Non-trivial deformation of an entire function abz + e^{bz} + c

Shunsuke Morosawa^{∗}

Department of Mathematics and Information Science, Faculty of Science, Kochi University, Kochi 780-8520, Japan and

Masahiko Taniguchi

Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

Abstract

In this note, we consider dynamics of the family of transcendental entire functions f(z; a, b, c) = abz + e^{bz} + c. In particular, studying the case where they have a Baker domains, we investigate the area of Julia sets and the Hausdorff convergence of Julia sets.

Keywords: transcendental entire function; Baker domain; Julia set; Hausdorff convergence

Classification Categories: AMS No. 30D05, 58F23

Consider the family

f(z; a, b, c) = abz + e^{bz} + c.

These f(z; a, b, c) with a, b ≠ 0 are structurally infinite ([7]), for they have infinitely many critical values. For the sake of simplicity, we assume that a, b and c are real.

Theorem 1 Let ab ≥ 1, a > 0, c ≤ −1 and

a log a − a + c < 0,

∗Partially supported by Grant-in-Aid for Scientific Research 09640199

then f = f(z; a, b, c) has a Baker domain B = B_{a,b,c}, which is completely invariant.

In particular, The Julia set is the boundary of B.

Proof. We set D_{0} = {Re z < 0}. Elementary calculation shows that D_{0} is mapped by f to a proper subdomain in D_{0}, and hence D_{0} is invariant. The Fatou component B containing D_{0} is a Baker domain, for if z ∈ D_{0}, then

Re f(z) < abRe z.

Also the line L_{k} = {z = x + iy | x ≥ 0, y = (2k + 1)π/b} for k ∈ ℤ is mapped by f into D_{0}, since the assumptions imply that

Re (abz + e^{bz} + c) = abx − e^{bx} + c < 0,

for every z ∈ L_{k}.

Since every preimage of D_{0} by f contains either D_{0} or some L_{k}, it intersects with D_{0}, which shows that B is completely invariant.

The set D_{0} in the proof is called a fundamental set of the Baker domain B (see [1]).

The function f = f(z; a, b, c) as above has a completely invariant Baker domain B containing the left half plane and the rays L_{k}. Since, for the set of all the critical points,

\{z_{m} = \frac{log a + (2m + 1)πi}{b}\}_{m∈ℤ} ⊂ D_{0} ∪ \{L_{k}\}_{k∈ℤ},

the orbits of every critical point have the imaginary parts tending to −∞. Hence, a theorem of Hinkkanen, Krauskopf and Kriete [2] (essentially due to Bergweiler [3]) gives the following

Theorem 2 For such an f(z; a, b, c) as in Theorem 1,

F (f) = B.

Proof. First, the boundary of the Baker domain B has a positive Euclidean distance, say σ > 0, from the post-critical set C^{+}, consisting of the forward orbits of all critical points of f.

On the other hand, since B contains all L_{k}, every other Fatou component U has a bounded distance, say η, from the boundary of B. Then a theorem of Hinkkanen, Krauskopf and Kriete gives the assertion.

But for the sake of convenience, we include a proof, following their preprint. Suppose that there were a Fatou component U, which should be either a wandering domain or a Baker domain disjoint from B. Fix a point ζ ∈ U, and set ζ_{n} = f^{n}(ζ). Then the disk D_{n} with center ζ_{n} and radius

r_{n} = σ + d(ζ_{n}, B)

satisfies that D_{n} ∩ C^{+} = ∅. Then there are a univalent branch ϕ_{n} of (f^{n})^{−1} which send ζ_{n} to ζ, and a point ξ_{n} ∈ J(f) such that

|ζ_{n} − ξ_{n}| = d(ζ_{n}, B),

for every n. Since functions

g_{n}(z) = ϕ_{n}(r_{n} z + ζ_{n})

on {|z| < 1} map 0 to ζ and the images are disjoint from C^{+}, {g_{n}} is a normal family. So assume that g_{n} converges to g uniformly on compact sets. Then the preimages w_{n} of ξ′_{n} = ϕ_{n}(ξ_{n}) by g_{n} satisfy that

1 > \frac{η}{σ + η} > |w_{n}| > d

with some d > 0 by Schwarz lemma, since ξ′_{n} ∈ J(f) can not tend to ζ. Thus g is non-constant and hence a univalent map, and the Carathéodory convergence theorem implies that, for an r < 1 sufficiently near to 1, {f^{n}} on D = g({|z| < r}) converges uniformly, and hence D ⊂ F (f). But this contradicts the fact that ξ′_{n} ∈ J(f) are contained in D for all but finite n.

Remark If every point z in a component U as in the above proof has such orbits that their real parts tending to +∞, then the standard argument due to Eremenko and Lyubich gives the same contradiction.

We further note the following facts.

Theorem 3 The Julia set of such an f(z; a, b, c) as in Theorem 1 has vanishing area

Proof. We can find an ϵ > 0 such that

(L_{k})_{ϵ} = {w ∈ ℂ | d(w, L_{k}) ≤ ϵ} ⊂ B

for every k, the assertion follows from McMullen-Stallard theorem in [6].

Proposition 4 Set a = e^{iα} and b = iβ. Then we have

f(z; a, b, c) = ∫_{0}^{z} 2be^{(βt+α)i/2} \frac{βt − α + π}{2} ∏_{k=1}^{∞} (1 − \frac{(βt − α + π)^{2}}{4k^{2}π^{2}} ) dt + d

Proof. Since

f′(z) = 2be^{(βz+α)i/2} cos \frac{βz − α}{2},

by using the famous equation

sin πz = πz ∏_{k=1}^{∞} (1 − \frac{z^{2}}{k^{2}} ) ,

we have

f′(z) = 2be^{(βz+α)i/2} \frac{βz − α + π}{2} ∏_{k=1}^{∞} (1 − \frac{(βz − α + π)^{2}}{4k^{2}π^{2}} ) .

Hence we obtain the desired equation.

Finally we consider convergence of Julia sets for two cases.

Theorem 5 Fix b > 0 and c < −3 satisfying 1 − bc + log b > 0 and let ab < 1. Then the sequence of the Julia sets f_{a}(z) = f(z; a, b, c) converges to the Julia set of f(z) = z + e^{bz} + c in the sense of the Hausdorff convergence as ab ↗ 1.

Proof. The assumption 1 − bc + log b > 0 implies F (f) consists of only one Baker domain, which can be seen by the same argument as in the proof of Theorem 1.

In the case ab < 1, there exists an attracting fixed point on the negative real axis, say −r_{a} (r_{a} > 0). It is easy to see that r_{a} tends to infinity monotonically as ab ↗ 1. We set

D_{a} = {|z + r_{a}| < r_{a}}.

For z ∈ D_{a}, we have

|f_{a}(z) + r_{a}| < abr_{a} + 2 < r_{a} + c + 3 < r_{a}.

This means D_{a} is a collapsing disk for the attracting fixed point −r_{a}. The sequence {D_{a}} is monotonically increasing and converges to D_{0} = {Re z < 0} in the sense of the Carathéodory convergence. To see the sequence of the basins B_{a} of the attracting fixed points −r_{a} to F (f) in the sense of the Carathéodory convergence, we need to show only that, for an arbitrary compact set E ⊂ F (f), there exists an A > 0 such that E ⊂ B_{a} for all a > A with ab < 1.

There exists an n such that f^{n}(E) ⊂ D_{0}, for D_{0} is a fundamental set. Since {f_{a}} converges to f uniformly on compact sets as ab ↗ 1, we have f^{n}(E) ⊂ D_{a}, and hence E is contained in B_{a} for all a > A for some A with ab < 1 (see [5]). Therefore the sequence of the complement {B_{a}^{c}} converges to the complement F (f)^{c} = J(f) in the sense of the Hausdorff convergence. Since J(f) has no interior points, we see that {J(f_{a})} converges to J(f) in the sense of the Hausdorff convergence.

The Fatou set of z ↦ z +e^{z} contains infinitely many Baker domains, each of which is contained in the strip

S_{m} = {|Im z − (2m − 1)π| < π}

for some m ∈ ℤ\{0} (see [4]). The orbits of every point in the Baker domain have the real parts tending to −∞.

Theorem 6 Let f_{c}(z) = z + e^{z} + c with c < 0, then F (f_{c}) consists of only one Baker Domain. Furthermore {J(f_{c})} converges to J(z + e^{z}) as c ↗ 0 in the sense of the Hausdorff convergence.

Proof. We set f(z) = z + e^{z}. Let B_{m} be the Baker domain in S_{m}. If z = x + i{(2m − 1)π ± (α + π/2)} with 0 ≤ α < π/2, then we have

Re f(z) = x + e^{x} sin α > x.

If z = x + i{(2m − 1)π ± α} with 0 ≤ α < π/2 and x < 0, then we have

Im f(z) = (2m − 1)π ± α ∓ e^{x} sin α

and thus

(2m − 1)π − α < Im f(z) < (2m − 1)π + α.

Hence, for z ∈ B_{m}, there exists an N such that

|Im f^{n}(z) − (2m − 1)π| < \frac{π}{2}

for all n > N. Setting

K_{m} = \{|Im z − (2m − 1)π| < \frac{π}{2} , Re z < 0\} ,

we see that K_{m} is a fundamental set of B_{m}.

We consider only the case c > −1. Let D_{0} = {Re z < log(−c)}, then it is a fundamental set of a Baker domain of f_{c}. Since f_{c}(z) = f(z) + c,

D_{0} ⋃ ( ⋃_{m∈ℤ\{0}}K_{m})

is also a fundamental set. By the argument similar to that in the proof of Theorem 1, we see that F (f_{c}) consists of only one Baker domain. Further by the argument similar to that in the proof of Theorem 5, we see that {J(f_{c})} converges to J(f) as c ↗ 0 in the sense of the Hausdorff convergence.

References

[1] C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc., 265(1981) 69-95.

[2] A. Hinkkanen, B. Krauskopf and H. Kriete, Growing a Baker domain from attracting islands I: The dynamics of the limit function, preprint.

[3] W. Bergweiler, Invariant domains and singularities, Math. Proc. Camb. Phil. Soc. 117 (1995) 525-532.

[4] S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda, Holomorphic Dynamics, Cambridge Univ. Press, 1999.

[5] S. Morosawa, The Carathéodory convergence of Fatou components of Polynomials to Baker domains or wandering domains, to appear in Proceedings of the Second Congress ISAAC

[6] G.M. Stallard Entire functions with Julia sets of zero measure, Math Proc. Camb. Phil. Soc. 108 (1990) 551-557.

[7] M. Taniguchi, Maskit surgery and structurally finiteness of entire functions, to appear