Dynamics of Generalized Nevanlinna Functions
Abstract.
In the early 1980’s, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in early work on rational maps. See [DH, BH] for example.
Here, we continue to study these ideas in the realm of transcendental functions.
In [KK1], it was shown that for the tangent family, , the way the hyperbolic components meet at a point where the asymptotic value eventually lands on infinity reflects the dynamic behavior of the functions at infinity. In the first part of this paper we show that this duality extends to a much more general class of transcendental meromorphic functions that we call generalized Nevanlinna functions with the additional property that infinity is not an asymptotic value. In particular, we show that in “dynamically natural” one dimensional slices of parameter space, there are “hyperboliclike” components with a unique distinguished boundary point whose dynamics reflect the behavior inside an asymptotic tract at infinity. Our main result is that every parameter point in such a slice for which the asymptotic value eventually lands on a pole is such a distinguished boundary point.
In the second part of the paper, we apply this result to the families , , to prove that all hyperbolic components of period greater than are bounded.
2010 Mathematics Subject Classification:
37F10, 30D05, 37F30, 30D30Introduction
In the early 1980’s, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in early work on rational maps. See [DH, BH] for example. Here, we continue to study these ideas in the realm of transcendental functions.
In [KK1], it was shown that for the tangent family, , the way the hyperbolic components meet at a point where the asymptotic value eventually lands on infinity, reflects the dynamic behavior of the functions at infinity. This is very different from what one sees for the exponential family (see [DFJ, RG, Sch]) where all the hyperbolic components are unbounded. A crucial difference between these families is that for the exponential family infinity is an asymptotic value whereas for the tangent family it is not.
In the first part of this paper we define a much more general class of transcendental meromorphic functions that we call generalized Nevanlinna functions. We show that families of this class with the additional property that infinity is NOT an asymptotic value exhibit this duality; that is, in a one dimensional slice of the parameter space, the way the hyperbolic components meet at a point where the asymptotic value eventually lands on infinity reflects the dynamical behavior of the functions at infinity.
To describe these functions and state our theorems, we need some background. The dynamical plane of a meromorphic map is divided into two sets: the Fatou or stable set on which the iterates are well defined and form a normal family, and the Julia set, its complement. The Julia set can be characterized as the closure of the set of repelling periodic points, or equivalently, the closure of the set of prepoles, points that map to infinity after finite iteration. A good introduction to meromorphic dynamics can be found in [Ber].
The points over which a meromorphic function is not a regular covering map are called singular values. There are two types of singular values: critical values (images of zeroes of , the critical points) and asymptotic values (points where as ). If an asymptotic value is isolated, the local inverse there is the logarithm. We denote the class of meromorphic functions with finitely many singular values by . This class is particularly tractable: families in have parameter spaces with natural embeddings into , where is a simple function of the number of singular values; all asymptotic values are isolated; their Fatou domains have a simple classification because there are no wandering domains (see Section 1 for definitions).
The focus in this paper is on the subclass of functions in for which infinity is not an asymptotic value. These necessarily have infinitely many poles and this behavior at infinity has consequences for the dynamical and parameter spaces (see e.g. [BK]). In particular, it is very different from that for entire functions or transcendental functions with finitely many poles.
A general principle in dynamics is that each stable dynamical phenomenon is “controlled” by a singular point. For example, each attracting or parabolic periodic cycle always attracts a singular value. Using this principle, one can define one dimensional slices of the parameter space of a family in that are “compatible with the dynamics” by keeping all but one of the dynamical phenomena fixed and letting the remaining one, controlled by the “free singular value ”, vary. Looking for regions in the slice where the free singular value is attracted to an attracting cycle gives a picture of how the “hyperboliclike” components of the slice fit together around the bifurcation locus.
Dynamically natural slices were defined in [FK] for families of functions in . It was shown there that for those slices, as for slices of parameter spaces of rational maps, the bifurcation locus contains parameters distinguished by functional relations. There are Misiurewicz points where lands on a repelling periodic cycle and centers where lands on a superattracting cycle. Another type of distinguished parameter, not seen for rational maps, is a virtual cycle parameter, where lands on a pole.
Dynamically natural slices contain two different kinds of “hyperboliclike” domains in which the functions are all quasiconformally conjugate on their Julia sets: capture components, where the free asymptotic value is attracted to one of the fixed phenomena — an attracting or superattracting cycle, and shell components, where the free asymptotic value is the only singular value attracted to an attracting cycle. In a shell component, the period of the attracting cycle is constant and the multiplier map is a welldefined universal covering map onto the punctured disk. Properties of shell components for general families in were studied in detail in [FK]. In particular, it was proved that the boundary of every shell component contains a special point, the virtual center where the limit of the multiplier map is zero. One of the main results proved there is
Theorem A.
For families in , a virtual center on the boundary of a shell component in a dynamically natural slice is a virtual cycle parameter and any virtual cycle parameter on the boundary is a virtual center.
In this paper, we concern ourselves with a special subclass in . We start with the Nevanlinna functions, the family of functions with asymptotic values and no critical values. We next form the family by pre and postcomposing by polynomials of degrees and respectively. We then look at , the subset of functions in all of whose asymptotic values are finite. We first define the embedding of into and then study shell components of dynamically natural slices for this embedding. We include the proof, given originally in [FK], that
Proposition A.
If is topologically conjugate to a meromorphic function in , and if is meromorphic, then it is also in ; that is there is a function and polynomials of degrees respectively such that .
A corollary is
Corollary A.
There is a natural embedding of into and hence an embedding of into .
Theorem A does not preclude the existence of virtual cycle parameters that are not on the boundary of a shell component. For example, such a parameter might be buried in the bifurcation locus. Our first new theorem says this cannot happen.
Theorem B.
In a dynamically natural slice in , every virtual cycle parameter lies on the boundary of a shell component.
Corollary B, which follows directly from Theorems A and B, says that in the notions of virtual center and virtual cycle parameter are equivalent.
Corollary B.
In a dynamically natural slice of , every virtual cycle parameter is a virtual center and vice versa.
In [FK] it was proved that for families in , shell components of period in dynamically natural slices are always unbounded and it was conjectured that, in contrast to families of entire functions, those of period greater than are always bounded. The conjecture was proved true in the tangent family in [KK1]. In this paper we prove it for the generalization of the tangent family, . This is a subfamily of .
Theorem C.
Every shell component of period greater than in the plane of the family is bounded.
The paper is organized as follows. Part I is a discussion of the general family and dynamically natural slices of its parameter space. In Section 1 we give a brief overview of the basic theory, set our notation and discuss the theorem of Nevanlinna, Theorem 1.1, which we use to define the class of Nevanlinna functions. Next, in Section 2, we define the classes and , and prove Proposition A and Corollary A. In Section 2.1 we recall the properties of Fatou components that we need and define dynamically natural slices of parameter spaces and the shell components in these slices. In Section 3, we define virtual cycle parameters and virtual centers, state Theorem A and prove Theorem B. Part II is a discussion of the special symmetric subfamily . In Section 4 we classify the shell components by period and discuss the special properties of the components of periods and . Finally, in Section 5 we prove Theorem C.
Part I The family
1. Basics and Tools
1.1. Meromorphic functions.
In this paper, unless we specifically say otherwise, we always assume that an entire or meromorphic map is transcendental and so has infinite degree. If we mean a map of finite degree we call it polynomial or rational. We also always assume infinity is an essential singularity. We need the following definitions:
Definition 1.
A point is called a singular value^{1}^{1}1In the literature these are sometimes called singularites of . of if, for some small neighborhood of , some branch of is not well defined. If is a zero of , it is a critical point and its image is a critical value. The branch with is not well defined so a critical value is singular. A point is also singular if there is a path such that and . The limit is singular and called an asymptotic value of . Singular values may be critical, asymptotic or accumulations of such points. We denote the set of singular points of by .
If an asymptotic value is isolated, and is an asymptotic path for , we can find a nested sequence of neighborhoods of , with , and a particular branch of such that is a nested sequence of neighborhoods containing and . Then can be chosen small enough so that the map is a universal covering map. In this case is called an asymptotic tract for the asymptotic value and is called a logarithmic singular value. The number of distinct asymptotic tracts of a given asymptotic value is called its multiplicity^{2}^{2}2Note this notion is not the same as multiplicity of a critical point..
Definition 2.
A ray approaching infinity is called a Julia ray or Julia direction for the meromorphic function if assumes all (but at most two) values infinitely often in any sector containing .
An example to keep in mind is with asymptotic values at and . The asymptotic tracts are the left and right half planes and the Julia directions are parallel to the positive and negative imaginary axes. Another example is with asymptotic values , asymptotic tracts the upper and lower half planes and Julia directions parallel to the positive and negative real axes.
In this paper, whenever we talk about the number of critical points and/or asymptotic values, we tacitly assume that we count with multiplicity.
An entire function always has an asymptotic value at infinity.
We will only be interested in meromorphic functions with . These are called functions of finite type. Note that all the asymptotic values 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 asymptotic values, finitely many critical points and a single essential singularity at infinity. (See [DK, KK1, EreGab] for further discussion.)
Recall that the Schwarzian derivative of a function is defined by
(1) 
Because Schwarzian derivatives satisfy the cocycle relation
and the Schwarzian derivative of a Möbius transformation is zero, solutions to equation (1) are determined only up to postcomposition by a Möbius transformation.
Nevanlinna’s theorem says
Theorem 1.1 (Nevanlinna).
Every meromorphic function with asymptotic values and critical points has the property that its Schwarzian derivative is a rational function of degree . If , the Schwarzian derivative is a polynomial . In the opposite direction, for every polynomial function of degree , the solution to the Schwarzian differential equation is a meromorphic function with exactly asymptotic values and no critical points.
Remark 1.1.
The proof of the first part of this theorem involves the construction of the function as a limit of holomorphic functions whose Schwarzians are rational of bounded degree. The proof of the second part of the theorem involves understanding the asymptotic properties of solutions to the equation . In particular, there are exactly “truncated solutions” that, for any , have asymptotic developments of the form
defined in the sector . Each is entire and tends to zero as tends to infinity along each ray of the sector and tends to infinity in the adjacent sectors. The rays separating the sectors are the Julia rays for . See [Hille, Nev] for details.
Definition 3.
We denote the family of meromorphic functions with asymptotic values and no critical values by and call the functions Nevanlinna functions.
One immediate corollary is that Nevanlinna functions cannot have exactly one asymptotic value. Moreover, since polynomials of degree depend on their coefficients, and the solutions are determined up to postcompostion by Möbius transformations which depend on three coefficents, another immediate corollary is
Corollary 1.2.
Nevanlinna functions with asymptotic values have a natural embedding into .
We are interested in the dynamical systems generated by these Nevanlinna functions and some generalizations of them. Infinity plays a special role so we separate the cases where infinity is an asymptotic value from those where it is not. For these functions, the dynamics are unchanged by conjugation by an affine transformation. Thus suitably normalized solutions have a natural embedding into
In his proof of the above theorem, (see also [DK], Section 1), Nevanlinna shows that for , a neighborhood of infinity is divided into exactly disjoint sectors, , each with angle . Each of these is an asymptotic tract for one asymptotic value. The sectors are separated by the rays that bound them, and these are Julia rays. Although two tracts may map to the same asymptotic value, tracts in adjacent sectors must map to different asymptotic values.
Another immediate corollary of Nevanlinna’s theorem is that the family is topologically closed. That is,
Corollary 1.3.
If is topologically conjugate to a meromorphic function in , and if is meromorphic, then it is also in .
2. The family
A family that is more general than is the family of functions
This family is also topologically closed. We have^{3}^{3}3This result is proved in [FK]. Since the proof is short and the ideas used are relevant we include it here.
Proposition A.
If is topologically conjugate to a meromorphic function in , and if is meromorphic, (with essential singularity at infinity), then it is also in ; that is there is a function and polynomials of degrees respectively such that .
The proof uses quasiconformal mappings and the Measurable Riemann 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 quasiconformally conjugate to . It then follows from Theorem 3.3 of [KK1] that it is true for topologically conjugate to .
Let be a quasiconformal homeomorphism with Beltrami coefficient such that
is meromorphic. We can use to pull back the complex structure defined by to obtain a complex structure such that the map
is holomorphic. Note that this is not a conjugacy since it involves two different homeomorphisms. Similarly, we can use to pull back the complex structure defined by to obtain a complex structure such that the map
is meromorphic. Again this is not a conjugacy. The function
is a composition of meromorphic functions and is meromorphic. Now set
Although is not conjugate to , we were given that
is meromorphic so that is also meromorphic.
The main point here is that although is not a conjugate of , since the quasiconformal maps and are homeomorphisms, the map is a meromorphic map with the same topology as ; that is, it has asymptotic values and no critical values so that by Corollary 1.3 belongs to . Similarly, although and are not respective conjugates of and , because the quasiconformal maps , and are homeomorphisms, the maps and are holomorphic maps with the same topology as and ; that is, they are respectively a degree and a degree branched covering of the Riemann sphere with the same number of critical points and the same branching as and . The maps , and are defined up to normalization. If we want to keep the essential singularity of at infinity, we normalize so that and are polynomials of degrees and respectively, (that is, infinity is a fixed critical point with respective multiplicities and ). ∎
A corollary to this theorem is
Corollary A.
The space of functions has a natural embedding into and the subspace has dimension .
Proof.
A polynomial of degree is determined by its coefficients. If , set . It is determined by its coefficients. It is straightforward to check that if are linearly independent solutions of the equation
then . Since the space of solutions to this linear differential equation has dimension , and the solution of the Schwarzian equation is the quotient of two such solutions, is determined by parameters. Requiring that infinity cannot be an asymptotic value restricts one parameter. Therefore has a natural embedding into and the subspace has dimension . ∎
2.1. Dynamics of meromorphic functions
Let be a transcendental meromorphic function and let denote the iterate of , that is for . Then is welldefined, except at the poles of , which form a countable set. These points have finite orbits that end at infinity.
The basic objects studied are the Fatou set and Julia set of the function . The Fatou set of is defined by
and the Julia set by
Note that the point at infinity is always in the Julia set. If is a meromorphic function with more than one pole, then the set of prepoles, is infinite. By the Picard theorem, is normal on . Since it is not normal on , , (see also [BKL1]).
A point is called a periodic point of , if and for any . The multiplier of the cycle is defined to be The periodic point is attracting if , superattracting if , parabolic if , where is rational number, and neutral if is not rational. It is repelling if .
If is a component of the Fatou set, then is either a component of the Fatou set or a component missing one point. For the orbit of under , there are only two cases:

there exist integers such that , and is called eventually periodic;

for all , , and is called a wandering domain.
Suppose that is a periodic cycle of Fatou components, then either:

The cycle is (super)attracting: each contains a point of a periodic cycle with multiplier and all points in each domain are attracted to this cycle. If , the critical point itself belongs to the periodic cycle and the domain is called superattracting.

The cycle is parabolic: the boundary of each contains a point of a periodic cycle with multiplier , , a divisor of , and all points in each domain are attracted to this cycle.

The components of the cycle are Siegel disks: that is, each contains a point of a periodic cycle with multiplier , where is irrational and there is a holomorphic homeomorphism mapping each to the unit disk , and conjugating the first return map on to an irrational rotation of . The preimages under this conjugacy of the circles , foliate the disks with forward invariant leaves on which is injective.

The components of the cycle are Herman rings: each is holomorphically homeomorphic to a standard annulus and the first return map is conjugate to an irrational rotation of the annulus by a holomorphic homeomorphism. The preimages under this conjugacy of the circles , foliate the disks with forward invariant leaves on which is injective.

is an essentially parabolic (Baker) domain: the boundary of each contains a point (possibly ) and for all , but is not holomorphic at . If , then .
Definition 4.
Define the postsingular set of as the closure of the orbits of the singular values; that is,
For notational simplicity, if a prepole is a singular value, is a finite set and includes infinity.
Each nonrepelling periodic cycle is associated to some singular point. In particular we have, see e.g. [M2], chap 811 or [Ber], Sect.4.3,
Theorem 2.1.
If is an attracting, superatttracting, parabolic or Baker periodic cycle of Fatou components, then for some , contains a singular value. If is a cycle of rotation domains (Siegel disks or Herman rings) the boundary of each contains the accumulation set of some singular value.
We use the following definition of a hyperbolic function in ^{4}^{4}4This definition is equivalent to standard definitions of hyperbolic functions because for these function is finite.
Definition 5.
An is called hyperbolic if
Note that if all the singular values of are attracted to attracting or superattracting cycles then is hyperbolic.
For a discussion of hyperbolicity for more general meromorphic functions, see the discussion in [Z] and the references therein.
Singularly finite maps may have Baker domains but
Theorem 2.2.
[BKL4] If is finite, then there are no wandering domains in the Fatou set.
2.2. Holomorphic families
In this paper, we are interested in the family of meromorphic functions in for which infinity is not an asymptotic value. For each choice of triples this is an example of a holomorphic family. Below we state the general definitions and results we need.
Definition 6 (Holomorphic family).
A holomorphic family of meromorphic maps over a complex manifold is a map , such that is meromorphic for all and is holomorphic for all .
Definition 7 (Holomorphic motion).
A holomorphic motion of a set over a connected complex manifold with basepoint is a map given by such that

for each , is holomorphic in ,

for each , is an injective function of , and,

at , .
A holomorphic motion of a set respects the dynamics of the holomorphic family if whenever both and belong to .
The following equivalencies are proved for rational maps in [McM] and extended to the transcendental setting in [KK1].
Theorem 2.3.
Let be a holomorphic family of meromorphic maps with finitely many singularities, over a complex manifold , with base point . Then the following are equivalent.

The number of attracting cycles of is locally constant in a neighborhood of .

There is a holomorphic motion of the Julia set of over a neighborhood of which respects the dynamics of .

If in addition, for , is are holomorphic maps parameterizing the singular values of , then the functions form a normal family on a neighborhood of .
Definition 8 (stability).
A parameter is a stable parameter for the family if it satisfies any of the above conditions.
The set of non stable parameters is precisely the set where bifurcations occur, and it is often called the bifurcation locus of the family , and denoted by . In families of maps with more than one singular value, however, it makes sense to consider subsets of the bifurcation locus where only some of the singular values are bifurcating, in the sense that the families are normal in a neighborhood of for some values of , but not for all. We define
In this paper we investigate dynamically natural one dimensional slices of the holomorphic family . In these slices, roughly speaking, all the dynamic phenomena but one are fixed and the last is determined by a “free asymptotic value”. We will study the components of the complement of the bifurcation locus in these slices. The parameters in them are stable.
Precisely, (see [FK])
Definition 9.
A one dimensional subset is a dynamically natural slice with respect to if the following conditions are satisfied.

is holomorphically isomorphic to the complex plane punctured at points where the function is not in the family; for example, points where the number of singular values is reduced. The removed points are called parameter singularities. By abuse of notation we denote the image of in by again, and denote the variable in by .

The singular values are given by distinct holomorphic functions , , and an asymptotic value that is an affine function of .^{5}^{5}5This is only for convenience of exposition and can be arranged by a holomorphic change of coordinates in . We call the free asymptotic value; we require that .

The poles (if any) are given by distinct holomorphic functions , , .

The critical values and some of the asymptotic values are attracted to an attracting or parabolic cycle^{6}^{6}6If the attracting cycle is parabolic, the functions in the slice are not hyperbolic but the results hold. See [FK] for further discussion whose period and multiplier are constant for all . We call these dynamically fixed singular values. The remaining singular values include ; if there is only one remaining value, it may have arbitrary behavior. If there are several, they satisfy a relation that persists throughout so that the remaining dynamical behavior is controlled by the behavior of . Examples of how this may work are discussed in Section 4.

Suppose is attracted to , the basin of attraction of an attracting cycle that does not attract any of the dynamically fixed singular values. Then the slice contains, up to affine conjugacy, all meromorphic maps that are quasiconformally conjugate to in and conformally conjugate to on .

is maximal in the sense that if where is a parameter singularity, then does not satisfy at least one of the conditions above.
In the components of the complement of the bifurcation locus in these slices is attracted to an attracting cycle of fixed period; this is the period of the component. We distinguish two cases:

does not attract one of the dynamically fixed singular values. We call these Shell components and denote the individual components by and the collection by .

attracts one of the dynamically fixed singular values in addition to attracting . We call these Capture components. Their properties are very different from those of the shell components and 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 is the restriction of a holomorphic family in to a dynamically natural slice and denote a function in by . We need the following definitions
Definition 10.
Suppose that for some , satisfies , where and ; that is, is a prepole of order . Then is called a virtual cycle parameter of order . This is justified by that fact that if is an asymptotic path for and if is the branch of the inverse of taking to the prepole then
We can think of the points as forming a virtual cycle.
Definition 11.
Let be a shell component in and let
be the attracting cycle of period that attracts . Suppose that as , and the multiplier Then is called a virtual center of .
Since the attracting basin of the cycle must contain , we will assume throughout that the points in the cycle are labeled so that and are in the same component of the immediate basin.
The next theorem collects the main the results in [FK] about shell components in a dynamically natural slice for a holomorphic family of transcendental functions with finite singular set, none of whose asymptotic values is at infinity. Note that Theorem A of the introduction is part (c) of the theorem.
Theorem 3.1.
Let be a shell component in . Then if

The map is a universal covering map. It extends continuously to and is piecewise analytic; either is simply connected and is infinite to one or is isomorphic a punctured disk and the puncture is a parameter singularity.

There is a unique virtual center on . If the period of the component is , the virtual center is at infinity.

Every (finite) virtual center of a shell component is a virtual cycle parameter and any virtual cycle parameter on the boundary of a shell component is a virtual center.
Here we prove a stronger theorem for slices of the family ; recall that all their asymptotic values are finite. Note that because the functions are of the form , if is an asymptotic value of , then is an asymptotic value of . There are distinct asymptotic tracts corresponding to each asymptotic tract of so that there are distinct asymptotic tracts at infinity separated by the Julia directions. The asymptotic values, tracts and Julia directions depend holomorphically on the parameters. At each pole of of order there are preasymptotic tracts and the pullbacks of the rays in the Julia directions separate them.
Theorem B.
Let be a dynamically natural slice for the meromorphic family consisting of meromorphic functions of the form all of whose asymptotic values are finite, and let be a virtual center parameter of order . Then is on the boundary of a shell component of order in . That is, in any neighborhood of there exists such that has an attracting cycle of period .
As an immediate corollary to Theorem 3.1 and Theorem B is
Corollary B.
In a dynamically natural slice of , every virtual cycle parameter is a virtual center and vice versa.
Remark 3.1.
The essence of the theorem is that the dynamic picture at the poles is reflected in the parameter picture at the virtual center parameters.^{7}^{7}7 This is analogous to the situation for Misiurewicz points in the parameter plane for quadratic polynomials. If infinity is an asymptotic value, both the dynamics and parameter pictures are different.
Proof.
Since is a virtual center parameter of order , has a virtual cycle . For each , the cycle uniquely determines a branch of the inverse of , such that . By abuse of notation, for readability, we drop the and denote all of these branches by . Because we are in a dynamically natural slice of a holomorphic family, the analytic continuations of the , denoted by are well defined.
Note however, that in a neighborhood of the asymptotic value , the inverse branch of is not uniquely determined. Since is part of a virtual cycle, a neighborhood of has at least one preimage that is in an asymptotic tract. If the multiplicity of is one, then by definition there is a unique asymptotic tract that is determined by the virtual cycle of ; we denote it by and take as the branch that maps to . Then taking the analytic continuation of this we obtain as the analytic continuation of . If the multiplicity of is greater than one there will be a choice among the tracts corresponding to (and hence the inverse branch ). Similarly, if there is more than one asymptotic value that varies with (and satisfies a functional relation with ), we choose the tract (or one of them if there is more than one) corresponding to the free asymptotic value^{8}^{8}8See the examples in Part 2.. In the argument below, we take , or choose one of the tracts as , along with the corresponding branch . Since, in general, the asymptotic value is not omitted, there may also be infinitely many inverse branches of the neighborhood that are bounded. We don’t need to concern ourselves with them here.
Because we are in a dynamically natural slice we have the following holomorphic functions:

is the free asymptotic value of .

If , then is the holomorphic function defining the pole of such that .

Note that for in a neighborhood of , the affine function determines a corresponding neighborhood of in the dynamic plane of and vice versa. Define the map by . Then if is small enough, both and are in , a small neighborhood of in the dynamic plane of .

In the dynamic plane of , set . Then is the preimage of the pole in a neighborhood of the asymptotic value .

Each has infinitely many inverses in the asymptotic tract ; denote these by , .
The main ideas for the proof are first to use the relation between and the free asymptotic value to carefully choose and fix a close to and then to construct a domain such that . It will then follow from Schwarz’s lemma that has an attracting fixed point in .
Choosing : (See figure 1.) The asymptotic tract lies between two Julia rays and these span an angle . We choose a small enough neighborhood of such that for each , the Julia rays of , lie within of respectively. Each has an asymptotic tract between these rays and we can find a set which lies inside the sector in bounded by the rays and such that is a sector in with vertex at infinity and angle .
As above, let be a neighborhood of and let be the corresponding neighborhood in parameter space. For each , and is defined. Define by . Since contains and is holomorphic, we can find such that Set and let be the “triangular” subset of with vertex at . Note that the number of such triangular sets is equal to the order of the pole. If it is one, the set is unique. If it is greater than one, we choose one arbitrarily.
Constructing : (See figure 2.) Now we work with a fixed and we set . Since we have . In we have and . Let be a circle with center and radius . Taking closer to if necessary, we may assume lies in a compact subset of ; that is the disk . Now and is an infinite to one cover so contains preimages , , of .
We want to approximate the distance from a point on to . Let and be a conformal homeomorphisms from the left half plane to and the punctured disk to respectively chosen such that .
Because is a homeomorphism, is bounded above and below for in a closed disk containing . Applying Koebe’s distortion theorem, there are positive constants such that for ,
We are assuming is small so we may assume it is less than . Thus we can find closed annuli in and containing and respectively.
We then have, for appropriate positive constants, ,
(2) 
This says that the lift of the circle lies in a vertical strip of bounded horizontal width in Let denote ; it is the lift of the annulus in to .
The covering group for the exponential map acting on pulls back to an infinite cyclic group generated by a map that is the covering group acting on under the map so that
Thus
This says that
and because is bounded in the closure of a fundamental domain for intersected with , it is bounded for all in . Now because is a conformal homeomorphism it preserves the hyperbolic density so that
Therefore the hyperbolic density in is invariant under the action of . Equation (2), says that is comparable to which gives us the estimate we want; that is, there are positive constants such that
Note that because we have a group action on , there will be a and a in each fundamental region. Now let be a triangular region in with one vertex at , two sides spanning an angle of , where is the order of the pole , joining to , and third side an arc of the circle ; the sides are chosen so that contains in its interior. Next set . First note that and is a doubly infinite curve in that stays a bounded distance from , where the bound depends on and . The inverse images of sides of the triangle form scallops “above” (further inside ).
Now look at . This is a triangular shaped region contained in with a boundary point ; it contains . Choose realizing the minimum above so that is contained in . Then, as is a pole of order , we have
(3) 
Since the derivative of along the orbit of from to is bounded and varies holomorphically with , the derivative of along the orbits of from to and to are also bounded for . Because and is small, when we map by the images of and points on are comparably close to the pole . Specifically, for some positive constant , we have
Comparing this with the estimate (3) it follows that
Therefore because is inside ,
so that by the Schwarz lemma has a fixed point in . ∎
Part II The Extended Tangent Family .
In this part of the paper we use the results above to prove that for family , every shell component of period is bounded. A corollary is that every capture component is bounded as well. The proof will follow from by studying the period and period components.
4. The shell components of
The family is a subfamily of with and and in the one dimensional slice of consisting of functions that fix and have symmetric asymptotic values. The functions in have one fixed critical point at and have either one asymptotic value with mulitplicity or two asymptotic values with multiplicity that are opposite in sign. Specifically, the map has two distinct asymptotic tracts and two asymptotic values, . Each of these tracts has preimages under . If is even, all of the asymptotic tracts map onto punctured neighborhoods of the single point which is the free asymptotic value . If is odd, of the asymptotic tracts map onto punctured neighborhoods of and the other tracts map onto punctured neighborhoods of . In this case we choose as the free asymptotic value. The other asymptotic value satisfies the relation .
The punctured plane is thus a dynamically natural slice in and the dynamics are determined by the forward orbit of .
The full set of shell components in this slice is denoted by and we divide it into subsets depending on the period of the cycle as follows:
Definition 12.
If is even
otherwise
For readability we only include the subscript on if the period is not obvious from the context.
Figure 3 shows the parameter plane for the family . The period shell components are yellow, the period shell components are cyan blue. The capture components are green.
4.1. Symmetries
Note that for any , . In addition, if , are the roots of unity,
It follows that if then