Chaotic composition operators on the classical holomorphic spaces

Abstract

We call the composition operator induced by a biholomorphic automorphism of the unit disc U a Möbius composition operator. Here, if ϕ has a fixed point in U, then we call ϕ elliptic. In this paper, we give an elementary proof to the fact that a Möbius composition operator is chaotic on the Hardy space H^{p} and on the Bergman space B^{p} for every p ∈ (0, +∞) if and only if the corresponding ϕ is non-elliptic. This result is a generalization of Hosokawa’s results in [4].

1 Introduction

Let X be a topological vector space and L : X → X a continuous linear operator. We say that L is hypercyclic if there is an element x of X such that the orbit {L^{k}(x)} is dense in X. Here L^{k} is the k-th iteration of L for every positive k ∈ ℤ. On the other hand, we say that L is chaotic if

1) the set of periodic points of L are dense in X,

2) L is transitive, and

3) L has sensitive dependence on initial condition. (See for instance, [7] and [9].) Here if X is (separable and) completely metrizable, it is well-known that the condition 1) and hypercyclicity of L imply that L is chaotic. Hence in the case of Banach spaces, the famous hypercyclicity criterion due to Kitai, Gethner, and Shapiro can be rewritten as a criterion for chaos.

Proposition 1 Let X be a separable complex Banach space and L : X → X a bounded linear operator. Suppose that there are a dense subset S of X, a convergent series ∑_{k=1}^{∞} a_{k} of positive numbers, and a positive constant A(x) for every x ∈ S such that L^{−1} exists on S and that

∥L^{k}(x)∥ ≤ A(x)a_{|k|}

for every k ∈ ℤ, where L^{−k} = (L^{−1})^{k} for every positive k ∈ ℤ. Then L is chaotic.

Proof. First, L is hypercyclic by the hypercyclicity criterion (see for instance,

[5] and [9]). Next, for every x ∈ S and positive k ∈ ℤ,

y_{k} = ∑_{m=−∞}^{+∞} L^{mk}(x) ∈ X,

which is periodic, i.e. L^{k}(y_{k}) = y_{k}. Also ∥y_{k} − x∥ ≤ 2A(x) ∑_{m=1}^{∞} a_{mk}, which tends to 0 as k → +∞. Since S is dense in X, periodic points are dense in X.

Now, let U be the unit disc in ℂ, and Aut(U) the group of Möbius transformations which preserve U. For every ϕ ∈ Aut(U), we define the composition operator C_{ϕ} by setting

C_{ϕ}f = f ∘ ϕ

for any function f on U, which we call the Möbius composition operator for ϕ.

Next, the classical holomorphic spaces are defined as follows.

Definition For every p with 0 < p < ∞, the Bergman space B^{p} is the space consisting of all holomorphic functions f on U such that

∥f∥_{B^{p}}^{p} = ∫_{U} |f|^{p}dA < ∞,

where dA is the normalized area measure dxdy/π.

The Hardy space H^{p} is the space consisting of all holomorphic functions f on U such that

∥f∥_{H^{p}}^{p} = lim_{r→1} ∫_{0}^{2π} |f(re^{iθ})|^{p} \frac{dθ}{2π} < ∞.

Here recall that the identical embedding of H^{p} into B^{p} is continuous, that every f ∈ H^{p} has the boundary function f(e^{iθ}) = lim_{r→1} f(re^{iθ}) almost all θ ∈ ℝ, and that

∥f∥_{H^{p}}^{p} = ∫_{0}^{2π} |f(e^{iθ})|^{p} \frac{dθ}{2π} .

Also, if p ≥ 1 then H^{p} and B^{p} are Banach spaces and even if p ∈ (0, 1), they are separable completely metrizable topological vector spaces, and the following fact is well-known and easily seen.

Proposition 2 For every p with 0 < p < ∞, every Möbius composition operator is continuous on B^{p} and on H^{p}.

In the sequel, we say that ϕ ∈ Aut(U) is hyperbolic or parabolic, respectively, if ϕ has exactly two or one fixed point(s) on the unit circle ∂U. If ϕ has a fixed point in U, then we call ϕ elliptic (including the identical map). Thus non-elliptic elements are either hyperbolic or parabolic. And it is known ([9]) that, for every non-ellipic element of Aut(U), the corresponding Möbius composition operator is chaotic on the complex topological vector space H(U), consisting of all holomorphic functions on U with the topology induced by uniform convergence on compact sets of U.

In this paper, we give an elementary proof of the following generalization of Hosokawa’s results in [4].

Theorem 3 For every ϕ ∈ Aut(U), the following conditions are equivalent. 1) ϕ is non-elliptic.

2) C_{ϕ} is chaotic on some Hardy space H^{p} with p ∈ (0, +∞).

3) C_{ϕ} is chaotic on every Hardy space H^{p} with p ∈ (0, +∞).

4) C_{ϕ} is chaotic on some Bergman space B^{p} with p ∈ (0, +∞).

5) C_{ϕ} is chaotic on every Bergman space B^{p} with p ∈ (0, +∞).

Here clearly 3) implies 2) and 5) implies 4). Also it is well-known that 2) or 4) implies 1). Actually, any elliptic ϕ ∈ Aut(U), C_{ϕ} is not hypercyclic on B^{p} and on H^{p} for every p ∈ (0, +∞). (cf. [1] and [8]. Indeed, fix p ∈ (0, +∞) and an element f in A^{p} or in H^{p} arbitrarily. Let α ∈ U be the fixed point of ϕ. Then (C_{ϕ})^{k}f(α) = f(α) for every k ∈ ℤ. Since the norm convergence in B^{p} or in H^{p} implies the local uniform convergence on U, the orbit of f is dense neither in B^{p} nor in H^{p}.) Thus we need to prove that 1) implies 3) and 5), which will be proved in the following sections.

Finally, the author would like to hearty thanks to Professor K. Matsuzaki and the referee for their valuable comments.

2 Möbius composition operators for hyperbolic automorphisms

In this section, we prove the following

Theorem 4 Suppose that 0 < p < ∞. Then for every hyperbolic ϕ in Aut(U), the composition operator C_{ϕ} on B^{p} and on H^{p} is chaotic.

Let H(\overline{U}) be the set of all functions holomorphic in a neighborhood of \overline{U}. Then it is well known that H(\overline{U}) is dense in H^{p} and in B^{p}. (See for instance,

[2] and [3].) In particular, H^{p} (as a subset) is dense in B^{p}. Thus we need to prove the assertion only for the Hardy spaces. Hence we fix p ∈ (0, +∞), and consider the Hardy space H^{p} only, though the same arguments give the assertion for B^{p}.

Lemma 5 Let α be a point on the unit circle ∂U, and N(α) be the set of all functions holomorphic in a neighborhood of the closed unit disc \overline{U} vanishing at α. Then N(α) is dense in H^{p}.

Furthermore, the subsets N(α^{K}), consisting of all functions holomorphic in a neighborhood of \overline{U} with a zero of order not less than K ≥ 2 at α, and N(α, β) = N(α) ⋂ N(β) with another β ∈ ∂U are also dense in H^{p}.

Proof. Set

g_{t}(z) = \frac{z − α}{z − tα}

for every real t > 1. Then g_{t} ∈ H(\overline{U}), and sup_{z∈\overline{U}} |g_{t}(z)| < 1. Hence by the Lebesgue convergence theorem, we have

lim_{t→1} ∥f − g_{t}f∥_{H^{p}} = 0

for every f ∈ H(\overline{U}). Since g_{t}f ∈ N(α), we have the first assertion. The other cases are similar.

To estimate the norm of (C_{ϕ})^{k}f for every k ∈ ℤ, we divide U into the Dirichlet fundamental regions for the cyclic group G generated by ϕ. Let D_{0} be the Dirichlet fundamental region for G with the center 0, i.e.

D_{0} = {z ∈ U | d(z, 0) < d(z, ϕ^{ℓ}(0)), ℓ ∈ ℤ − {0}},

where d(·, ·) is the Poincaré distance on U. Then D_{k} = ϕ^{k}(D_{0}) is the Dirichlet fundamental region for G with the center ϕ^{k}(0).

Remark When we map U to the upper half plane H so that 0 corresponds to i and ϕ to z ↦ λz with λ ∈ (0, 1), D_{0} corrsponds to the domain

{z ∈ H | λ^{1/2} < |z| < λ^{−1/2}}.

In the sequel, let α and β be the attracting and the repelling fixed point of ϕ, i.e. corresonding to 0 and ∞, respectively, on \overline{H}.

Lemma 6 There are constants C > 0 and λ ∈ (0, 1) such that the length L(L_{k}) = ∫_{L_{k}} dθ/(2π) of L_{k} = \overline{D_{k}} ∩ ∂U is bounded from above by Cλ^{|k|} for every k ∈ ℤ.

Proof. There is a constant λ ∈ (0, 1) as in the above remark, or equivalently, such that

ϕ^{k}(z) = \frac{(α − λ^{k}β)z − (1 − λ^{k})αβ}{(1 − λ^{k})z + αλ^{k} − β}

for every k ∈ ℤ. Hence

(ϕ^{k})′(z) = \frac{λ^{k}(α − β)^{2}}{((1 − λ^{k})z + αλ^{k} − β)^{2}} .

Here,

(1 − λ^{k})z + αλ^{k} − β = (z − α) (\frac{z − β}{z − α} − λ^{k})

if k > 0 and

λ^{−k}((1 − λ^{k})z + αλ^{k} − β) = −(z − β) (\frac{z − α}{z − β} − λ^{−k})

if k < 0. Since the image of \overline{D_{0}} by (z − α)/(z − β) is disjoint from the non-negative real axis {x ∈ ℝ | x ≥ 0}, |(ϕ^{k})′(z)| is bounded by C_{0}λ^{|k|} with a suitable C_{0} on \overline{D_{0}} for every k ∈ ℤ.

Lemma 7 Fix f ∈ N(α, β). Then there is a constant C′ such that

sup_{z∈\overline{D_{k}}} |f(z)| ≤ C′λ^{|k|}

for every k. Proof. Since there is a constant M such that

|f(z)| ≤ M|z − α||z − β|

on \overline{U} and since

|ϕ^{k}(z) − α| ≤ 2λ^{k} |\frac{z − α}{z − β}| ,

we can find a constant M_{1} such that

|f(ϕ^{k}(z))| ≤ M_{1}λ^{k}

on \overline{D_{0}} for every positive k. The case that k is negative is similar by considering |ϕ^{k}(z) − β|. Lemma 8 Under the same circumstance as above, there are constants C_{0} and c_{0} ∈ (0, 1) such that

∥(C_{ϕ})^{k}f∥_{H^{p}} ≤ C_{0}(|k| + 1)^{1/p}c_{0}^{|k|}

for every k ∈ ℤ. Proof. Set p̃ = min{p, 1}. Then lemmas 6 and 7 imply that

∥(C_{ϕ})^{k}f∥_{H^{p}}^{p}

≤ ∑_{m∈ℤ} ( sup_{z∈\overline{D_{k+m}}} |f(z)|^{p}) · L(L_{m}) ≤ ∑_{m∈ℤ} C(C′)^{p}λ^{|m|+p|k+m|}
≤ C(C′)^{p} (  ∑_{|m+k|≤|k|, |m|≤|k|} (λp̃)^{|m|+|k+m|}

+ ∑_{|m+k|>|k|} (λp̃)^{|m|+|k+m|} + ∑_{|m|>|k|} (λp̃)^{|m|+|k+m|})

≤ C(C′)^{p} ((|k| + 1)(λp̃)^{|k|} + \frac{4}{1 − (λp̃)}(λp̃)^{|k|})

for every k ∈ ℤ. Hence we have the assertion with c_{0} = λ^{min{1,1/p}}.

Remark We can show Theorem 4 and several corollaries such as in [4] directly by using Proposition 1. For instance, the proof of Lemma 8 implies that αC_{ϕ} is chaotic on H^{p} if p ≥ 1 and λ^{1/p} < |α| < λ^{−1/p}.

Finally, recall that the hypercyclicity criterion can be applied to the case of separable completely metrizable topological vector spaces. In particular, Lemma 8 implies the well-konwn fact that C_{ϕ} with a hyperbolic ϕ is hypercyclic on H^{p}. Thus Theorem 4 follows from the following lemma due to K. Matsuzaki [6].

Lemma 9 For every f ∈ N(α, β) and every positive odd integer 2k + 1 with k > 0, set

g_{k} = ∑_{m=−∞}^{∞}(C_{ϕ})^{(2k+1)m}f.

Then there is a constant C′′ such that |g_{k}| ≤ C′′ on \overline{U} for every k. Moreover, g_{k} converges to f in H^{p} as k → ∞.

Proof. First set D_{∗}(k) = ⋃_{j=−k}^{k} \overline{D_{j}}, then by Lemma 7, we have

|g_{k}(z)| ≤ ∑_{m=−∞}^{∞} |f(ϕ^{(2k+1)m}(z))| ≤ ∑_{m=−∞}^{∞} sup_{w∈ϕ^{(2k+1)m}(D_{∗}(k))} |f(w)|

≤ C′ (1 + 2 ∑_{m=1}^{∞}(λ^{k})^{2m−1}) = C′ (1 + \frac{2λ^{k}}{1 − λ^{2k}} )

for every z ∈ \overline{U}. Hence we have the first assertion.

Next, for an arbitrary ϵ > 0, choose a positive integer k_{0} so that D_{∗}(k_{0}) contains the set

I = {e^{iθ} | ϵ ≤ |θ − θ_{1}| ≤ π} ⋂{e^{iθ} | ϵ ≤ |θ − θ_{2}| ≤ π}

where we set α = e^{iθ_{1}} and β = e^{iθ_{2}}. If z ∈ D_{∗}(k_{0}) and if k ≥ k_{0}, then we have

|f(z) − g_{k}(z)| ≤ ∑_{m∈ℤ−\{0\}}|f(ϕ^{(2k+1)m}(z))| ≤ \frac{2C′λ^{k}}{1 − λ^{2k}} .

Hence we conclude that

∥f − g_{k}∥_{H^{p}}^{p}
≤ ∑_{j=1}^{2} ∫_{\{|θ−θ_{j}|<ϵ\}} |f(e^{iθ}) − g_{k}(e^{iθ})|^{p} \frac{dθ}{2π} + ∫_{I} |f(e^{iθ}) − g_{k}(e^{iθ})|^{p} \frac{dθ}{2π}

≤ 2\frac{ϵ}{π} (C′ + C′′)^{p} + ( \frac{2C′λ^{k}}{1 − λ^{2k}} )^{p} ,

which shows the second assertion.

3 Möbius composition operators for parabolic automorphisms

In this section, we prove the following

Theorem 10 Suppose that 0 < p < ∞. Then for every parabolic ϕ in Aut(U), the composition operator C_{ϕ} on B^{p} and on H^{p} is chaotic.

As before, we need to prove Theorem 10 for H^{p} only. Fix p ∈ (0, +∞) and let α be the fixed point of ϕ. Let D_{0} be the Dirichlet fundamental region for the group generated by ϕ with the center 0. Then again, D_{k} = ϕ^{k}(D_{0}) is the Dirichlet fundamental region for G with the center ϕ^{k}(0) for every k ∈ ℤ.

Remark When we map U to the upper half plane H so that 0 corresponds to i and ϕ to z ↦ z + 1, D_{0} corrsponds to the domain

{z ∈ H | |Re z| < 1/2}.

Lemma 11 There is a constant C such that

L(L_{k}) ≤ C(max{1, |k|})^{−2}

for every k ∈ ℤ.

Next, fix f ∈ N(α^{K}) with K ≥ 2. Then there is a constant C′ > 0 such that

sup_{z∈\overline{D_{k}}} |f(z)| ≤ C′(max\{1, |k|\})^{−K}

for every k ∈ ℤ.

Proof. First, since

ϕ^{k}(z) = α + \frac{z − α}{1 + kγ(z − α)}

with a suitable γ ∈ ℂ,

(ϕ^{k})′(z) = \frac{1}{(1 + kγ(z − α))^{2}} ,

and we have the first assertion.

Next, since

|ϕ^{k}(z) − α| = \frac{|z − α|}{|1 + kγ(z − α)|},

we see that there is a constant M such that D_{k} is contained in a disc with center α and radius M(max{1, |k|})^{−1} for every k ∈ ℤ. Also as before, there is a constant M′ such that

|f(z)| ≤ M′|z − α|^{K}

on \overline{U}, and hence

|f(ϕ^{k}(z))| ≤ M′ sup_{z∈\overline{D_{0}}} |ϕ^{k}(z) − α|^{K} ≤ M′M(max\{1, |k|\})^{−K}

on \overline{D_{0}} for every k. Thus we have the second assertion.

Hence as in §2, we have the following estimates.

Lemma 12 Under the same circumstance as above, fix a positive integer K such that Kp ≥ 2, and f ∈ N(α^{K}). Then there is a constant C_{1} such that

∥(C_{ϕ})^{k}f∥_{H^{p}} ≤ C_{1}(max{1, |k|})^{−2/p}

for every k ∈ ℤ.

Proof. As before, we have

∥(C_{ϕ})^{k}f∥_{H^{p}}^{p} ≤ ∑_{m∈ℤ} CC′(max{1, |m|})^{−2}(max{1, |k + m|})^{−Kp}

≤ CC′ (2(max{1, |k|})^{−2} + ∑_{|k+m|<|k|, |m|<|k|} |m|^{−2}|k + m|^{−2}

+ ∑_{|k+m|>|k|} |m|^{−2}|k + m|^{−2} + ∑_{|m|>|k|} |m|^{−2}|k + m|^{−2}) .

Here the second term in the right hand side is 0 if k = 0, ±1, and is

∑_{m=1}^{|k|−1} m^{−2}(|k| − m)^{−2}

≤ ∑_{m=1}^{[|k|/2]} m^{−2}(|k|/2)^{−2} + ∑_{m=[|k|/2]+1}^{|k|−1} (|k|/2)^{−2}(|k| − m)^{−2}

≤ 16|k|^{−2} ∑_{m=1}^{∞} m^{−2}

if k ∉ {−1, 0, 1}. Hence the right hand side is bounded from above by

CC′(max{1, |k|})^{−2} (2 + 16 ∑_{m=1}^{∞} |m|^{−2} + 4 ∑_{m=1}^{∞} |m|^{−2}) ,

which implies the assertion.

Remark For f ∈ N(α^{K}) with Kp ≥ 3, we can show similarly as above that there is a constant C′_{1} such that

∥(C_{ϕ})^{k}f∥_{B^{p}} ≤ C′_{1}(max{1, |k|})^{−3/p}

for every k ∈ ℤ. Hence in the parabolic case, we can show Theorem 10 directly by using Proposition 1 for H^{p} with 1 ≤ p < 2 and B^{p} with 1 ≤ p < 3.

Finally, Lemma 12 and the hypercyclicity criterion imply that C_{ϕ} with a parabolic ϕ is hypercyclic on H^{p}. Hence Theorem 10 follows from the following lemma.

Lemma 13 For every f ∈ N(α^{K}) with Kp ≥ 2 and every positive odd integer 2k + 1 with k > 0, set

g_{k} = ∑_{m=−∞}^{∞}(C_{ϕ})^{(2k+1)m}f.

Then there is a constant C′′ such that |g_{k}| ≤ C′′ on \overline{U} for every k. Moreover, g_{k} converges to f in H^{p} as k → ∞.

Proof. First set D_{∗}(k) = ⋃_{j=−k}^{k} \overline{D_{j}}, then by Lemma 11, we have

|g_{k}(z)| ≤ ∑_{m=−∞}^{∞} |f(ϕ^{(2k+1)m}(z))| ≤ ∑_{m=−∞}^{∞} sup_{w∈ϕ^{(2k+1)m}(D_{∗}(k))} |f(w)|

≤ C′ (1 + 2 ∑_{m=1}^{∞} \frac{1}{k^{2}(2m − 1)^{2}} )

for every z ∈ \overline{U}. Hence we have the first assertion.

Next, for an arbitrary ϵ > 0, choose a positive odd integer 2k_{0} + 1 so that D_{∗}(k_{0}) contains the arc I = {e^{iθ} | ϵ ≤ |θ − θ_{0}| ≤ π}, where we set α = e^{iθ_{0}}. If z ∈ D_{∗}(k_{0}) and if k ≥ k_{0}, then we have

|f(z) − g_{k}(z)| ≤ ∑_{m∈ℤ−\{0\}}|f(ϕ^{(2k+1)m}(z))| ≤ \frac{2C′}{k^{2}} ∑_{m=1}^{∞} \frac{1}{(2m − 1)^{2}} .

Hence we conclude that

∥f − g_{k}∥_{H^{p}}^{p}
= ∫_{\{|θ−θ_{0}|<ϵ\}} |f(e^{iθ}) − g_{k}(e^{iθ})|^{p} \frac{dθ}{2π} + ∫_{I} |f(e^{iθ}) − g_{k}(e^{iθ})|^{p} \frac{dθ}{2π}

≤ \frac{ϵ}{π} (C′ + C′′)^{p} + (\frac{2C′}{k^{2}} ∑_{m=1}^{∞} \frac{1}{(2m − 1)^{2}} )^{p} ,

which shows the second assertion.

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