Ergodicity of some classes of meromorphic functions ^{∗}

Linda Keen^{†} Mathematics Department CUNY Lehman College Bronx, NY 10468, U.S.A.

Janina Kotus^{‡} Institute of Mathematics Polish Academy of Sciences 00-950 Warsaw, Poland.

Abstract

We consider the class of meromorphic functions whose set of singular values is bounded and for which the ω−limit set of the post singular set is a compact repeller. We show that if two simples growth conditions are satisfied, then the function is ergodic on its Julia set.

1 Introduction

The dynamical phenomena of systems generated by conformal mappings of the plane are controlled by the behavior of the forward orbits of the singular values. The closure of these orbits is called the post-singular set, or in the case of rational maps, the post-critical set. For rational maps, it is known that outside the post-critical set the function is expanding and this expansion leads to the following dichotomy: a rational map either acts ergodically with respect to Lebesgue measure on the sphere and the Julia set is the full sphere or the postcritical set behaves as a measure theoretic attractor. (See [16], [18]). For transcendental meromorphic functions, the essential singularities make the situation more complicated. For example, in [15] Lyubich proved that for the exponential function, the set of points whose ω-limit set contains the ω-limit set of the post-singular set may have positive but not full measure and in [17] McMullen proved that for the sine family the set of points attracted to infinity always has positive measure and many ergodic components.

In this paper, motivated by our study of the tangent family [14], we find sufficient conditions for a transcendental meromorphic function to be ergodic on its Julia set. We prove Theorem 1. Let f be a transcendental meromorphic function whose singular values lie in a bounded set. If f satisfies two simple growth conditions, if the ωlimit set of the post singular set, ω_{f}, is compact and if for some k > 1, |f′| > k on ω_{f}, then f is ergodic with respect to Lebesgue measure on its Julia set.

∗1991 Mathematics Subject Classification: 30D05, 58F08

†Supported in part by NSF and PSC-CUNY

‡Supported in part by Polish KBN Grant No 2 P03A 025 12 ”Iteracje funkcji holomorficznych”

As examples of the theorem we see first that if the omitted values of the tangent map λ tan z land on repelling periodic cycles, the map is ergodic and second that that there are values of λ such that the map λe^{−z^{2}} sin z is ergodic. These special cases also follow from the results in [8].

Our techniques are an adaptation of those developed in [11] to control the set attracted to the essential singularities. The new ideas involve dealing with the poles.

The paper is organized as follows. In section 2 we set our notation and summarize the basic definitions and theory. In section 3 we prove our theorem and in section 4 we discuss applications and examples.

We would like to thank I.N. Baker for helpful comments on the original version and for pointing out that our proof of the main theorem holds more generally than we had originally stated it.

2 Preliminaries

2.1 Julia sets of meromorphic functions

If f : ℂ → ℂ̂ is a transcendental meromorphic function, the orbits of points fall into three categories: they may be infinite, they may become periodic and hence consist of a finite number of distinct points, or they may terminate at a pole of f. To study the dynamics, we define the stable set, or Fatou set, Ω_{f} as the set of those points z such that the sequence f^{∘n}(z) is defined and meromorphic for all n and forms a normal family in a neighborhood of z. The unstable set, or Julia set, J_{f} is the complement of the stable set. We assume ∞ ∈ J_{f}. Thus, Ω_{f} is open, J_{f} is closed. It is easy to see that Ω_{f} is completely invariant and z ∈ J_{f} if and only if f(z) ∈ J_{f} or z = ∞. As in the rational case J_{f} ⊂ \overline{(⋃_{n≥0} f^{∘−n}(z))} for all z ∈ ℂ̂ − E_{f}, where E_{f} consists of at most two exceptional values with finite inverse orbits. For all transcendental entire functions ∞ is an omitted value, so it is also exceptional. For generic transcendental meromorphic functions, the set of prepoles P_{f} = ∪_{n≥0}f^{∘−n}(∞) is infinite and J_{f} = \overline{P_{f}}. Moreover, the Julia set J_{f} is the closure of the repelling periodic points ([1, 3, 9]).

The singular set S = S_{f} of a meromorphic function f consists of those values in ℂ at which f is not a regular covering. Therefore at a singular value v there is a branch of the inverse which is not holomorphic but has an algebraic or transcendental singularity. If the singularity is algebraic, v is a critical value, whereas if it is transcendental, there is a path α : [0, ∞) → ℂ such that lim_{t→∞} |α(t)| = ∞ and lim_{t→∞} f(α(t)) = v, and v is called an asymptotic value for f. If we can associate to a given asymptotic value v an asymptotic tract, that is, a simply connected unbounded domain A such that f(A) is a punctured neighborhood of v and f : A → f(A) is an unramified covering, then v is called a logarithmic singularity.

We define the post-singular set as

P S = P S_{f} = \overline{⋃_{n≥0} f^{n}(S)}

and denote its ω−limit set by ω_{f}.

We distinguish the following classes of meromorphic functions. For f meromorphic and p ≥ 1 define

M_{p} = {f : card S = p}, M = ⋃_{p≥1} M_{p}, and B = {f : S is bounded}.

Of course M ⊂ B. Some basic examples of entire functions in M are:

λe^{z}, a sin(z) + b, f(z) = ∫ ^{z} h(η) exp(p(η))dη,

where λ, a, b ∈ ℂ and h(η), p(η) are polynomials. The class M also includes meromorphic functions with polynomial Schwarzian derivative. An example is λ tan(z) whose Schwarzian derivative is constant. Examples of functions in B − M are:

f(z) = λsin(z)/z and f(z) = λe^{−z^{2}} sin(z);

an example of a function not in B is f(z) = λz sin z.

We call the set ω_{f} a repeller if, for all z ∈ ω_{f} such that f(z) is defined, f(z) ∈ ω_{f} and if there exists k > 1 such that, for all z ∈ ω_{f}, |f′(z)| ≥ k. It is a compact repeller if f(z) is defined for all z ∈ ω_{f} and if ω_{f} is compact.

In particular, if f ∈ M and all critical and asymptotic values are eventually mapped to repelling periodic points, then ω_{f} is a compact repeller. Note that for transcendental meromorphic functions with poles, ω_{f} may include points with finite forward orbits. For example, the singular set S of the function f(z) = \frac{πi}{2} tan(z) consists of the omitted values \{±\frac{π}{2} \}. Since they are poles and have no forward orbit, ω_{f} = S_{f} is a repeller but is not a compact repeller. Clearly, if ω_{f} is not finite and contains prepoles it cannot be a compact repeller.

2.2 Classification of stable behavior

Let D be a component of the stable set; f will map D to a component, but if the image contains an asymptotic value, the map might not be onto. In any case, we call the image f(D) and note that either there exist integers m ≠ n > 0 such that f^{∘n}(D) = f^{∘m}(D), and D is called eventually periodic with period p = min(|m − n|), with the minimum taken over all such m, n, or for all m ≠ n, f^{∘n}(D) ∩ f^{∘m}(D) = ∅, and D is called a wandering domain.

The qualitative and quantitative description of the eventually periodic behavior is slightly more complicated than in the rational case because of the transcendental singularity at ∞ and the possibility that f^{p} may not be defined at some values. As for rational maps, eventually periodic domains may be attracting, parabolic or rotation domains. In addition, however, an eventually periodic domain D may be an an essentially parabolic or Baker domain; that is, the boundary of D contains a point z_{0} (possibly ∞) such that f^{np}(z) → z_{0} for z ∈ D and f^{p} is not holomorphic at z_{0}. If p = 1, then the only possible boundary point is ∞.

2.3 Two propositions

We shall need the following proposition proved in [13]

Proposition 2.1. Let f be a meromorphic function and let D be an open disk such that D ∩ J_{f} ≠ ∅. Suppose that there are branches g_{n_{k}} ∈ f^{−n} holomorphic and univalent on D with n_{k} → ∞. Then (g′_{n_{k}}) → 0 uniformly on every compact set in D.

We shall also need the following version of Koebe’s distortion theorem.

Proposition 2.2. If g : D(0, 1) → ℂ is a univalent map normalized so that g(0) = 0, then for every 0 < k < 1 and z, w ∈ D(0, k), there is a constant T (k) such that and |g′(z)|/|g′(w)| lies between 1/T (k) and T (k).

2.4 Orbits that tend to infinity

In this section we adapt the discussion of Eremenko and Lyubich [11] on entire functions in B to meromorphic functions in B. They give a sufficient condition for the measure of the set

I_{∞}(f) = {z ∈ ℂ : f^{n}(z) → ∞}

to be zero. We need additional condition on the Laurent expansions about the poles to prove this in the meromorphic case.

At each pole p of order m_{p}, form the Laurent expansion of f about p, f_{p}(z) = c_{p}/(z − p)^{m_{p}}(1 + ϕ_{p}(z − p)) where ϕ(z) is analytic and ϕ(z − p) = o((z − p)^{m_{p}}).

Suppose f ∈ B. Let D(q, r) denote the disk of radius r centered at q, A_{r} the annulus A_{r} = {z : r < |z| < ∞}.

Because S(f) is bounded, we can find R_{0} > 0 such that S(f) ⊂ D(0, R_{0}). Fix R > R_{0}. Without loss of generality we may assume that f is analytic at 0 and |f(0)| < R_{0}/2.

Suppose w ∈ A_{R} and f(z) = w. Since there are no singular values in A_{R}, any branch g of f^{−1} such that g(w) = z can be continued analytically throughout A_{R}. Let g be some branch of f^{−1} and set V = g(A_{R}). By [20], ∞ is either a logarithmic branch point of g with asymptotic value a, V = V_{a} is simply connected and f : V_{a} → A_{R} is a universal covering, or ∞ is an algebraic branch point of g of order m_{p} −1 for some integer m_{p}, V = V_{p} is conformally equivalent to a disk punctured at a pole p of f and f : V_{p} → A_{R} is a regular m_{p} to 1 covering.

Let m denote Lebesgue measure in ℂ and let Θ_{R}(r, f) be the linear measure of the set {θ : |f(re^{iθ})| < R}.

With this notation we have

Proposition 2.3. Let f ∈ B and suppose that there is a positive integer J and positive constants b, B, C_{1}, C_{2} > 0 such that for every p, and z ∈ V_{p}, the multiplicities m_{p} are bounded by J and the coefficients c_{p} and functions ϕ_{p}(z−p) and ϕ′_{p}(z − p) satisfy

b < |c_{p}| < B, |ϕ_{p}(z − p)| < C_{1}, |ϕ′_{p}(z − p)| < C_{1}, C_{2} < |1 + ϕ_{p}(z − p)| (∗)

Moreover suppose

lim inf_{r→∞} \frac{1}{log r} ∫_{1}^{r} Θ_{R}(t, f)\frac{dt}{t} > 0. (∗∗)

Then m(I_{∞}(f)) = 0.

Proof: We summarize the proof in [11] that m(I_{∞}) = 0 for entire functions satisfying condition (∗∗), indicating the necessary changes to account for the poles. We remark that the proof there is valid if there are only finitely many poles. We therefore assume here that our functions have infinitely many poles

The first step in the proof that m(I_{∞}) = 0 is to obtain a uniform expansion estimate on |f′| restricted to each component V_{a} or V_{p}.

For simply connected V_{a}, the argument is the same as in [11] using Teichmüller’s construction of a logarithmic coordinate. We outline that argument here. If ∞ is a logarithmic branch point of g, we first define a logarithmic coordinate w = log z, z ∈ V_{a} and set U_{a} = log V_{a}; then U_{a} ⊂ H = {w : ℜw > 0} has infinitely many simply connected unbounded components U_{a,n} each contained in a vertical strip of width 2π and the exponential maps each U_{a,n} univalently onto V_{a}. We have a commutative diagram:

U_{a}  →^{F} H_{R}
exp ↓  ↓ exp
V_{a}  →^{f} A_{R}

where H_{R} = {w : ℜw > log R} and F maps U_{a,n} ⊂ H univalently onto H_{R}.

If w ∈ H, the disk D = D(F (w), δ(w)) centered at F (w) with radius δ(w) = ℜF (w) − log R is contained in H_{R}. Denote the branches of F^{−1} by G_{n} : H_{R} → U_{a,n}. By the Koebe \frac{1}{4}-theorem G_{n}(D) contains a disk D_{n} of center w and radius \frac{1}{4}δ(w)|(G_{n})′(F (w))|. Since U_{a,n} cannot contain any vertical segment of length 2π, for w ∈ U_{a,n} we have

|F′(w)| > \frac{1}{4π} (ℜF (w) − log R).

It follows there are constants k, K > 1 such that if ℜF (w) > k, then |F′| > K. By the chain rule, we have |f′(z)| > K|f(z)|/|z|.

To obtain an estimate for |f′| = |f′_{p}| on the punctured disk domain V_{p} we use a different argument.

Note that since ϕ_{p} and ϕ′_{p} are analytic in V_{p}, |ϕ_{p}| and |ϕ′_{p}| achieve their maxima on ∂V_{p}. Increasing R clearly decreases these maxima so that by taking R large enough, we may assume C_{1}, C_{2} and C_{1}/C_{2} are small relative to R.

If z ∈ V_{p}, by conditions (∗)

R ≤ |f_{p}(z)| = |\frac{c_{p}}{(z − p)^{m_{p}}} (1 + ϕ_{p}(z − p))| ≤ \frac{B}{|z − p|^{m_{p}}} (1 + C_{1})

so that |z − p| < M_{p} = \frac{B(1+C_{1})}{R})^{\frac{1}{mp}} < 1.

For t > R, set A_{R,t} = A_{R} \ A_{t}. Choose R′ >> R and such that R′ > C_{2}b; for each pole p set Ṽ_{p} = f_{p}^{−1}(A_{R,R′}). Then if z ∈ Ṽ_{p}, by conditions (∗) we have

\frac{bC_{2}}{|z − p|^{m_{p}}} ≤ |f_{p}(z)| = |\frac{c_{p}}{(z − p)^{m_{p}}} (1 + ϕ_{p}(z − p))| ≤ R′

so that |z −p| > ρ_{p} = \frac{bC_{2}}{R′} ^{1/m_{p}}. The annulus ρ_{p} < |z −p| < M_{p} is thus contained in Ṽ_{p}. Note that since for all p, 1 ≤ m_{p} ≤ J, the moduli of these annuli are uniformly bounded.

Next compute that

|f′_{p}(z)| = |f_{p}(z)| |\frac{m_{p}}{z − p} + \frac{ϕ′_{p}(z − p)}{1 + ϕ_{p}(z − p)}| .

In Ṽ_{p} we have

|f′_{p}(z)| ≤ R′ |J (\frac{R′}{bC_{2}} )^{\frac{1}{mp}} + \frac{C_{1}}{C_{2}} |

and

|\frac{m_{p}}{z − p} + \frac{ϕ′_{p}(z − p)}{1 + ϕ_{p}(z − p)}| > ||\frac{m_{p}}{z − p}| − |\frac{ϕ′_{p}(z − p)}{1 + ϕ_{p}(z − p)}|| ≥ |\frac{1}{M_{p}^{m_{p}}} − \frac{C_{1}}{C_{2}} | .

Thus by choosing R and R′ large enough, we see that there are constants, K′, K, independent of p, such that for every |p| > R, in Ṽ_{p} we have

K′ > |f′_{p}(z)| > K > 1.  (1)

To show that m(I_{∞}) = 0, by the Lebesgue density theorem it is sufficient to show that for any point z ∈ I_{∞},

lim sup_{δ→0} \frac{m(D(z, δ) ∩ I_{∞})}{m(D(z, δ))} < 1.  (2)

In order to do this we first restate condition (∗∗) as follows: Let V = f^{−1}(A_{R}). Then there is a κ > 0 such that

lim sup_{t→∞} \frac{area(A_{R,t} ∩ V)}{area(A_{R,t})} ≤ 1 − κ.  (3)

In terms of the logarithmic coordinates log z = s + iθ, let S_{t} = {log R < s < t, 0 ≤ θ < 2π} and set U = log V. Then there is a δ > 0 such that

lim sup_{t→∞} \frac{area(S_{t} ∩ U)}{area(S_{t})} ≤ 1 − δ.  (4)

For z_{0} ∈ I_{∞}, set z_{n} = f^{n}(z_{0}). We may assume without loss of generality that |z_{n}| > R is so large that the estimates (1) on |f′| hold. Then either there is some N such that for all n > N, z_{n} ∈ V_{a} for some simply connected component V_{a} or there is a subsequence z_{n_{j}} → ∞ and a sequence of poles p_{j} → ∞ such that z_{n_{j}} ∈ V_{p_{j}}.

In the first case the argument is just as in [11]. For w_{0} = log z_{0} in a fixed strip of width 2π, let w_{n} = F (w_{n−1}), where F is the lift defined above. We may assume that for all n, ℜw_{n} is large enough that |F′(w_{n})| > K > 1. Let F_{n}^{−1}(w_{n}) = w_{n−1}. Then, there is a constant d, independent of n such that F_{n}^{−1} is univalent on D_{n} = D(w_{n}, d). Moreover, by the expansion of F, we may choose m such that F (D_{n}) contains a vertical segment of width bigger than 2π. Fix such an m and let B_{j} = F^{−j}(D(w_{m}, d/4)) for j = 1, … m, where F^{−j} is the composition of the appropriate inverse branches. Applying the Koebe distortion theorem to the function F^{−m} we have B_{m} ⊂ D(w_{0}, d/4K^{m}). Moreover if s_{m} is the radius of the smallest disk centered at w_{0} containing B_{m}, there is a constant t < 1, independent of m such that D(w_{0}, ts_{m}) ⊂ B_{m} ⊂ D(w_{0}, s_{m}). Clearly s_{m} → 0 as m → ∞ so that applying (4) we obtain (2).

In the second case we need to modify the argument. Since the moduli of the annulii Ṽ_{p} have the same bounds, there is a constant 0 < τ < 1, independent of V_{p}, such that \frac{area(Ṽ_{p})}{area(V_{p})} ≤ 1 − (\frac{bC_{2}R}{B|1+C_{1}|R′})^{\frac{2}{mp}} < 1 − τ. We want to show that there is a constant μ > 0 independent of p satisfying

\frac{area(V_{p} ∩ f_{p}^{−1}(V))}{area(V_{p})} ≤ 1 − μ.  (5)

With the constants of inequality (1), set L = \frac{K′}{K}. We claim that μ = \frac{κ}{L^{2}} τ where κ is the constant in inequality (3).

We rewrite \frac{area(V_{p}∩f_{p}^{−1}(V))}{area(V_{p})} and estimate

\frac{area([(V_{p} \ Ṽ_{p}) ∩ f_{p}^{−1}(V)] ∪ [Ṽ_{p} ∩ f_{p}^{−1}(V)])}{area(V_{p})} ≤

\frac{area([(V_{p} \ Ṽ_{p}) ∩ f_{p}^{−1}(V)])}{area(V_{p})} + \frac{area(Ṽ_{p})}{area(V_{p})} =

\frac{area([(V_{p} \ Ṽ_{p}) ∩ f_{p}^{−1}(V)])}{area([V_{p} \ Ṽ_{p}])} × \frac{area([V_{p} \ Ṽ_{p}])}{area(V_{p})} + \frac{area(Ṽ_{p})}{area(V_{p})} ≤

(1 − \frac{κ}{L^{2}} ) (1 − \frac{area(Ṽ_{p})}{area(V_{p})}) + \frac{area(Ṽ_{p})}{area(V_{p})}

= 1 − \frac{κ}{L^{2}} − \frac{area(Ṽ_{p})}{area(V_{p})} + \frac{κ}{L^{2}} \frac{area(Ṽ_{p})}{area(V_{p})} + \frac{area(Ṽ_{p})}{area(V_{p})}

≤ 1 − \frac{κ}{L^{2}} + \frac{κ}{L^{2}} (1 − τ)

= 1 − \frac{κ}{L^{2}} τ = 1 − μ.

Now fix n_{j} such that z_{n_{j}} ∈ V_{n_{j}} and choose a univalent branch of f_{p_{n_{j} −1}}^{−1} so that f_{p_{n_{j} −1}}^{−1} : z_{n_{j}} ↦ z_{n_{j}−1}. Let m = n_{j} and set D_{m} = D(z_{m}, M_{m}). For large m, |z_{m} − p_{m}| << ρ_{m} and V_{p_{m}} ≈ D(z_{m}, M_{m}). We define B_{m} = f^{−m}(z_{m}, M_{m}/4) where again f^{−m} is a composition of the appropriate inverse branches so that z_{0} ∈ B_{m}. Then f^{−m} is univalent on V_{p_{m}} and we can control distortion on D_{m}. Now for any trajectory from z_{n} to z_{n+k}, by the chain rule and the estimates on factors |f′|, whether the corresponding domain containing z_{j} is a simply connected V_{a} or is an annular Ṽ_{p}, we have |f^{k}(z_{n+k})′| > K^{k} \frac{|z_{n+k}|}{|z_{n}|}. We apply

this with n = 0 and k = n_{j} and we may certainly assume \frac{|z_{n_{j}} |}{|z_{0}|} > 1. In the

annular domains we also have |f^{k}(z_{n_{j}})′| < K′^{k}.

We may assume |z_{n_{j}}| > |z_{n_{j−1}}| so that the contraction over this part of the orbit is at least 1/K^{m}. Then by the Koebe theorems B_{m} ⊂ D(z_{0}, M_{m}/K^{m}), its distortion is bounded independent of m and its diameter goes to zero as m goes to infinity. Thus applying (5) to D_{m} and pulling back we obtain (2).

Thus we have shown that (2) holds at any point in m(I_{∞}) and hence that its Lebesgue measure is zero.

3 The results

3.1 Two lemmas

For a given f define the sets

L = L_{f} = {z : f^{n}(z) → ω_{f}}
K_{ϵ} = K_{ϵ}(f) = {z : dist(z, ω_{f}) < ϵ}

where we omit the f unless confusion results.

If ω_{f} is a compact repeller it is obvious that it contains no critical points. In fact, the post-singular set does not accumulate on ω_{f} but actually lands on it. Precisely, Lemma 3.1. If ω_{f} is a compact repeller then there is an ϵ > 0 and an integer N > 0 such that if c ∈ S, n > N and f^{n}(c) ∈ K_{ϵ/2} then f^{n}(c) ∈ ω_{f}.

Proof: If P S is finite, the lemma is obviously true so assume P S is not finite. Since ω_{f} is compact and P S is infinite, ω_{f} contains no prepoles. Moreover, since it is a repeller, there exist constants k > 1 and ϵ > 0 such that |f′(z)| ≥ k > 1 on the sets K_{ϵ} and K_{ϵ/2}.

Set V = f(\overline{K_{ϵ/2}}). Since \overline{K_{ϵ/2}} contains no poles V is compact. Moreover, since f|_{K_{ϵ/2}} is expanding, \overline{K_{ϵ/2}} ⊂ int(V) and there is a regular branch g of f^{−1} that maps V to \overline{K_{ϵ/2}}. Thus, the annuli A_{0} = V − g(V), A_{n+1} = g^{n}(A_{0}), n ∈ ℕ are nested and mod(A_{n}) ≠ 0 for all n ∈ ℕ.

We claim that for each n ∈ ℕ

f^{n}(S) ∩ (\overline{K_{ϵ}} − ω_{f}) = ∅.

If not, there is a sequence n_{k} ∈ ℕ, n_{k} → ∞ such that for each n_{k}, there is some c_{k} ∈ S (not necessarily distinct) satisfying v_{k} = f^{n_{k}}(c_{k}) ∈ \overline{K_{ϵ}} − ω_{f}. By the compactness of \overline{K_{ϵ}}, as k → ∞, the sequence v_{k} accumulates and by definition, any accumulation point belongs to ω_{f}. For each k, there is an i(k) such that v_{k} ∈ A_{i(k)} and w_{k} = f^{i_{k}}(v_{k}) ∈ A_{0}. Since \overline{A_{0}} is compact, the sequence w_{k} has an accumulation point in \overline{A_{0}} which by definition, also belongs to ω_{f}, This is a contradiction because A_{0} is separated from ω_{f} by the annuli A_{n}, n > 0 and P S ∩ (\overline{K} − ω_{f}) = ∅ as required.

Next, for the set L_{f} we have,

Lemma 3.2. If f ∈ B and ω_{f} is a compact repeller then the Lebesgue measure of L_{f} is 0.

Proof: Since ω_{f} is a compact repeller, it is clear by the classification of stable domains that L_{f} ⊂ J_{f}.

Let ϵ be chosen as in lemma 3.1 and define

L = ⋂_{n≥0} f^{−n}(K_{ϵ/2}).

Then z ∈ L − P_{f} if its full forward trajectory belongs to K_{ϵ/2}. We will prove that m(J_{f} ∩ (L − P_{f})) = 0. Since L ⊂ ⋃_{n=0}^{∞} f^{−n}(L) and P_{f} is countable, this will imply m(L) = 0.

Suppose that m(L−P_{f}) > 0 and let z_{0} be a density point of L−P_{f}. Since ω_{f} is compact, the orbit \{f^{n}(z_{0})\} has a finite accumulation point y_{0} ∈ \overline{K_{ϵ/2}} ⊂ K_{ϵ}. It follows that there exists a sequence n_{k} → ∞ such that z_{k} = f^{n_{k}}(z_{0}) → y_{0}. For k ∈ ℕ, let D_{k} = D(z_{k}, ϵ/4) and let g_{k} be the branch of f^{−n_{k}} that maps z_{k} to z_{0}. Since z_{k} ∈ K_{ϵ/2}, D_{k} ⊂ K_{ϵ} and by lemma 3.1 g_{k} is univalent on D_{k}.

Now by the definition of a density point and by propositions 2.1 and 2.2

\frac{m(g_{k}(D_{k}) ∩ L)}{m(g_{k}(D_{k}))} → 1.

Again by proposition 2.2

\frac{m(D_{k} ∩ f^{n_{k}}(L))}{m(D_{k})} → 1

and m(D_{k} ∩ f^{n_{k}}(L)) = m(D_{k}).

Let U be an open set with compact closure contained in ℂ−\overline{K}. Since z_{k} ∈ J_{f} there exists an integer N such that f^{N}(D_{k}) ⊃ \overline{U} so that m(f^{N+n_{k}}(L) ∩ U) > 0. By definition however, for all k ∈ ℕ, f^{k}(L) ⊂ K_{ϵ/2} so f^{N+n_{k}}(L ∩ U) = ∅. This contradiction finishes the proof.

3.2 The main theorem

We are now ready to prove the main theorem.

Theorem 3.3. Suppose f ∈ B. If ω_{f} is a compact repeller and if conditions (∗) and (∗∗) hold then f is ergodic with respect to Lebesgue measure on its Julia set.

Proof: Suppose that E ⊂ J_{f} is an f-invariant measurable set of positive measure and let z be a density point of E. Since P_{f} is countable we may assume z ∉ P_{f}. Under the hypotheses that f ∈ B and that both (∗) and (∗∗) hold, by proposition 2.3 we may assume z ∉ I_{∞} and thus that its orbit has a finite accumulation point y; let z_{k} = f^{m_{k}}(z) → y, k → ∞.

Under the hypotheses that ω_{f} is a compact repeller, by lemma 3.2 we may assume z ∉ L and hence y ∉ ω_{f}. Now suppose that either y = f^{n}(v) for some v ∈ S or that for some sequence c_{k} ∈ S and iterates n_{k}, f^{n_{k}}(c_{k}) → y. By lemma 3.1, in the first case f^{N}(y) ∈ ω_{f} and in the second, the n_{k} are bounded and f^{n_{k}+N}(c_{k}) ∈ ω_{f}. In either case there is an N such that f^{N}(z_{k}) → ω_{f}. Arguing as in the proof of lemma 3.1, we can find a sequence k_{j} → ∞ such that w_{j} = f^{N+k_{j}}(z_{k}) lie in a compact annulus A_{0} separated from ω_{f}; since f^{N+k_{j}}(y) ∈ ω_{f}, we have a contradiction. Thus η = dist(y, P S) > 0 and we can define a univalent branch g_{k} of f^{−m_{k}} on D_{k} = D(z_{k}, η/4) such that g_{k}(z_{k}) = z.

As above, by the definition of a density point and by propositions 2.1 and 2.2 we obtain

\frac{m(g_{k}(D_{k}) ∩ E)}{m(g_{k}(D_{k}))} → 1 and \frac{m(D_{k} ∩ f^{m_{k}}(E))}{m(D_{k})} → 1.

Since E is forward invariant we have m(D_{k} ∩ E) = m(D_{k}).

If f is not ergodic, we can find another forward invariant positive measure set F ⊂ J_{f}, m(E ∩ F) = 0, and a density point z′ for F for which we can construct disks D′_{l} of fixed radius such that m(F ∩ D′_{k′}) = m(D′_{k′}). Because z_{k} ∈ D_{k}, z_{k′} ∈ D′_{k′} defined as above belong to J_{f} there is some N ∈ ℕ such that f^{N}(D′_{k′}) ⊃ D_{k}. It follows that m(E ∩ F) > m(f^{N}(D′_{k′} ∩ F) ∩ (D_{k} ∩ E)) > 0 and we obtain a contradiction.

4 Examples

1. Examples of functions that the main theorem applies to are found in the class of meromorphic functions with polynomial Schwarzian derivative. By Nevanlinna’s theorem, if the Schwarzian of f is a polynomial of degree p ≥ 0 then f ∈ M_{p+2} and all singularities of f^{−1} are logarithmic. Moreover, at least half of the singularities are finite so it is not hard to show directly that condition (∗∗) holds.

Now note from [19] that if the Schwarzian derivative is polynomial is degree p − 2, then in a sector of width 2π/p − 2ϵ about each of the p Julia directions, the function has the asymptotic form

\frac{AG_{ν} + BG_{ν+1}}{CG_{ν} + DG_{ν+1}}

where

G_{ν} ≈ exp (−1)^{ν+1}z^{p/2}.

Therefore in any given Julia direction the coefficients c_{p} and functions ϕ_{p} are all approximately equal and condition (∗) holds.

The dynamical properties of these functions were described in [9]. In particular, it is shown that they have no wandering domains and no Baker domains. Thus, if each singular value lands on a repelling cycle, J = ℂ̂ and the function is ergodic.

A special subclass contains the functions with constant Schwarzian derivative. To this class belong the families λe^{z}, λ tan(z), λ ∈ ℂ where it is easy to construct examples for which the singular values land on repelling cycles. For instance: if f_{1}(z) = πi tan(z), then S(f_{1}) = {±π}, P S = {±π, 0}, ω_{f} = {0} and |f′_{1}(0)| = π; and if f_{2}(z) = πi exp(z) then S = {0}, P S = {0, πi, −πi}, ω_{f} = {−πi} and |f′_{2}(−πi)| = π.

2. Functions in the above class where theorem 3.3 does not apply are e^{z} and \frac{πi}{2} tan(z). For e^{z} the orbit of the singular value 0 tends to ∞ and for \frac{πi}{2} tan(z) the singular values are the poles ±π/2; thus, although ω_{f} is a repeller, it is not compact. In fact, it is proved in [15] that e^{z} is not ergodic. The same is probably true for πi/2 tan(z).

Another simple example where theorem 3.3 does not apply is the family a sin(z)+b studied in [17]. Although the singular set consists of two points, and a, b can be chosen so that the critical values are mapped to repelling periodic orbits, condition (∗∗) fails and J_{f} = \overline{I_{∞}(f)} = ℂ̂ has many ergodic components.

3. It is hard to find examples of maps with infinitely many singular values satisfying the hypotheses of our theorem because it is hard to tell whether ω_{f} is a compact repeller and to check the conditions on the Laurent expansions. Here is one such example.

Consider the entire function in f_{λ} = λe^{−z^{2}} sin z. Since it is entire condition (∗) is vacuous. To see that it satisfies condition (∗∗) note that for λ real, if | arctan z| < 1/2 then |f′_{λ}(z)| is bounded. Its infinitely many critical values are bounded and accumulate on the asymptotic value 0 hence it belongs to B. Moreover, λ may be chosen so that ω_{f_{λ}} is a compact repeller. Thus theorem 3.3 applies so that f_{λ} is ergodic on its Julia set.

In this case we also know there are no eventually periodic domains and, since m(I_{∞}) = 0, no Baker domains. There are also no wandering domains. If there were a multiply connected wandering domain D, it would contain a homotopically non-trivial loop γ and by [2] the winding number of (a subsequence of) f^{n}(γ) would be non-zero for n sufficiently large and the diameter of f^{n}(γ) would tend to infinity. Thus γ would have to intersect the real axis; since the positive and negative real axes are asymptotic curves such domains cannot exist. Since m(I_{∞}) = 0, no wandering domain can escape to infinity. Moreover, since by [7] all limit functions of f^{n}|_{D} would have to be constants in the compact repeller ω_{f_{λ}}, we conclude J_{f_{λ}} = ℂ̂.

References

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[15] M. Lyubich. The measurable dynamics of the exponential map. Siberian J. Math. 28(5): 111-127, 1987.

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[19] R. Nevanlinna. Über Riemannsche Fläschen mit endlich vielen Windungspunkten. Acta Math. 58, 1932.

[20] R. Nevanlinna. Analytic functions. Springer Verlag, 1970.

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