Stable Components in the Parameter Plane of Transcendental Functions Of Finite Type

Abstract

We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call shell components, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the virtual center, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. Our dynamical plane results apply without the restriction of finite type.

Appendix

Any meromorphic function that is a branched cover of the sphere of finite degree d is a rational function and so can be expressed as a quotient of relatively prime polynomials at least one of which has degree d. The $2d+1$ coefficients of these polynomials define a natural embedding of the space of such functions into ${{\mathbb {C}}}^{2d+1}$. Up to affine conjugation then, these functions are represented as a complex analytic manifold of dimension $2d-2$. This family is denoted by $Rat_d$ in the literature.

Not all holomorphic families of meromorphic functions have such an obvious representation as a complex manifold. In this appendix we consider some transcendental families that do. These are the families that most of our examples are drawn from. The functions have explicit expressions that involve complex constants. We show that under quasiconformal deformation, the deformed function has a similar expression in which only the constants are changed. Thus the constants determine the embedding of the manifold into ${{\mathbb {C}}}^n$ for the appropriate n.

We begin with families of meromorphic functions with $p< \infty$ asymptotic values, no critical points and a single essential singularity at infinity. We denote these by $\mathcal {F}_p$. (See [30, 32, 41] for further discussion.)

These functions have a particularly nice characterization. Recall that the Schwarzian derivative of a function g is defined by

$$\begin{aligned} S(g)(z)=(g''/g')' - \frac{1}{2}(g''/g')^{2}. \end{aligned}$$

(8.1)

Nevanlinna, [46, 47], Chap X1, proved

Theorem 8.1

Every meromorphic function g with $p < \infty$ asymptotic values and $q < \infty$ critical points has the property that its Schwarzian derivative is a rational function of degree $p +q -2$. If $q=0$, the Schwarzian derivative is a polynomial P(z). In the opposite direction, for every polynomial function P(z) of degree $p-2$, the solution to the Schwarzian differential equation $S(g)=P(z)$ is a meromorphic function with exactly p asymptotic values and no critical points.

Since the theorem is classical and the proof is not well known, we sketch the proof here and refer the reader to the literature for details.

Proof

The proof follows from the construction of a function with p asymptotic values and no critical points as a limit of rational functions with p branch points. Letting the order of the branching at some or all of these p points increase, one obtains a sequence of rational functions. In the limit, the images of the branch points whose order goes to infinity become logarithmic singularities and the limit function has finitely many branch points and finitely many logarithmic singularities. The limit function is a parabolic covering map from the plane to itself. The Schwarzian derivatives of the rational functions in the sequence are again rational functions with degree determined only by the number of branch points, not their order. The limit of the Schwarzian derivatives is the Schwarzian derivative of the limit and so must be rational.

If all the branch points become asymptotic values in the limit, the limit function has no critical points or critical values and hence its derivative never vanishes. It follows from the definition of the Schwarzian that in the limit, it must be a polynomial. $\square$

It is classical, ( see e.g., [38]), that solutions to the Schwarzian equation $S(g)=P(z)$ are related to solutions of the linear second degree ordinary differential equation

$$\begin{aligned} w'' + \frac{1}{2} P(z) w = 0. \end{aligned}$$

(8.2)

Such an equation has a two dimensional space of holomorphic solutions. If $w_{1}, w_{2}$ are a pair of linearly independent solutions, and

$$\begin{aligned} g= \frac{aw_{2} + bw_{1}}{cw_{2}+dw_{2}}, \,\, a,b,c,d \in {{\mathbb {C}}}, \, ad-bc = 1, \end{aligned}$$

it is easy to check that $S(g)=P(z)$. Moreover, if g is any solution of the Schwarzian equation, $w=\sqrt{1/g'}$ is a solution of the linear equation.

Remark 8.2

One of the basic features of the Schwarzian derivative is that it satisfies the following cocycle relation: if f, g are meromorphic functions then

$$\begin{aligned} S(g ( f))(z) = S(g(f)) f'(z)^2 + S(f(z)). \end{aligned}$$

(8.3)

In particular, if T is a Möbius transformation, $S(T(z))=0$ and $S(T\circ g(z) )= S((g(z))$ so that post-composing by T does not change the Schwarzian. Under pre-composition by a Möbius transformation the Schwarzian behaves like a quadratic differential. In particular, pre-composing by an affine transformation multiplies the Schwarzian by a constant.

This means that if we find a specific pair $(w_1^*,w_2^*)$ of solutions to the second order linear equation and set $g^* = w_2^*/w_1^*$, then every solution of the Schwarzian equation $S(g) = P(z)$ has the formula

$$\begin{aligned} g= \frac{ag^* + b }{cg^*+d }, \,\, a,b,c,d \in {{\mathbb {C}}}, \, ad-bc = 1, \end{aligned}$$

In particular, if $p=2$, P(z) is identically constant. Using one of the constants in an affine conjugation, we may assume $P \equiv -1/2$; then a specific pair of solutions to equation (8.2) is $w_1^*=e^{-\frac{z}{2}}, w_2^*=e^{\frac{z}{2}}$ so we have $g^*=w_2^*/w_1^*=e^{z}$. The functions in the family $\mathcal {F}_2$ have the form

$$\begin{aligned} g(z)= \frac{ae^{z} + b }{ce^{z}+d }, \,\, a,b,c,d \in {{\mathbb {C}}}, \, ad-bc = 1. \end{aligned}$$

Since there is one more degree of freedom from the affine conjugation, we see this is a three dimensional family. We have discussed several dynamically natural slices of $\mathcal {F}_2$ in this paper.

For $\mathcal {F}_3$, P(z) is linear and two specific solutions to the second order linear equation

$$\begin{aligned} w''+\frac{1}{2} \zeta w = 0 \end{aligned}$$

are given by the Airy functions²$Ai(\zeta ), Bi(\zeta )$. Setting $g^*(\zeta )=Ai(\zeta )/Bi(\zeta )$ we obtain a solution with three asymptotic values, 0, i and $-i$. Since we are interested in the dynamics, we may conjugate by the affine transformation $\zeta =rz+s$. We may thus transform any function whose Schwarzian is linear, and hence any function in $\mathcal {F}_3$, up to affine conjugation, to one given by the formula

$$\begin{aligned} g(\zeta )= \frac{a g^*(\zeta ) + b }{cg^*(\zeta )+d }, \,\, a,b,c,d \in {{\mathbb {C}}}, \, ad-bc = 1. \end{aligned}$$

The asymptotic values of g are $\frac{b}{d}, \frac{ai +b }{ci+d}, \frac{-ai +b }{-ci+d }$.

We could define a dynamically natural slice, for example, by choosing the coefficients a, b and d so that

$$\begin{aligned} g(1)=1, \text{ and } g'( 1)= 1/2. \end{aligned}$$

It is not hard to compute that the remaining coefficient c is an affine function of the asymptotic value $\frac{b}{d}$.

There are standard functions that solve the second order equation when the coefficient polynomial is of degree two or three, and they give formulas for functions in $\mathcal {F}_4$ and $\mathcal {F}_5$.

The following families of holomorphic functions have rational Schwarzian derivatives and are determined by their topological covering properties. We list them here and refer the reader to the cited literature for further discussion of them.

• Functions of type $Re^Q$ for R rational and Q polynomial. See [19, 55] for discussions with R polynomial.

• Functions of type $\int _{z_0}^{z} P(t)e^Q(t) dt$ for P, Q polynomials. See [2, Th. 6.2] and [54, Prop. 2]).

1.1 Compositions

The following theorem shows how we can find more families for which the functions have explicit formulas. (See [23] for further discussion of these functions).

Theorem 8.3

Let $f_0 \in \mathcal {F}_p$

• 1. Suppose $g_0 = f_0 \circ Q_0$ is a function such that $Q_0$ is a polynomial of degree d and suppose that g is a meromorphic function quasiconformally conjugate to $g_0$. Then g can be expressed as $g=f \circ Q$ for some function $f \in \mathcal {F}_p$ and some polynomial Q of degree d.

• 2. Suppose $R_0 \in Rat_d$ is rational of degree d and $h_0 = R_0 \circ f_0$. Then if h(z) is quasiconformally conjugate to $h_0$, h can be expressed as $h=R \circ f$ for functions $R \in Rat_d$ and $f \in \mathcal {F}_p$.

Remark 8.4

The proof of part 1 of the theorem works if $Q_0$ is replaced by a rational function $R_0 \in Rat_d$ but then the composed function $g_0$ has essential singularities at the poles of $R_0$.

Remark 8.5

Note that in part 1, $g = f \circ Q$ is a function with rational Schwarzian. For each asymptotic value of f of multiplicity m, g has an asymptotic value of multiplicity dm; moreover g has the same critical points as Q, namely $2d-2$ critical points counted with multiplicity, $d-1$ of which are at infinity. In part 2, however, it is no longer true that the Schwarzian of the composed function $h=R \circ f$ is a rational function. For example, it may have infinitely many critical points.

Remark 8.6

Observe that if $f\in \mathcal {F}_p$ and Q is a degree d polynomial then $Q\circ f$ and $f\circ Q$ are semiconjugate by a degree d polynomial – in fact by Q.

Proof

The proof of both parts of the theorem is essentially the same. We therefore carry it out only for part 1.

In part 1, let $\phi ^{\mu }$ be a quasiconformal homeomorphism with Beltrami coefficient $\mu$ such that

$$\begin{aligned} g_{\mu } = \phi ^{\mu } \circ g_0 \circ (\phi ^{\mu })^{-1} \end{aligned}$$

is meromorphic. We can use $f_0$ to pull back the complex structure defined by $\mu := \bar{\partial }\phi ^\mu / \partial \phi ^\mu$ to obtain a complex structure $\nu =f_0^*\mu$ such that the map

$$\begin{aligned} f_{\mu } = \phi ^{\mu } \circ f_0 \circ (\phi ^{\nu })^{-1} \end{aligned}$$

is meromorphic. Note that this is not a conjugacy since it involves two different homeomorphisms.

We can now write

$$\begin{aligned} g_{\mu } = \phi ^{\mu } \circ f_0 \circ (\phi ^{\nu })^{-1} \circ \phi ^{\nu } \circ Q_0 \circ (\phi ^{\mu })^{-1} \end{aligned}$$

and set

$$\begin{aligned} Q_{\mu }=\phi ^{\nu } \circ Q_0 \circ (\phi ^{\mu })^{-1}. \end{aligned}$$

Again this is not a conjugacy but it is meromorphic since $g_{\mu }$ and $f_{\mu }$ are homeomorphisms.

The main point here is that although $f_{\mu }$ is not a conjugate of $f_0$, since the quasiconformal maps $\phi ^{\mu }$ and $\phi ^{\nu }$ are homeomorphisms, the map $f_{\mu }$ is a meromorphic map with the same topology as $f_0$; that is, it has p asymptotic values and no critical values. By Nevanlinna’s theorem, Theorem 8.1, $f_{\mu }$ belongs to $\mathcal {F}_p$. Similarly, although $Q_{\mu }$ is not defined as a conjugate of $Q_0$, since the quasiconformal maps $\phi ^{\nu }$ and $\phi ^{\eta }$ are homeomorphisms, the map $Q_{\mu }$ is a meromorphic map with the same topology as $Q_0$; that is, it is a degree d branched covering of the Riemann sphere with the same number of critical points and the same branching as $Q_0$ and thus it must be a polynomial of degree d. $\square$