Note on the iteration of f_{μ}(z) = z exp(z + μ)

Shunsuke Morosawa

Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780, Japan

Abstract

We show that the set of real parameters μ’s for each of which the Julia set of f_{μ} is the whole complex plain has positive Lebesgue measure.

1 Introduction and the result

In this note, we consider a family of transcendental entire functions

{f_{μ}(z) = z exp(z + μ) | μ ∈ R}.

We denote the Fatou set of f_{μ} and the Julia set of f_{μ} by F (μ) and J(μ), respectively. It is easy to see that if μ < 0, then z = 0 is an attracting fixed point of f_{μ} and therefore F (μ) ≠ ∅. Conversely, as a classical problem, an existence of μ for which F (μ) = ∅, or equivalently J(μ) = C, was considered. Baker ([1]) proved an existence of such μ. Further, Jang ([6]) showed that there are infinitely many such μ. Successively, Kuroda and Jang ([7]) constructed a sequence {μ_{n}} of such μ with μ_{n} → ∞ as n → ∞. In this note, we prove the following.

Theorem 1 Let

E = {μ ∈ R | J(μ) = C}.

Then the one-dimensional Lebesgue measure of E is positive.

We now state briefly some properties of the iteration of transcendental entire functions. A cyclic component of the Fatou set is one of the following four components;

(a) an attracting component

(b) a parabolic component

(c) the Siegel disk or

(d) the Baker domain.

A singular value of a function is either a critical value of it or an asymptotic value of it. Singular values play important roles of the study of the iteration theory. The immediate basin of an attracting cycle contains at least one singular value. Same is true for a parabolic cycle. The boundary of a Siegel disk is contained in the closure of the forward orbits of the singular values. No existence of wandering component for the iteration of rational functions was proved by Sullivan ([8]). If a transcendental entire function has finitely many singular values, it is called of finite type. Eremenko and Lyubich ([3]) extended a Sullivan’s result for transcendental entire functions of finite type. That is, the Fatou set of a transcendental entire function of finite type contains no wandering domain. Further, they also proved that the Fatou set of a transcendental entire function of finite type contains no Baker domain.

2 Some properties of f_{μ}(z) = z exp(z + μ)

The singular values of f_{μ} are f_{μ}(−1) = − exp(−1 + μ) and 0. Hence f_{μ} is of finite type and therefore F (μ) contains neither wandering domain nor the Baker domain. The fixed points of f_{μ} in R are 0 and −μ. If μ < 0 then 0 is an attracting fixed point. If μ = 0 then 0 is a rationally indifferent fixed point. If μ > 0 then 0 is a repelling fixed point. Using the graphical analysis, we see that f_{μ}^{n}(−1) → 0 as n → ∞ for μ < 0. Hence if μ < 0 then the component of F (μ) containing 0 is the only one invariant component. We also see that −μ is an attracting fixed point for 0 < μ < 2 and that −μ is a rationally indifferent fixed point for μ = 2. Hence we have J(μ) ≠ C for μ ≤ 2.

Hereafter, we consider the case for μ > 2. Note that the orbit of f_{μ}(−1) is contained in the negative real axis, furthermore

{f_{μ}^{n}(−1)}_{n=1}^{∞} ⊂ [f_{μ}(−1), 0].

Hence F (μ) does not contain any Siegel disk.

3 Unimodal map

We recall that f_{μ} is a unimodal map on (−∞, 0]. There is a famous Jakobson’s Theorem on unimodal maps.

Theorem 2 ([5]) Let

h_{a}(x) = ax(1 − x) (0 < a ≤ 4)

defined on [0, 1] and

Λ = \{a | \begin{matrix}h_{a} has an invariant measure on [0, 1] which is absolutely \\ continuous with respect to the Lebesgue measure. \end{matrix} \}

Then the Lebesgue measure of Λ is positive.

Consider h_{a} as a complex variable function. If h_{a} has an attracting cyclic point or a parabolic cyclic point in C, then the intersection of its basin and the real axis is not empty, since its finite critical value is in R. Moreover, the intersection of its basin and [0, 1] is an open and dense set in [0, 1] . Hence, in this case, there is an invariant measure on [0, 1] which is not absolutely continuous with respect to the Lebesgue measure. Thus, for each a ∈ Λ, the Fatou set of h_{a} coincide with the immediate basin of the super-attracting fixed point ∞.

After Jakobson, there have been many works on generalization of his result and simplification of its proof. We use one of these results proved by Tsujii([10]).

Let k_{α}(x) = −αx and α = exp(−1 + μ). Since μ > 2, we have α > 1. Set

g_{α}(x) = k_{α}^{−1}f_{μ}k_{α}(x) = αx exp(−αx + 1).

Then g_{α} is a unimodal map from [0, 1] to itself. Hence we consider g_{α} as a real valued function on [0, 1]. The set of the singular values of g_{α} is {0, 1} and the set of the fixed points of g_{α} is {0, (1 + log α)/α}. Since the Schwarzian derivative of g_{α} is

Sg_{α}(x) = −\frac{α^{2}}{2(1 − αx)^{2}} \{(αx − 2)^{2} + 2\} < 0,

g_{α} is an S-unimodal map.

Furthermore, by induction, we have the following;

g_{α}^{n}(x) = α^{n}x exp \{α ∑_{k=0}^{n−1} (\frac{1}{α} − g_{α}^{k}(x))\}  (1)

(g_{α}^{n})′(x) = g_{α}^{n}(x)\frac{g′_{α}(x)}{g_{α}(x)}α^{n−1} ∏_{k=1}^{n−1} (\frac{1}{α} − g_{α}^{k}(x))  (2)

We now prepare a theorem on unimodal maps we need to prove Theorem 1. Let h_{t}(x) = H(x, t), x ∈ [0, 1], t ∈ [0, 1] and each h_{t} is a C^{2}-unimodal map from [0, 1] to itself. Set h = h_{0} and let c be a critical point of h. We consider the following conditions;

(ND) h′′(c) ≠ 0.

(CE) There are constants a and b such that, for all n ≥ 0,

(i) |dh^{n}(f(c))| > exp(an − b)
(ii) |dh^{n}(x)| > exp(an − b) for x ∈ h^{−n}(c)

(Hyp) All the cyclic points of h are repelling.

(W) lim inf_{n→∞} \frac{1}{n} log |dh(h^{n}(c))| = 0

(NV_{t}) ∑_{j=0}^{∞} \frac{∂H(h_{t}^{j}(c), t)}{dh_{t}^{j}(h_{t}(c))} ≠ 0

Further, we define

Λ = {t | h_{t} satisfies the conditions (ND), (CE), (Hyp), (W) and (NV_{t}) holds.}.

Theorem 3 ([10]) Let Λ be the set above. Then the Lebesgue measure of Λ is positive and, for each t ∈ Λ, h_{t} has an invariant measure which is absolutely continuous with respect to the Lebesgue measure.

In fact, Tsujii([10]) proved not only families of unimodal maps but also families of more general functions.

4 Proof of Theorem 1

Using Theorem 3, we prove Theorem 1. Let u be the maximal one satisfying that after three iteration of f_{u}, the critical point 1/u mapped to the fixed point (1 + log u)/u for the first time. That is,

g_{u}^{3} (\frac{1}{u}) = \frac{1}{u}(1 + log u) and
g_{u}^{j} (\frac{1}{u}) ≠ \frac{1}{u}(1 + log u) (j = 0, 1, 2)

hold. In [7], it was proved the existence of such u and it satisfies u > 3. We investigate whether these conditions (ND), (CE), (Hyp), (W) and (NV_{t}) are satisfied by this u.

(ND) It is clear.

(CE) First we show (i). We have

g_{u}^{n} (\frac{1}{u}) = \frac{1}{u}(1 + log u)

for n ≥ 3. Since g_{u}(1/u) = 1, by using (2),

(g_{u}^{n})′(1) = \frac{1}{u}(1 + log u)\frac{g′_{u}(1)}{g_{u}(1)}u^{n−1} (\frac{1}{u} − g_{u}(1)) ∏_{k=2}^{n−1} \frac{− log u}{u}

holds for n ≥ 3. Easy calculation shows that (g_{u}^{0})′(1) ≠ 0, (g_{u}^{1})′(1) ≠ 0 and (g_{2}^{2})′(1) ≠ 0. It follows that (i). Facts that g_{α} is S-unimodal and that (i) holds imply that (ii) holds.

(Hyp) Since the asymptotic value 0 is a repelling fixed point and the critical value −1 is preperiodic, it follows that all the cyclic points are repelling.

(W) Since g_{u}(1/u) = (1 + log u)/u for all n ≥ 3, the equality holds.

(NV_{t}) From the equations (1) and (2), we obtain

∑_{n=0}^{∞} \frac{∂_{α}g_{u} (g_{u}^{n} (\frac{1}{u}))}{dg_{u} (\frac{1}{u})} = \frac{1}{u} + \frac{1}{u(1 − u)} ∑_{n=0}^{∞}(− log u)^{−n}

= \frac{1}{u} + \frac{1}{u(1 − u)} \frac{log u}{1 + log u}.

Here, A fact u > 3 implies that

|\frac{1}{1 − u}| < \frac{1}{2} and 0 < \frac{log u}{1 + log u} < 1.

Hence, we have

|∑_{n=0}^{∞} \frac{∂_{α}g_{u} (g_{u}^{n} (\frac{1}{u}))}{dg_{u} (\frac{1}{u})}| > 0.

Therefore, all the conditions of Theorem 3 are satisfied for this u. Hence the set of the parameters μ’s for each of which f_{μ} has an invariant measure with respect to Lebesgue measure has a positive measure. By the argument similar to that stated just after Theorem 2, there is neither an attracting cycle nor a parabolic cycle concerning with the critical value 1. Further, asymptotic value 0 is a repelling fixed point of g_{α} for α > 1. Hence, our claim is obtained.

5 Remarks

If we pay attention to the graphs of f_{μ} in (−∞, 0], the Kuroda-Jang Theorem remind us a problem on bifurcations of real quadratic polynomials. We have a problem similar to one of real quadratic polynomials.

Problem 1 Is the set of μ’s for each of which f_{μ} is hyperbolic open and dense in R ?

It is easy to see that the set is open. In the case of the family of real quadratic polynomials, Świa̧tek([9]) proved such set is dense.

Similarities between the quadratic family and our family considered in this note let us think some problem in the case that μ is complex. Set

f_{λ}(z) = λz exp(z).

Drawing the set of λ for which {f_{λ}^{n}(−1)}_{n=1}^{∞} is a bounded set by a computer, we can see two Mandelbrot-like sets stretched to the infinity along the real axis adjoining at the origin. In fact, Fagella studied this family in [4].

References

[1] I. N. Baker, Limit functions and sets of non-normality iteration theory, Ann. Acad. Sci. Fenn., A.I. Math., 467(1970), 1-9.

[2] A. F. Beardon, Iteration of Rational Functions, GTM 132, Springer, 1991.

[3] A. E. Eremenko & M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier Grenoble, 42(1992), 989-1020.

[4] N. Fagella, Limiting dynamics for the complex standard family, Int. J. Bifurcation and Chaos, 5(1995), 673-699.

[5] M. V. Jakobson, Absolutely continuous invariant measures for one parameter families of one dimensional maps, Commun. Math. Phys., 122(1989), 293-320.

[6] C. M. Jang, Julia set of the function z exp(z +μ), Tohoku Math. J., 44(1992), 271-277.

[7] T. Kuroda & C. M. Jang, Julia set of the function z exp(z + μ) II, preprint.

[8] D. Sullivan, Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains, Ann. Math., 122(1985), 401-418.

[9] G. Świa̧tek Hyperbolicity is dense in the real quadratic family, preprint.

[10] M. Tsujii, Positive Lyapunov exponents in families of one dimensional dynamical systems, Invent.math., 111(1993), 113-137.