (cid:96) ± k , v λ = sη ± k , s > .Moreover, for λ ∈ (cid:96) ± k ∩ Ω ± :(i) When both p, q are even: f λ has a single attracting fixed point z λ and arg z λ = arg v λ ;(ii) When p is odd and q is even, again f λ has a single attractingfixed point z λ and arg z λ = arg v λ or arg( − v λ ) ;(iii) When pq is odd: f λ either has two attracting fixed points, z λ and − z λ or a single attracting period two cycle { z λ , − z λ } . Moreover,if λ ∈ (cid:96) + k , f λ has two attracting fixed points, z λ on (cid:96) + k and − z λ on (cid:96) − k , which attract v λ and − v λ respectively; if λ ∈ (cid:96) − k ,then f λ has one attracting cycle of period two, { z λ , − z λ } whichattracts both v λ and − v λ and we can label the points so that arg z λ = arg v λ .Proof. If f λ ( z ) ∈ S and has an attracting fixed point z = z λ , thenusing the relation(4) λ = z tan p z q , we compute that the multiplier µ ( z ) is µ ( z ) = pqz q − tan p − z q sec z q = 2 pq z q sin 2 z q . Set u = 2 z q , so that µ ( z ) = h ( u ) = pqu/ sin u . The locus | h ( u ) | = 1in the u = x + iy plane consists of two branches, one in the upperhalf plane and one in the lower half plane. Call the unbounded regionsdefined by these curves U ± respectively. Each is symmetric with respectto both the real and imaginary axes. Moreover the regions intersectthe imaginary axis in the intervals u = ± ir q , r > r , respectively,where h ( ± ir q ) = 1. Inside these domains | h ( u ) | <
1. As | y | → ∞ , thebranches are asymptotic to the curves ± e | y | pq + iy. For each u in U ± there are q corresponding z = z λ ’s and these form2 q regions V ± k , k = 0 , . . . , q − C . The rays in the directions η ± k are axes of symmetry for the V ± k . We give the argument for η k = η + k ;the argument for η − k follows similarly. That is, we want to show that if z ∈ V k , then z = η k ¯ z is also in V k . Now if z ∈ V k , then z q ∈ U + andso z q = − ¯ z q is also in U + . Taking appropriate q th -roots, the symmetryfollows. From equation (4) the images Ω ± k of the V ± k are in S . First, thesymmetry of each of the V ± k ’s translates into a symmetry of the corre-sponding Ω ± k . Suppose z , z = ( η + k ) z ∈ V + k . If p is even we have λ ( z ) = η + k ¯ z tan p ( − ¯ z q ) = η + k λ ( z )which implies that the image of V + k is symmetric about the lines indirection η + k . Similarly for V − k about the lines in direction η − k . If,however, p is odd, λ ( z ) = η ± k ¯ z tan p ( − ¯ z q ) = − η ± k λ ( z )so that the images of V ± k are symmetric about the rays in directions iη ± k .To see that there are 2 q distinct domains Ω ± k , assume that z is on aline of symmetry for V ± k ; that is, z = sη ± k , s > r . By equation (4) wehave λ ( iη ± k ) = sη ± k tan p ( ± is q ) = sη ± k ( ± i ) p tanh p ( s q ) . If pq is even, on the rays where z = sη + k and z = sη − k , the argumentsof the corresponding λ ’s under the relation (4) are different, so that theimages of V + k and V − k are different. Therefore there are 2 q componentsin S corresponding to V ± k ; these are denoted by Ω ± k . Moreover, if z = sη + k and p is even, then arg v λ = arg z . If however p is oddand q is even, then if z = sη + k , arg v λ = arg z but if z = sη − k , thenarg v λ = arg z + π . In either case, both z and v λ are perpendicular to λ . If pq is odd, then λ ( z ) is an even function, z tan p z q = − z tan p ( − z ) q , so that both V ± k have the same image Ω + k . This gives us q componentsin S . For λ in Ω + k , f λ has two fixed points z and z and they both havethe same multiplier, µ ( λ ). Moreover on the symmetry lines, z = sη ± k ,we have arg v λ = arg sη ± k . Consider the q components Ω − k = { λ | − λ ∈ Ω + k } . As we saw in the discussion of Proposition 4.1, when λ ∈ Ω − k , f λ has one attracting cycle with period 2 and multiplier µ ( λ ) so thesecomponents are also in S and there are 2 q components in S in thiscase as well. (cid:3) YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 25
The proof of Theorem B and the statement of Theorem 4.2 tell ussomething about the structure of the parameter plane for the family F λ in a neighborhood of a virtual center. We have Corollary 4.3.
In the parameter plane of the family F λ , there are pq shell components of period k that meet at every virtual center parameterof order k > . That is, the virtual center parameter is a commonboundary point of these pq components. Infinity is a common boundarycomponent of q components.Proof. In the proof of Theorem B we used that fact that the virtualcenter parameter λ ∗ corresponds to a particular prepole p λ ∗ of f λ ∗ andused one of the asymptotic tracts at p λ ∗ in our construction to find theshell component at λ ∗ . There are 2 pq tracts at each prepole; these cor-respond to the asymptotic values tied to v λ by the functional relation.We could have used any one of these in the argument above to obtaina shell component at the virtual center. (cid:3) Separating Lines.
In this section we discuss the rays in the λ plane spanned by the roots of v qλ = 1 and their negatives.Let ω ± k , k = 0 , . . . , q − ω qk = 1 and ( ω − k ) q = − v λ = i p λ is the free asymptotic value, the rays in the λ planewe are interested in are those spanned by i − p ω ± k . Denote them by˜ (cid:96) k = i − p rω ± k , r > Lemma 4.4.
For the family F λ ,(i) If pq is even, none of the rays in the set R (cid:48) = { i − p rω ± k , r > } qk =1 intersects any component of S ; that is S ∩ R (cid:48) = ∅ for all k .(ii) if pq is odd, at each virtual cycle parameter λ on the ray ˜ (cid:96) k ,such that v λ is a pre-pole of f λ of order n , n is odd and thereare two components of S n +1 intersecting the ray. In one of thesecomponents there are two periodic cycles of order n + 1 and inthe other there is a single cycle of order n + 1) attracting bothasymptotic values.Proof. Set λ = ri − p ω ± k for some real r so that v qλ is real. Notice thatthis implies f λ ( v λ ) lies in the same line as λ . We have to look at theparities of p and q .To prove (i), suppose first that p is even. Then λ and v λ are inthe same line. Moreover, since λ q is real, it follows that λ , v λ , andall its images under f λ , lie on the same line, although perhaps onopposite sides of the origin. Therefore this line is invariant under f λ and the restriction of f λ to this line is conjugate to a real-valued function f r ( t ) = r tan p t q .Next suppose p is odd and q is even. Then λ and v λ are in perpen-dicular lines. Since q is even, λ q is real so ( f λ ( v λ )) q is real, and f λ ( v λ )is also on the line through λ , as is the rest of its orbit. That is, f λ is invariant on the line through λ and the orbit of v λ eventually landson the line. Again the restriction of f λ to this line is conjugate to areal-valued function f r ( t ) = r tan p t q .We claim that in these cases, f λ cannot have an attracting cycle otherthan the super-attracting fixed point 0. We consider the family of realvalued functions f r ( t ) = r tan p t q . On the interval (0 , q (cid:112) π/ f r , f (cid:48) r and f (cid:48)(cid:48) r are all positive and f r goes from 0 to infinity. Therefore there is onefixed point z inside this interval. Because 0 is a superattracting fixedpoint, its basin contains the interval (0 , z ). By symmetry, about theimaginary axis if p is even and about the origin if p is odd, the basinof 0 contains the interval I = ( − z , z ). Since f (cid:48)(cid:48) r ( t ) > f (cid:48) r ( t ) > t in ( z , q (cid:112) π/ t has period π , | f (cid:48) r ( t ) | < I , the unionof the translates of the interval I .We claim there are no other attracting periodic cycles in R (cid:48) . If therewere such a cycle, { z , . . . , z n } , its points would have to be outside I and its multiplier would have to satisfy Π n | f r ( z n ) | <
1. This cannothappen since none of these factors can be less than 1.To prove (ii), we have both p and q odd. Then λ q and ( f λ ( v λ )) q arepure imaginary. Therefore f λ ( v λ ) = λ tan p ( f λ ( v λ )) q = ± i tanh p ( f λ ( v λ ) q is in the line containing v λ . The orbit of v λ thus alternates betweentwo perpendicular lines, so if it approaches an attracting cycle, thatcycle must have even period and hence λ does not belong to S .Suppose λ is on the ray spanned by ri − p ω k , and is inside one of thecomponents Ω k intersecting it. The multiplier of the cycle is real andmonotonic in Ω k . The virtual center λ ∗ of Ω k is therefore on the ray.By Corollary 4.3, there are 2 pq components at λ ∗ and since pq is odd,as we move around λ ∗ , they alternate between those with two cycles ofperiod n +1 and one cycle of period 2( n +1) attracting both asymptoticvalues. Again since pq is odd, there is one of each type intersecting theray. (cid:3) Components of S . By Theorem 3.1, each component in S issimply connected and the multiplier map is a universal covering. More-over, each component has a virtual center on its boundary where thelimit of the multiplier is zero, and if this is finite, it is a virtual center YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 27 parameter so that the asymptotic value is a pre-pole. For complete-ness, we include a proof of the last statement for components of S n sothat we can use it to characterize the virtual centers of S . Lemma 4.5.
If the shell component Ω ∈ S n , is bounded, then thevirtual center λ ∗ satisfies f n − λ ∗ ( v λ ∗ ) = ∞ . That is, the asymptoticvalue is a pre-pole of order n and the virtual center is a virtual cycleparameter.Proof. Without loss of generality, we assume f λ has an attracting peri-odic cycle of period n ; the proof is similar if f λ has one attracting cycleof period 2 n . Let { z , z , · · · , z n − } be the attracting cycle of f λ , andsuppose z lies in an asymptotic tract. The multiplier of the cycle is n − (cid:89) i =0 f (cid:48) λ ( z i ) = n − (cid:89) i =0 λpqz q − i tan p − z qi cos ( z qi ) = (2 pq ) n n − (cid:89) i =0 z qi sin(2 z qi )If the multiplier tends to 0 as λ → λ ∗ , then at least one of the factors z q / sin(2 z q ) tends to 0, which in turn implies that the imaginary part z q is unbounded. Since z is in the asymptotic tract, it follows that as λ → λ ∗ , | Im z q | → ∞ and z = λ tan p z q → v λ ∗ . Because z = λ tan p z qn − and λ is bounded, it follows that z qn − → kπ + π/ λ → λ ∗ and f n − λ ∗ ( v λ ∗ ) = ∞ . (cid:3) Lemma 4.6. If Ω is a component of S , then its virtual center is asolution of v qλ = kπ + π/ for some integer k .Proof. By Lemma 4.5, we only need to show that each component of S is bounded.Suppose there is an unbounded component Ω ∈ S and f λ has aperiodic cycle { z , z } . We will show that if Ω is unbounded then bothpoints lie in asymptotic tracts, and therefore that p and q are both odd.This in turn implies that f λ has only one attracting cycle of period 2so that Ω ∈ S and not in Ω ∈ S .Let λ ( t ) → λ ∗ , t → ∞ be a path in Ω such that the multiplier of theperiodic cycle µ ( λ ( t )) →
0. If Ω is unbounded, the multiplier of theperiodic cycle has absolute value 1 for any finite point on the boundary.Therefore, “virtual center” λ ∗ of Ω is at infinity. From the proof ofLemma 4.5, the point z ( λ ( t )) of the cycle must be in the asymptotictract and | Im z ( λ ( t )) q | → ∞ and thus z ( λ ( t )) = λ tan p z ( λ ( t )) q (cid:16) i p λ is also goes to infinity.Suppose z ( λ ( t )) q = x ( t ) + iy ( t ) (cid:16) i pq λ ( t ) q . We claim that if y ( t )goes to infinity, then z ( λ ( t )) is also in an asymptotic tract. If it is not,then x ( t ) → ∞ but y ( t ) stays bounded. Take a sequence z ( t k ) such that x ( t k ) = kπ . Thentan( x ( t k ) + iy ( t k )) = tan( iy ( t k )) = i tanh( y ( t k )) = B k i, where B k is real and bounded.Since z ( t k ) = λ tan p ( z ( t k )) q = λ ( B k i ) p , z ( t k ) q /z ( t k ) q = B pqk ∈ R ;that is, z ( t k ) is asymptotically parallel to z ( t k ) with a bounded ratioso that the imaginary part of z q ( t ) remains bounded. This contradictsour assumption that z q ( t ) is unbounded, so it follows that Ω is boundedas claimed. (cid:3) Theorem C
We now have all the ingredients to prove Theorem C.
Theorem C.
All the components of S n , n > are bounded.Proof. We proved in Theorem 4.2 and Lemma 4.4 that for the family F λ , S , consists of 2 q unbounded components, {± Ω , · · · , ± Ω q } . Theseare symmetric about the rays R = {± rη k , r > r } qk =1 and separated bythe rays R (cid:48) = {± rω k , r > } qk =1 . The boundary of each Ω k is an analytic curve defined by the relation | f λ ( z ) | = 1 where z is the fixed point or, if pq is odd and there isa single period 2 cycle, { z , z } , the relation | f λ ( z ) f (cid:48) λ ( z ) | = 1. Weassume now that there are two fixed points. The discussion for theperiod 2 cycle is essentially the same. Along ∂ Ω k there is a sequenceof points ν k,m where f λ ( z ) = − ν k,m where a newcycle of period two appears and so there is a bud component Ω ,k,m of S tangent to Ω k there. By Lemma 4.6, Ω ,k,m is bounded and has avirtual center, say s m,k on the ray in R separating Ω k from the nextone in order around the origin; for argument’s sake assume it is − Ω k +1 .If we now look at the points on the boundary of − Ω k +1 , there is asequence ν k +1 ,m (cid:48) where the multiplier of the cycle is − ,k +1 ,m (cid:48) with virtual center s m (cid:48) ,k +1 on the same ray of R separating Ω k and − Ω k +1 . Choose ν k +1 ,m (cid:48) so that s m (cid:48) ,k +1 = s m,k . Wecan do this since the boundaries of Ω k and − Ω k +1 are both asymptoticto the ray containing the centers.At s m,k there are 2 pq shell components and Ω ,k,m and W ,k +1 ,m (cid:48) aretwo of them. Now we draw a curve in Ω ,k,m from ν k,m to s m,k andanother from s m,k to ν k +1 ,m (cid:48) in Ω ,k +1 ,m (cid:48) . We then choose another par-abolic point ν k +1 ,m (cid:48)(cid:48) on the boundary of − Ω k +1 whose bud componentΩ ,k +1 ,m (cid:48)(cid:48) has a center s m (cid:48)(cid:48) ,k +1 on the next ray in order around the origin.We draw a simple curve in − Ω k +1 from ν k +1 ,m (cid:48) to ν k +1 ,m (cid:48)(cid:48) . Continu-ing around, in the next component Ω k +2 , we find the bud component YNAMICS OF GENERALIZED NEVANLINNA FUNCTIONS 29 that shares s m (cid:48)(cid:48) ,k +1 as its center and draw a simple curve through thebud components joining − Ω k +1 and Ω k +2 . We continue in this manneruntil we get back around to the original Ω k . We join all the curves inthe ± Ω k and the bud components between them. The result, γ m is asimple closed around the origin.In this way, choosing the ν k,m carefully and systematically, we cancreate a nested sequence of curves γ m around the origin. Any othercomponent of S is disjoint from the components ± Ω k and the budcomponents Ω ,k,m and so must lie inside one of the γ m and is thereforebounded. (cid:3) An immediate corollary of the above proof is
Corollary 5.1.
All the capture components in the dynamically naturalslice are bounded.
References [A] L. Ahlfors,
Lectures on quasiconformal mappings , Van Nostrand, New York,1966[BKL1] I. N. Baker, J. Kotus and Y. L¨u,
Iterates of meromorphic functions II:Examples of wan-dering domains, , J. London Math. Soc. (2) (1990), 267-278.[BKL2] I. N. Baker, J. Kotus and Y. L¨u, Iterates of meromorphic functions I ,Ergodic Th. and Dyn. Sys (1991), 241-248.[BKL3] I. N. Baker, J. Kotus and Y. L¨u, Iterates of meromorphic functions III:Preperiodic Domains , Ergodic Th. and Dyn. Sys (1991), 603-618.[BKL4] I. N. Baker, J. Kotus and Y. L¨u, Iterates of meromorphic functions IV:Critically finite functions , Results in Mathematics (1991), 651-656.[Ber] W. Bergweiler, Iteration of meromorphic functions , Bull. Amer. Math. Soc. (1993), 151-188.[BK] Walter Bergweiler and Janina Kotus, On the Hausdorff dimension of theescaping set of certain meromorphic functions , Trans. Amer. Math. Soc. ,(2012), 5369–5394.[BF] B. Branner, N. Fagella.
Quasiconformal surgery in holomorphic dynamics ,Cambridge University Press, 2014.[BH] X. Buff and C. Henriksen,
Julia sets in parameter spaces , Comm. Math. Phys. (2001) no.2, 333–375.[CG] L. Carleson and T. Gamelin,
Complex dynamics , Universitext: Tracts inMathematics, Springer-Verlag, New York, 1993[DFJ] R. L. Devaney, N. Fagella and X. Jarque
Hyperbolic components of the com-plex exponential family , Fundamenta Mathematicae, ,(2002), 193–215.[DK] B. Devaney and L. Keen,
Dynamics of Meromorphic Maps: Maps with Poly-nomial Schwarzian Derivative , Annales Scientifiques de l’Ecole Normale Su-perieure, (1989), 55-79.[DH] A. Douady and J.H. Hubbard, tude dynamique des polynmes complexes I &II. Publ. Math. d’Orsay, (1984-1985) [FG] N. Fagella, A. Garijo.
The parameter planes of λz m e z for m ≥
2, Commun.Math. Phys., 273(3), 755-783, 2007.[EreGab] A. Eremenko and A. Gabrielov,
Analytic continuation of eigenvalues ofa quartic oscillator,
Comm. Math. Phys. (2009), no. 2, 431–457. MR2481745 (2010b:34189)[FK] N. Fagella, L. Keen.
Stabel comonents in the parameter plane of meromorphicfunctions of finite type.
Submitted. ArXiv http://arxiv.org/abs/1702.06563.[Hille] E. Hille,
Ordinary differential equations in the complex domain , Wiley-Interscience, NewYork-London-Sydney, 1976, Pure and Applied Mathematics.MR 0499382[K] L. Keen, Complex and Real Dynamics for the family λ tan z . Proceedings ofthe Conference on Complex Dynamic , RIMS Kyoto University, 2001.[KK1] L. Keen and J. Kotus, Dynamics of the family of λ tan z . Conformal Geom-etry and Dynamics , Volume (1997), 28-57.[KK2] L. Keen and J. Kotus, Ergodicity of some classes of meromorphic functions ,Ann. Acad. Sci. Fenn. Math. , 1999, 133-145[KK3] L. Keen and J. Kotus, On Period Doubling and Sharkovskii type orderingfor the family λtanz , Value distribution theory and complex dynamics (HongKong, 2000), 51–78, Contemp. Math., 303, Amer. Math. Soc., Providence, RI,2002.[LV] O. Lectho and K. Virtanen,
Quasiconformal mappings in the plane
Berlin-Heidelberg-New York: Springer-Verlag, 1973[1] [RG] Lasse Rempe-Gillen,
Dynamics of exponential maps , Ph.D. thesis,Christian-Albrechts-Universit¨at Kiel, 2003.[Tan] Tan Lei,
Similarity between the Mandelbrot set and Julia sets.
Comm. Math.Phys. 134(3), 587–617 (1990)[McM] C.T. McMullen,
Complex dynamics and renormalization , Annals of Math-ematics Studies, Vol. 135, Princeton University Press, 1994.[M1] J. Milnor,
Geometry and dynamics of quadratic rational maps , ExperimentalMathematics, Volume (1993), 37-83.[M2] J. Milnor, Dynamics in One Complex Variable , Third Edition, AM(160),Princeton University Press, 2006.[MU] V. Mayer and M. Urba´nski,
Thermodynamical formalism and multifrac-tal analysis for meromorphic functions of finite order,
Memoirs of AMS,203(2010), no. 954[Nev] R. Nevanlinna, ¨Uber Riemannsche Fl¨achen mit endlich vielen Win-dungspunkten,
Acta Math. (1932), no. 1 295–373. MR 1555350.[Nev1] R. Nevanlinna, Analytic functions , Translated from the second edition byPhilip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162,Springer-Verlag, New York-Berlin, 1970. RM 0279280 (43
Iteration of a class of hyperbolic meromor-phic functions , Proc. Amer. Math. Soc., (1999), 3251-3258[Sch] D. Schleicher,
Attracting dynamics of exponential maps , Ann. Acad. Sci. Fenn.Math. ,(2003), 3–34.[Y] J. C. Yoccoz, Th´eor`eme de Siegel, nombres de Bruno et polynˆomes quadra-tiques. (French) [Siegel theorem, Bryuno numbers and quadratic polynomials]Petits diviseurs en dimension 1.
Astrisque No. 231 (1995), 3–88.[Z] J. Zhang,
Dynamics of Hyperbolic Meromorphic Functions , arXiv:1209.1683