ON SOME HOLOMORPHIC DYNAMICAL SYSTEMS

Wojciech Chojnacki

0. Introduction

Let D be a domain in ℂ^{N} and φ a holomorphic automorphism of D. Let 𝒞 be the measure class of the Lebesgue measure in D, i.e., the set of all positive regular Borel measures on D whose null sets coincide with the Lebesgue null sets. Let φ_{∗} be the automorphism of 𝒞 given by

(φ_{∗}μ)(B) = μ(φ^{−1}(B)) (μ ∈ 𝒞 , B ∈ ℬ(D)),

where ℬ(D) denotes the Borel σ-algebra of D. Adopting the terminology introduced in [4], we will say that φ_{∗} is finite if it has a fixed point among probability measures.

Let L^{2}H(D) be the Hilbert space of all square Lebesgue integrable holomorphic functions on D. Suppose that L^{2}H(D) ≠ {0}. Let U_{φ} be the unitary operator in L^{2}H(D) defined by

U_{φ}f = (f ∘ φ)J_{φ} (f ∈ L^{2}H(D)),

where

J_{φ} = det (\frac{∂φ_{i}}{∂z_{j}} )_{1≤i,j≤N} (φ = (φ_{1}, …, φ_{N})).

The purpose of this paper is to exhibit various relations between φ, φ_{∗}, and U_{φ}. A fundamental result is that U_{φ} has either pure point spectrum or purely continuous spectrum, the first case occurring exactly when φ_{∗} is finite. We prove that any of the following two conditions ensures the finiteness of φ_{∗}: 1^{o} the existence of a φ-invariant probability measure absolutely continuous with respect to Lebesgue measure; 2^{o} the existence of a relatively compact orbit of φ. Of course, the first condition is also necessary. We show the necessity of a stronger version of the second condition (embeddability of φ in a compact transformation group) provided some mild restrictions on D are imposed. Assuming some hypotheses on D, we prove also that if a point in D is wandering, then U_{φ} has purely absolutely continuous spectrum, and, conversely, if U_{φ} has a non-zero absolutely continuous component in the spectrum, then all points in D are wandering. In particular, the spectrum of U_{φ} is either pure point, or purely absolutely continuous, or purely singular continuous. We show that if D is in a class of domains containing among others all bounded analytic polyhedra, then the spectrum of U_{φ} cannot be purely singular continuous.

1. A purity theorem

The starting point of our discussion is the following

Theorem 1.1. Suppose there exists a φ-invariant probability measure ℙ that is absolutely continuous with respect to Lebesgue measure. Then U_{φ} has pure point spectrum.

Proof. Given k ∈ ℤ and a bounded Borel measure μ on the unit circle 𝕋, let μ̂(k) be the kth Fourier coefficient of μ, i.e.,

μ̂(k) = ∫_{𝕋} t^{−k} dμ(t).

By the spectral theorem for unitary operators, there is a unique projection-valued measure P on 𝕋, taking values in a Boolean algebra of projections in L^{2}H(D), such that for each k ∈ ℤ,

U_{φ}^{k} = ∫_{𝕋} t^{−k} dP (t),  (1.1)

where the integral is to be interpreted in the sense of strong convergence. For any f, g ∈ L^{2}H(D), let π_{f,g} be the complex measure on 𝕋 such that

π_{f,g}(B) = (P (B)f, g) (B ∈ ℬ(𝕋)),

where (·, ·) stands for the scalar product in L^{2}H(D). Given k ∈ ℕ, let

φ^{k} = \underbrace{φ ∘ ⋯ ∘ φ}_{k times}, φ^{−k} = \underbrace{φ^{−1} ∘ ⋯ ∘ φ^{−1}}_{k times},

and let φ^{0} stand for the identity map of D. Since U_{φ}^{k} = U_{φ^{k}} for each k ∈ ℤ, it follows from (1.1) that

(U_{φ}^{k}f, g) = π̂_{f,g}(−k).  (1.2)

Let

L^{2}H(D) = H_{pp} ⊕ H_{c}

be the orthogonal decomposition of L^{2}H(D) in which H_{pp} is the closure of the linear span of the eigenvectors of U_{φ}, and H_{c} consists of those f ∈ L^{2}H(D) for which π̂_{f,f} is a continuous measure. As is known, this decomposition reduces P.

Suppose that the spectrum of U_{φ} is not pure point, that is, there exists a nonzero f in H_{c}. Let g be any element of L^{2}H(D) with the decomposition g = g′ + g′′ with respect to H_{pp} and H_{c}. Clearly, π_{f,g} = π_{f,g′′}. Since

π_{f,g′′} = \frac{1}{4} ∑_{k=0}^{3} i^{k}π_{f+i^{k}g′′,f+i^{k}g′′},  (1.3)

the measure π_{f,g} is continuous. By a theorem of Wiener [17, p. 108],

lim_{n→∞} \frac{1}{2n + 1} ∑_{k=−n} |π̂_{f,g}(k)|^{2} = ∑_{t∈𝕋} |π_{f,g}(\{t\})|^{2} = 0.

Thus in view of (1.2)

lim_{n→∞} \frac{1}{2n + 1} ∑_{k=−n}^{n} |(U_{φ}^{k}f, g)|^{2} = 0.  (1.4)

Let {φ_{n}: n ∈ ℕ} be any complete orthonormal set in L^{2}H(D). The function

K_{D}(z, w) = ∑_{n=1}^{∞} φ_{n}(z)\overline{φ_{n}(w)} (z, w ∈ D),  (1.5)

called the Bergman function of D, does not depend on the particular choice of {φ_{n}: n ∈ ℕ} (cf. [1, p. 21]). For each z ∈ D, let χ_{z} be the element of L^{2}H(D) such that

χ_{z} = K_{D}(w, z) (w ∈ D).  (1.6)

A fundamental property of K_{D} is that, given f ∈ L^{2}H(D) and z ∈ D,

f(z) = (f, χ_{z}).  (1.7)

Let K be a compact subset of D such that ℙ(K) > 0. It is well known that

sup{∥χ_{z}∥: z ∈ K} ≤ π^{−N/2}dist(K, ∂D)^{−1},  (1.8)

where ∥ · ∥ stands for the norm in L^{2}H(D) and the distance dist(K, ∂D) between K and the boundary ∂D of D refers to the supremum norm in ℂ^{N}. From this estimate it follows that the restrictions to K of the functions

z ↦ \frac{1}{2n + 1} ∑_{k=−n}^{n} |(U_{φ}^{k}f, g)|^{2} (n ∈ ℕ)

are uniformly bounded. On account of (1.4), we find that

lim_{n→∞} \frac{1}{2n + 1} ∑_{k=−n}^{n} ∫_{K} |(U_{φ}^{k}f, g)|^{2} dλ(z) = 0,  (1.9)

where λ stands for the Lebesgue measure in D. By (1.7) for each k ∈ ℤ,

∫_{K} |(U_{φ}^{k}f, g)|^{2} dλ(z) = ∫_{K} |(f ∘ φ^{k})J_{φ^{k}}|^{2} dλ = ∫_{φ^{k}(K)} |f|^{2} dλ,  (1.10)

so if we denote by 1_{B} the characteristic function of a subset B of D, then (1.9) can be rewritten in the form

lim_{n→∞} (\frac{1}{2n + 1} ∑_{k=−n}^{n} 1_{φ^{k}(K)}) |f|^{2} dλ = 0.

As f ≠ 0 λ-a.e., the last identity and Fatou’s lemma imply

lim inf_{n→0} \frac{1}{2n + 1} ∑_{k=−n}^{n} 1_{φ^{k}(K)} = 0 λ-a.e.  (1.11)

Let 𝔐 be the σ-algebra {B ∈ ℬ(D): 1_{B} = 1_{φ(B)} ℙ-a.e.} and 𝔼^{𝔐} be the corresponding conditional expectation operator. By Birkhoff’s ergodic theorem [14, p. 25]

lim_{n→0} \frac{1}{2n + 1} ∑_{k=−n}^{n} 1_{φ^{k}(K)} = 𝔼^{𝔐}(1_{K}) ℙ-a.e.

Comparing this equality with (1.11) and taking into account that ℙ is absolutely continuous with respect to λ, we find that

𝔼^{𝔐}(1_{K}) = 0 ℙ-a.e.

Hence

ℙ(K) = 𝔼[𝔼^{𝔐}(1_{K})] = 0,

a contradiction.

The proof is complete.

Theorem 1.2. If U_{φ} has a non-zero eigenvector, then φ_{∗} is finite.

Proof. Let h be an eigenvector of U_{φ} of unit norm. Setting

ℙ(B) = ∫_{B} |h|^{2} dλ (B ∈ ℬ(D))

defines a φ-invariant probability measure on D. Since h ≠ 0 λ-a.e., it follows that ℙ ∈ 𝒞.

The proof is complete.

As a corollary to Theorems 1.1 and 1.2, we obtain the following generalization of a result of [11].

Theorem 1.3. The spectrum of U_{φ} is either pure point or purely continuous according as φ_{∗} is finite or not.

We close this section with two simple examples.

Example 1.4. Let D be an open ball in ℂ^{N} centered at 0 and let φ be defined as

φ(z_{1}, …, z_{N}) = (e^{iθ_{1}}z_{1}, …, e^{iθ_{N}} z_{N}) ((z_{1}, …, z_{N}) ∈ D),

where θ_{1}, …, θ_{N} ∈ ℝ. Then the normalized Lebesgue measure in D is φ-invariant, and so, by Theorem 1.3, U_{φ} has pure point spectrum.

Example 1.5. Let D = {(z_{1}, …, z_{N}) ∈ ℂ^{N}: Im z_{1} > 0, …, Im z_{N} > 0} and let φ be defined as

φ(z_{1}, …, z_{N}) = (z_{1} + t_{1}, …, z_{N} + t_{N}) ((z_{1}, …, z_{N}) ∈ D),

where (t_{1}, …, t_{N}) ∈ ℝ^{N} \ {(0, …, 0)}. Let U be an open ball in D with sufficiently small radius so that φ^{m}(U)∩φ^{n}(U) = ∅ for any distinct integers m and n. Suppose that there exists a φ-invariant probability measure ℙ ∈ 𝒞. If 𝒩 is any finite subset of ℤ with cardinality n, then

n · ℙ(U) = ∑_{m∈𝒩} ℙ(φ^{m}(U)) = ℙ( ⋃_{m∈𝒩} φ^{m}(U)) ≤ 1.

Hence ℙ(U) = 0, which is incompatible with ℙ ∈ 𝒞. Therefore φ_{∗} is not finite, and consequently, by Theorem 1.3, U_{φ} has purely continuous spectrum.

2. Pure point spectrum

For each w ∈ D, let 𝒪(w) denote the orbit {φ^{k}(w): k ∈ ℤ} of w.

The following theorem generalizes some results of [11, 12, 13].

Theorem 2.1. Suppose that there exists w ∈ D such that the closure of 𝒪(w) in D is compact. Then U_{φ} has pure point spectrum.

Proof. Given a multiindex α = (α_{1}, …, α_{N}) ∈ (ℕ ∪ {0})^{N}, let |α| = α_{1} + ⋯ + α_{N}, and, given t ∈ D, let ¯∂^{α}χ_{t} or ∂_{}_{z̄_{1}^{α_{1}}}_{···}_{}_{z̄_{N}^{α_{N}}} denote the function

w ↦ \frac{∂^{|α|}K_{D}}{∂z̄_{1}^{α_{1}} ⋯ ∂z̄_{N}^{α_{N}}} (w, z)|_{z=t} .

It is easily seen that for each α ∈ (ℕ ∪ {0})^{N} and each z ∈ D, ¯∂^{α}χ_{z} is in L^{2}H(D) and

\frac{∂^{|α|}f}{∂z_{1}^{α_{1}} ⋯ ∂z_{N}^{α_{N}}} (z) = (f, ¯∂^{α}χ_{z}) (f ∈ L^{2}H(D)).

Since L^{2}H(D) ≠ {0}, it follows immediately from the last equality that for each z ∈ D there is a multiindex α such that ¯∂^{α}χ_{z} ≠ 0.

Let α be a multiindex such that ¯∂^{α}χ_{w} ≠ 0 and ¯∂^{β}χ_{w} = 0 for every multiindex β with |β| < |α|. Then

U_{φ^{−1}}(¯∂^{α}χ_{w}) = ∑ ∂_{z̄_{i_{1}} ···z̄_{iα_{1}} z̄_{i_{α_{1}+1}} ···z̄_{i_{α_{1}+α_{2}}} ···}χ_{φ(w)}

× \overline{\frac{∂φ_{i_{1}}}{∂z_{1}} (w)} · · · · · \overline{\frac{∂φ_{i_{α_{1}}}}{∂z_{1}} (w)} \overline{\frac{∂φ_{i_{α_{1}+1}}}{∂z_{1}} (w)} · · · · · \overline{\frac{∂φ_{i_{α_{1}+α_{2}}}}{∂z_{1}} (w)} ⋯

· \overline{J_{φ}(w)},

where the sum extends over all multiindices (i_{1}, …, i_{|α|}) with 1 ≤ i_{1} ≤ N, …, 1 ≤ i_{|α|} ≤ N. A similar formula is valid with φ replaced by φ^{k} (k ∈ ℤ\{0}), α replaced by γ such that |γ| = |α|, and w replaced by z ∈ 𝒪(w). In particular, it follows from that formula that the dimension d(z) of the linear span H_{z} of {¯∂^{γ}χ_{z}: |γ| = |α|} takes on a constant value d for z ∈ 𝒪(w). Obviously, d(z) ≤ d for all z in the closure \overline{𝒪(w)} of 𝒪(w) in D. Being compact, \overline{𝒪(w)} contains a minimal φ-invariant subset K (cf. [7, Thm. 2.22]). Clearly, if w′ ∈ K, then \overline{𝒪(w′)} = \overline{𝒪(z)} for each z ∈ \overline{𝒪(w′)} .

Replacing w by any w′ in K if necessary, we may safely assume that \overline{𝒪(z)} = \overline{𝒪(w)} for each z ∈ 𝒪(w). Then d(z) = d for all z ∈ \overline{𝒪(w)} . Now it is clear that z ↦ H_{z} is a continuous mapping from \overline{𝒪(w)} into the Grassmannian of d-dimensional linear subspaces of L^{2}H(D), and that the image of \overline{𝒪(w)} by that mapping is compact. Since, for each k ∈ ℤ, U_{φ^{k}}(¯∂^{α}χ_{w}) is an element of H_{φ^{−k}(w)} of norm ∥¯∂^{α}χ_{w}∥, it follows that the closure of {U_{φ^{k}}(¯∂^{α}χ_{w}): k ∈ ℤ} is compact. Let H be the closure of the linear span of {U_{φ^{k}}(¯∂^{α}χ_{w}): k ∈ ℤ}. H is a reducing subspace for U_{φ} and, by Weyl’s theorem [10, p. 456], the unitary representation k ↦ U_{φ^{k}}|H of Z in H is a direct sum of finite dimensional unitary representations (here U_{φ^{k}}|H stands for the restriction of U_{φ^{k}} to H). Since a unitary operator in a finite dimensional complex Hilbert space has pure point spectrum, it follows that U_{φ} has an eigenvector. By Theorem 1.3, U_{φ} actually has pure point spectrum.

The proof is complete.

3. Property (A)

We shall say that (D, φ) has property (A) if for each w ∈ D, there exist relatively compact open neighbourhoods U and V of w in D such that if U ∩ φ^{n}(U) ≠ ∅ for some n ∈ ℤ, then φ^{n}(U) ⊂ V.

Theorem 3.1. Suppose that D is a domain in ℂ and that U_{φ} has pure point spectrum. Then (D, φ) has property (A).

Proof. Since L^{2}H(D) ≠ {0}, it follows from a result of Wiegerinck [16] that L^{2}H(D) is in fact infinite dimensional. In particular, there exist two linearly independent eigenvectors h_{1}, h_{2} of U_{φ}. Let w be a point in D. Clearly, at least one of the meromorphic functions h_{1}/h_{2} and h_{2}/h_{1}, say h_{1}/h_{2} is holomorphic at w. Let D′ denote D with the poles of h_{1}/h_{2} deleted. For any δ > 0, let

W_{δ} = {z ∈ D′: |h_{1}(z)/h_{2}(z) − h_{1}(w)/h_{2}(w)| < δ},

and let V_{δ} be the (open) component of W_{δ} containing w. As is well known [6, p. 10], there exist k ∈ ℕ, an open neighbourhood Ω′ ⊂ D′ of w, an open neighbourhood Δ of 0, and one-to-one holomorphic functions g_{1} : Ω′ → Δ and g_{2} : Δ → Δ such that

h_{1}(z)/h_{2}(z) − h_{1}(w)/h_{2}(w) = g_{2}([g_{1}(z)]^{k})

for each z ∈ Ω′. It follows from this representation that there exists δ > 0 such that the closure of V_{3δ} in D is compact. The proof will be complete once we show that if V_{δ} ∩ φ^{n}(V_{δ}) ≠ ∅ for some n ∈ ℤ, then φ^{n}(V_{δ}) ⊂ V_{3δ}.

Suppose that φ^{n}(z_{1}) = z_{2} for z_{1}, z_{2} ∈ V_{δ} and n ∈ ℤ. Then, for each z ∈ V_{δ},

|h_{1}(φ^{n}(z))/h_{2}(φ^{n}(z)) − h_{1}(w)/h_{2}(w)|

≤ |h_{1}(φ^{n}(z))/h_{2}(φ^{n}(z)) − h_{1}(φ^{n}(z_{1}))/h_{2}(φ^{n}(z_{1}))|
+ |h_{1}(z_{2})/h_{2}(z_{2}) − h_{1}(w)/h_{2}(w)|

= |h_{1}(z)/h_{2}(z) − h_{1}(z_{1})/h_{2}(z_{1})| + |h_{1}(z_{2})/h_{2}(z_{2}) − h_{1}(w)/h_{2}(w)| < 3δ.

Thus φ^{n}(V_{δ}) ⊂ W_{3δ}. Since φ^{n}(V_{δ}) is connected and intersects V_{δ} ⊂ V_{3δ}, it follows that φ^{n}(V_{δ}) ⊂ V_{3δ}.

The proof is complete.

We shall say that a domain Ω in ℂ^{N} has property 𝒜 if, for each w ∈ Ω, χ_{w} ≠ 0 and the set comprising χ_{w} and all the ∂z̄_{i}χ_{w} (1 ≤ i ≤ N) is linearly independent.

Theorem 3.2. Suppose that D has property 𝒜 and that U_{φ} has pure point spectrum. Then (D, φ) has property (A).

Proof. Let w be a point in D. As χ_{w} ≠ 0, formulae (1.5) and (1.6) applied to a complete orthonormal set in L^{2}H(D) consisting of eigenvectors of U_{φ} show that h(w) ≠ 0 for some eigenvector h of U_{φ}. Let D′ denote D with the zeros of h deleted. For any δ > 0, let

W_{δ} = \{z ∈ D′: ∥(\overline{h(z)})^{−1}χ_{z} − (\overline{h(w)})^{−1}χ_{w}∥ < δ\},

and let V_{δ} be the component of W_{δ} containing w. Since the set

{χ_{w}} ∪ {∂z̄_{i}χ_{w}: 1 ≤ i ≤ N}

is linearly independent, it follows that an open neighbourhood of w can be diffeomorphically embedded in L^{2}H(D) by the map z ↦ (\overline{h(z)})^{−1}χ_{z}. Accordingly, we can find δ > 0 such that the closure of V_{3δ} in D is compact. The proof will be complete once we show that if V_{δ} ∩ φ^{n}(V_{δ}) ≠ ∅ for some n ∈ ℤ, then φ^{n}(V_{δ}) ⊂ V_{3δ}.

Suppose that φ^{n}(z_{1}) = z_{2} for z_{1}, z_{2} ∈ V_{δ} and n ∈ ℤ. Then, for each z ∈ V_{δ},

∥(\overline{h(φ^{n}(z))})^{−1}χ_{φ^{n}(z)} − (\overline{h(w)})^{−1}χ_{w}∥ ≤ ∥U_{φ^{n}}[(\overline{h(z)})^{−1}χ_{z} − (\overline{h(z_{1})})^{−1}χ_{z_{1}}]∥

+ ∥(\overline{h(z_{2})})^{−1}χ_{z_{2}} − (\overline{h(w)})^{−1}χ_{w}∥ < 3δ.

Thus φ^{n}(V_{δ}) ⊂ W_{3δ}. Since φ^{n}(V_{δ}) is connected and intersects V_{δ} ⊂ V_{3δ}, it follows that φ^{n}(V_{δ}) ⊂ V_{3δ}.

The proof is complete.

We shall say that a domain Ω in ℂ^{N} has property ℬ if, for each w ∈ Ω, χ_{w} ≠ 0 (or, equivalently, K_{Ω}(w, w) > 0) and the Bergman metric tensor

g_{Ω}(w) = ∑_{i,j=1}^{N} \frac{∂^{2}}{∂z_{i}∂z̄_{j}} log K_{Ω}(w, w) dz_{i}dz̄_{j}

is positive-definite. As is known (cf. [8, p. 296], every domain in ℂ^{N} which is biholomorphically equivalent to a bounded domain has property ℬ.

Theorem 3.3. Suppose that D has property ℬ. Then (D, φ) has property (A).

Proof. Let d_{D} be the geodesic distance relative to g_{D}. Let w be a point in D. For each δ > 0, let B_{D}(w, δ) denote the open ball relative to d_{D} centered at w with radius δ. It is an elementary result from differential geometry that there is δ > 0 such that the closure of B_{D}(w, 3δ) in D is compact. To end the proof, it suffices to show that if

φ^{n}(B_{D}(w, δ)) ∩ B_{D}(w, δ) ≠ ∅

for some n ∈ ℤ, then

φ^{n}(B_{D}(w, δ)) ⊂ B_{D}(w, 3δ).

Suppose that φ^{n}(z_{1}) = z_{2} for z_{1}, z_{2} ∈ B_{D}(w, δ) and n ∈ ℤ. As is well known (cf. [8, p. 299], φ is an isometry with respect to d_{D}. Therefore, for each z ∈ B_{D}(w, δ),

d_{D}(φ^{n}(z), w) ≤ d_{D}(φ^{n}(z), φ^{n}(z_{1})) + d_{D}(z_{2}, w)
= d_{D}(z, z_{1}) + d_{D}(z_{2}, w) < 3δ,

which yields the desired conclusion.

After this paper had been completed, the author proved that the properties 𝒜 and ℬ are equivalent [3]. Accordingly, Theorem 3.2 is subsumed in Theorem 3.3. Also Theorem 3.1 is a consequence of Theorem 3.3, for if D is a domain in ℂ such that L^{2}H(D) ≠ {0}, then D has property ℬ (cf. [3, Thm. 2]). Alternatively, Theorem 3.1 can be deduced from Theorem 3.2 by applying the result of [2].

4. Pure point spectrum (continued)

Let ℋ (D) be the space of all holomorphic mappings of D into itself equipped with the compact open topology. With composition as semigroup operation, ℋ (D) is a topological semigroup. The identity mapping is the identity of ℋ (D).

Theorem 4.1. Suppose that (D, φ) has property (A) and that U_{φ} has pure point spectrum. Then the closure of {φ^{k}: k ∈ ℤ} in ℋ (D) is a compact topological group.

Proof. We first show that if the closure G of {φ^{k}: k ∈ ℤ} in ℋ (D) is compact, then it is a topological group. It is clear that G is a topological semigroup. Let ψ be an element of G. Since the topology of ℋ (D) is metrisable, there exists a sequence (k_{n})_{n∈ℕ} in ℤ such that ψ = lim_{n→∞} φ^{k_{n}}. By the compactness of G, there is a subsequence (k_{n_{m}})_{m∈ℕ} of (k_{n})_{n∈ℕ} such that the sequence (φ^{−k_{nm}})_{m∈ℕ} converges to an element of G. Of course, lim_{m∈∞} φ^{−k_{nm}} = ψ^{−1}. Thus G is a group. Since G is a compact group which is a topological semigroup, it follows from a theorem of Ellis [5] that G is a topological group.

Suppose now that G is not compact. Then, by Montel’s theorem, there exists z ∈ D such that for each open neighbourhood Ω of z, the closure of ⋃_{k∈ℤ} φ^{k}(Ω) in D is not compact. Let U and V be relatively compact neighbourhoods of z in D such that if U ∩ φ^{n}(U) ≠ ∅ for some n ∈ ℤ, then φ^{n}(U) ⊂ V. As easy exhaustion argument shows the existence of a maximal subset ℳ of ℤ such that φ^{m}(U) ∩ φ^{n}(U) = ∅ for distinct m and n in ℳ. We claim that ℳ is infinite.

In fact, given k ∈ ℤ, choose m ∈ ℳ so that φ^{m}(U) ∩ φ^{k}(U) ≠ ∅. Then U ∩ φ^{k−m}(U) ≠ ∅ whence φ^{k−m}(U) ⊂ V and further φ^{k}(U) ⊂ φ^{m}(V). Consequently,

⋃_{k∈ℤ} φ^{k}(U) ⊂ ⋃_{m∈ℳ} φ^{m}(V)

and as the closure of ⋃_{k∈ℤ} φ^{k}(U) in D is not compact, ℳ must be infinite, as claimed.

Applying now an argument from Example 1.5, we see that φ_{∗} is not finite. Hence, in view of Theorem 1.3, the spectrum of U_{φ} cannot be pure point. This contradiction completes the proof.

As a corollary to Theorem 3.1, 3.2 and 4.1, we obtain

Theorem 4.2. Let D be either a domain in ℂ or a domain in ℂ^{N} (N ≥ 2) having property 𝒜. Then the closure of {φ^{k}: k ∈ ℤ} in ℋ (D) is a compact topological group.

Theorem 4.2 generalizes some results of [11, 12]. The proofs to the theorems of which Theorem 4.2 is a consequence rely on a modification of an argument due to T. Mazur [12]. Mzaur’s original argument used differential geometry and involved an assumption on D stronger than property ℬ, namely property 𝒞 which will be introduced below.

Theorem 4.3. Suppose that D has property 𝒜. If the orbit of some point in D has compact closure in D, then the orbits of all points in D have compact closure in D.

The above theorem fails if all the assumptions about D and φ are dropped. This is shown by the following

Example 4.4. Let D = ℂ^{2} and let φ be defined as

φ(z_{1}, z_{2}) = (z_{1}, z_{1} + z_{2}) (z_{1}, z_{2} ∈ ℂ).

Then all elements of {0} × ℂ are fixed points for φ, and for each w ∈ ℂ \ ({0} × ℂ), the closure 𝒪(w) is not compact.

5. Absolutely continuous spectrum

We shall say that a point w in D is wandering if there exists an open neighbourhood U of w such that U ∩ φ^{n}(U) = ∅ for each n ∈ ℕ. Of course, we may equally well assume in this definition that U ∩ φ^{n}(U) = ∅ holds for all n ∈ ℤ \ {0}.

Suppose that w ∈ D is not wandering, and let U be an open neighbourhood of w. Then, U ∩ φ^{n}(U) ≠ ∅ for infinitely many n ∈ ℕ. In fact, if there is n_{0} ∈ ℕ such that U ∩φ^{n}(U) ≠ ∅ for n ≥ n_{0}, then w ≠ φ^{n}(w) for n ≥ n_{0} and hence for all n ∈ ℕ. Now, if V is an open neighbourhood of w contained in U such that V ∩ φ^{n}(V) = ∅ for 1 ≤ n ≤ n_{0}, then, clearly, V ∩ φ^{n}(V) = ∅ for all n ∈ ℕ, a contradiction.

Let

H_{c} = H_{ac} ⊕ H_{sc}

be the orthogonal decomposition of H_{c} in which H_{ac} consists of those f ∈ H_{c} for which the measure π_{f,f} is absolutely continuous with respect to the Lebesgue measure in 𝕋, and H_{sc} consists of those f ∈ H_{c} for which the measure π_{f,f} is singular with respect to the Lebesgue measure in 𝕋. As is known, this decomposition reduces the projection-valued measure P.

Theorem 5.1. Suppose that a point w in D is wandering. Then χ_{w} ∈ H_{ac}.

Proof. Let U be an open neighbourhood of w such that U ∩ φ^{n}(U) = ∅ holds for all n ∈ ℤ \ {0}. Then, clearly, φ^{m}(U) ∩ φ^{n}(U) = ∅ for distinct m and n in ℤ.

By (1.10) and Levi’s monotone convergence theorem,

∫_{U} ∑_{k∈Z} |(U_{φ^{k}}χ_{w}, χ_{z})|^{2} dλ(z) = ∑_{k∈Z} ∫_{φ^{k}(U)} |χ_{w}(z)|^{2} dλ(z) ≤ ∥χ_{w}∥^{2}.

This jointly with (1.2) and Plancherel’s theorem shows that for λ-almost all z in U, the measure π_{χ_{w},χ_{z}} is absolutely continuous with respect to the Lebesgue measure in 𝕋. Since the mapping D ∋ z ↦ χ_{z} ∈ L^{2}H(D) is continuous, it follows that for all z in U, the measure π_{χ_{w},χ_{z}} is absolutely continuous with respect to the Lebesgue measure in 𝕋. In particular, this is the case of π_{χ_{w},χ_{w}}.

The proof is complete.

Theorem 5.2. If there is a wandering point in D, then H_{ac} ≠ {0}.

Proof. The set W of wandering points in D is clearly open. If W ≠ ∅, then, in view of (1.6), there exists w ∈ W such that χ_{w} ≠ 0. Now the theorem follows upon applying Theorem 5.1.

Theorem 5.3. Suppose that (D, φ) has property (A) and H_{ac} ≠ {0}. Then every point in D is wandering.

Proof. Suppose that f ∈ H_{ac} \ {0}. Let w be a point in D. Let U and V be open neighbourhoods of w with compact closure in D such that if U ∩ φ^{n}(U) ≠ ∅ for some n ∈ ℤ, then φ^{n}(U) ⊂ V. Given z ∈ V, let χ_{z} = g′_{z} + g′′_{z} be the decomposition of χ_{z} with respect to H_{pp} ⊕ H_{sc} and H_{ac}. Clearly, π_{f,χ_{z}} = π_{f,g′′_{z}}. Hence, in view of (1.3), the measure π_{f,χ_{z}} is absolutely continuous with respect to the Lebesgue measure in 𝕋. By (1.2) and the Riemann–Lebesgue lemma,

lim_{k→±∞}(U_{φ^{k}}f, χ_{z}) = 0.  (5.1)

In view of (1.8), the restriction to V of the functions z ↦ (U_{φ^{k}}f, χ_{z}) (k ∈ ℤ) are uniformly bounded. Applying Lebesgue’s dominated convergence theorem and taking into account (1.10) and (5.1), we get

lim_{k→±∞} ∫_{φ^{k}(V)} |f|^{2} dλ = lim_{k→±∞} ∫_{V} |(U_{φ^{k}}f, χ_{z})|^{2} dλ.  (5.2)

Suppose that w is not wandering. Then there exists a sequence (k_{n})_{n∈ℕ} of positive integers such that lim_{n→∞} k_{n} = ∞ and U ∩ φ^{k_{n}}(U) ≠ ∅ for each n ∈ ℕ. It is clear that given n ∈ ℕ, we have φ^{k_{n}}(U) ⊂ V. Hence U ⊂ φ^{−k_{n}}(V) and further

∫_{U} |f|^{2}dλ ≤ ∫_{φ^{−kn} (V)} |f|^{2}dλ.

The last inequality together with (5.1) yields

∫_{U} |f|^{2}dλ = 0.

Since f ≠ 0 λ-a.e. and λ(U) > 0, we get a contradiction.

The proof is complete.

Theorem 5.4. Suppose that (D, φ) has property (A) and that U_{φ} has a non-zero absolutely continuous component in the spectrum. Then the spectrum of U_{φ} is purely absolutely continuous.

Proof. Suppose that H_{ac} ≠ {0}. Then, by Theorem 5.3, every point in D is wandering, and, by Theorem 5.1, χ_{w} ∈ H_{ac} for each w ∈ D. In view of (1.7), the Hahn–Banach theorem, and the Riesz theorem, the linear span of {χ_{w}: w ∈ D} is dense in L^{2}H(D). Therefore H_{ac} coincides with L^{2}H(D).

The proof is complete.

As a consequence of Theorems 5.2, 5.3, and 5.4, we obtain

Theorem 5.5. Suppose that (D, φ) has property (A). If some point in D is wandering, then all points in D are so.

Notice that the above theorem fails if all the assumptions about D and φ are dropped. In fact, in D and φ are as in Example 4.4, then the set of wandering points in D coincides with ℂ^{2} \ ({0} × ℂ).

We shall say that a domain Ω has property 𝒞 if it has property ℬ and is complete with respect to the geodesic distance d_{Ω}. As is known (cf. [9]), every domain in ℂ^{N} which is biholomorphically equivalent to a bounded analytic polyhedron has property 𝒞.

Theorem 5.6. Suppose that D has property 𝒞. Then the spectrum of U_{φ} is either pure point or purely absolutely continuous.

Proof. Let ℌ(D) be the group of all holomorphic automorphisms of D, the group operation being composition of mappings. Since D has property 𝒞 and in particular has property ℬ, it follows that under the compact open topology, ℌ(D) is a Lie group (cf. [8, 300]). Let G be the closure of {φ^{n}: n ∈ ℤ} in ℌ(D). Clearly, G is a locally compact monothetic group. Therefore, either G is compact or G = {φ^{n}: n ∈ ℤ} and Z ∋ φ ↦ φ^{n} ∈ G is topological isomorphism of ℤ and G [15, p. 39]. In the first case, by virtue of Theorem 2.1, U_{φ} has pure point spectrum. In the other case, it turns out that U_{φ} has purely absolutely continuous.

Indeed, suppose that {φ^{n}: n ∈ ℤ} is topologically isomorphic to ℤ. If the spectrum of U_{φ} is not purely absolutely continuous, then, by Theorems 5.2 and 5.4, no point in D is wandering. Fix w ∈ D. Since D is complete with respect to d_{D}, it follows from the Hopf–Rinov (cf. [8, p. 56]) theorem that for each δ > 0, the closure of B_{D}(w, δ) in D is compact. Choose any δ_{0} > 0. Since w is not wandering, there is an unbounded sequence (k_{n})_{n∈ℕ} of positive numbers such that

B_{D}(w, δ_{0}) ∩ φ^{k_{n}}(B_{D}(w, δ_{0})) ≠ ∅

for each n ∈ ℕ. Consequently, for each δ ≥ δ_{0} and each n ∈ ℕ,

B_{D}(w, δ) ∩ φ^{k_{n}}(B_{D}(w, δ)) ≠ ∅

whence, as in the proof of Theorem 3.3,

φ^{k_{n}}(B_{D}(w, δ)) ⊂ B_{D}(w, 3δ).

By Montel’s theorem and the fact that each compact subset of D is contained in some B_{D}(w, δ) (δ ≥ δ_{0}), we conclude that the closure of {φ^{k_{n}}: n ∈ ℕ} in ℌ(D) is compact. Since {φ^{n}: n ∈ ℤ} is topologically isomorphic to ℤ, this cannot be the case unless (k_{n})_{n∈ℕ} is bounded. This contradiction completes the proof.

We conclude with a simple application of the theorem established. Let D and φ be as in Example 1.5. Then, clearly, D is biholomorphically equivalent to a polydisc which, being a bounded analytic polyhedron, has property 𝒞. Since U_{φ} has purely continuous spectrum, it follows from Theorem 5.5 that the spectrum of U_{φ} is in fact purely absolutely continuous.

Acknowledgement

The author would like to thank Professor T. Mazur for a stimulating discussion during the IX Conference on Analytic Functions, 1–8 June 1986, Lublin, Poland. The present note was written while the author was staying at McMaster University, Hamilton, Ontario, Canada. The author would like to thank Professor H. Heinig for making the stay possible and for warm hospitality.

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