CHAOS FOR FUNCTIONS OF DISCRETE AND

CONTINUOUS WEIGHTED SHIFT OPERATORS

Ralph deLaubenfels and Hassan Emamirad

Scientia Research Institute and Université de Poitiers

ABSTRACT. For L equal to the unilateral or bilateral shift on a weighted sequence space, we characterize, in terms of f and the weight function, those f holomorphic on the spectrum of L for which f(L) is a chaotic operator. For B equal to \frac{d}{dx}, the generator of left translation, on weighted L^{p} spaces on [0, ∞) or R, we similarly characterize those polynomials Q for which the differential operator Q(B) generates a chaotic semigroup.

1. Introduction and preliminaries Throughout this paper, all operators are linear, on a Banach space X. The operator T ∈ B(X), the space of bounded operators from X to itself, is said to be hypercyclic if there exists x ∈ X whose orbit {T^{n}x | n − 1 ∈ N} is dense in X. Note that this implies the existence of a dense set of vectors with dense orbits. A periodic point for T is a vector x such that T^{n}x = x for some n ∈ N. Finally, T is said to be chaotic if T is hypercyclic and there exists a dense set of periodic points for T.

Note that, since T is linear, being hypercyclic is equivalent to the more familiar condition of being topologically transitive: for any pair of nonempty open sets U and V, there exists a natural number n so that T^{n}(U) ∩ V is nonempty ([5, Theorem 1.2]; for a continuous analogue, see [3, Theorem 2.2]). The definition of chaotic given here also implies sensitive dependence on initial conditions; in [5, Proposition 6.1], it is shown that then {(T^{n}x − T^{n}y) | n − 1 ∈ N} is dense in X, for all x, y in a dense set; see also [1]. Thus the definition of chaos given here is equivalent, for T linear, to that given in [4, Definition 8.5, page 50].

Similarly, the strongly continuous semigroup of operators {T (t)}_{t≥0} is hypercyclic if there exists x ∈ X such that the orbit {T (t)x | t ≥ 0} is dense in X, and is chaotic if, in addition, there is a dense set of periodic points: x such that T (t)x = x for some t > 0.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 47B37, 47D06, Secondary 58F13.

Let X := ℓ^{p}(N, β⃗) (1 ≤ p < ∞), the weighted sequence space with the weight vector β⃗, equipped with the norm:

∥x⃗∥^{p} := ∑_{k=1}^{∞} |x(k)|^{p}β(k).

We choose the weight vector β⃗ in such a way that the left shift L on X defined by:

(Lx⃗) (k) := x(k + 1) (k ∈ N).

becomes a bounded operator on X. For f holomorphic on a neighborhood of the spectrum of L, let f(L) be defined by the usual Riesz functional calculus. Let D be the open unit disc in the complex plane, R the radius of convergence of the power series

z ↦ ∑_{k=1}^{∞} z^{k}β(k).

Then we show (Theorem 2.3) that, for f nonconstant, when ∑_{k=1}^{∞} R^{k}β(k) = ∞ and R > 0, then f(L) is a chaotic operator if and only if f(R^{\frac{1}{p}} D) intersects the unit circle ∂D if and only if f(L) has a nontrivial periodic point. To characterize g(L) generating a chaotic semigroup (Theorem 2.13), replace ∂D with iR; this is immediately seen by choosing f(z) := e^{tg(z)}. For f or g an affine function, even simpler characterizations are given. When g is a polynomial or f is a polynomial other than a complex multiple of z ↦ z^{n}, then the first equivalence holds without the hypothesis of divergence on the boundary of the disc of convergence (Lemma 2.7 and Theorems 2.8 and 2.14).

Section 3 considers the left shift on the bilateral weighted sequence space ℓ^{p}(Z, β⃗); the results are the same as in Section 2, with discs replaced by annuli (since a Laurent series replaces a power series).

In Sections 4 and 5 we characterize differential operators Q(d/dx), Q a polynomial, that generate chaotic semigroups on weighted L^{p} (1 ≤ p < ∞) spaces. On L^{p}([0, ∞), ρ), when we have divergence on the boundary of the half plane of convergence of the Laplace transform of the weight function, a strongly continuous semigroup generated by Q(B) is chaotic if and only if the image, under Q, of an appropriate half plane intersects the imaginary axis if and only if the semigroup has a nontrivial periodic point (Theorem 4.6). On L^{p}(R, ρ), the result is the same, with the half plane replaced by a vertical strip (Theorem 5.1). For first-order differential operators we have simpler characterizations (Corollaries 4.7 and 5.2).

In the sequel we show that many results from [3], [5], [12], [13] and [14] are immediate special cases of the results in this paper.

We will use the following very useful result.

Lemma 1.1 ([5, Corollary 1.5]). Suppose A ∈ B(X), Y_{1}, Y_{2} are dense subsets of X and Z : Y_{1} → Y_{1} such that

(1) AZy = y, ∀y ∈ Y_{1};

(2) lim_{n→∞} Z^{n}y = 0, ∀y ∈ Y_{1}; and

(3) lim_{n→∞} A^{n}w = 0, ∀w ∈ Y_{2}. Then A is hypercyclic.

In many of the classes of operators considered in this paper, an operator T (not a multiple of the identity) is chaotic if and only if it has a nontrivial periodic point. A similar result is obtained in [10] for the unilateral backward shift on Köthe echelon spaces and generalized by Grosse-Erdmann [8] for an arbitary Fréchet-sequence space in which canonical unit vectors e_{n} (e_{n}(z) = z^{n}) form an unconditional basis. Given all that chaos means, this is quite surprising. This result also has immense practical value, because finding a (single nontrivial) periodic point seems so much easier than demonstrating all the properties of chaos directly.

What is also surprising in our results is the abundance of chaos, in such a standard, linear setting. For example, we show (Corollary 5.6) that, if there exists real μ_{1} < μ_{2} so that ∫_{−∞}^{∞} e^{μ_{j}s}ρ(s) ds (j = 1, 2) is finite, then d^{2}/dx^{2}, on L^{p}(R, ρ), generates a strongly continuous chaotic semigroup. Thus we have numerous chaotic diffusion semigroups, under the hypothesis that the point spectrum of d/dx have nonempty interior.

We also show (Corollary 2.16) that, if L has an eigenvalue (clearly necessary for the existence of chaos), then for any nonconstant f holomorphic on σ(L), there exists a positive constant c such that cf(L) is chaotic. Similarly, if d/dx on L^{p}([0, ∞), ρ) has an eigenvalue, then for any strongly continuous semigroup {T (t)}_{t≥0} generated by a differential operator Q(d/dx) (Q a polynomial) there exists a real constant c such that {e^{ct}T (t)}_{t≥0} is a chaotic semigroup (Corollary 4.10); see also Remark 3.3(iv) and Corollary 5.5 for the bilateral case.

Thus, in considering the algebra of operators

{f(L) | f is holomorphic in a neighborhood of the spectrum of L},

it is natural to wonder how typical the behaviour we have seen is. Specifically, for what algebras of operators A is the following true?

CONJECTURE. Suppose A is a commutative algebra of operators that contains a chaotic operator, T ∈ A and T is not a multiple of the identity. Then there exists a constant c > 0 such that cT is chaotic.

Given the results of Theorems 2.3, 3.2, 4.4 and 5.1, it is also natural to ask the following.

OPEN QUESTION. For what algebras of operators A is it the case that T ∈ A is chaotic if and only if T has a nontrivial periodic point?

Throughout this paper, σ(B) is the spectrum of the operator B and P σ(B) is the point spectrum of B.

2. DISCRETE WEIGHTED ONE-SIDE SHIFTS. Throughout this section, we will denote by ℓ^{p}(N, β⃗) (1 ≤ p < ∞) the Banach space X of complex-valued sequences x⃗ := {x(k)}_{k=1}^{∞} such that

∥x⃗∥^{p} := ∑_{k=1}^{∞} |x(k)|^{p}β(k)

is finite. Here the sequence β⃗ satisfies

sup_{k≥1} \frac{β(k)}{β(k + 1)} < ∞.  (2.0)

We will denote by ℓ^{p}(N) ℓ^{p}(N, β⃗) with β(k) := 1.

Denote by L the one-side left shift on ℓ^{p}(N, β⃗)

(Lx⃗) (k) := x(k + 1) (k ∈ N).

Note that (2.0) is equivalent to L being bounded; the supremum in (2.0) is ∥L∥^{p}.

Let D denote the open unit disc in the complex plane, ∂D the unit circle, f be holomorphic in rD, for some r > ∥L∥, f(L) defined by the usual Riesz functional calculus

f(L) := \frac{1}{2πi} ∫_{|z|=r} f(z) (z − L)^{−1} dz.

We also denote by

h⃗_{μ} := (μ, μ^{2}, μ^{3}, ...)

the eigenvector for L corresponding to the eigenvalue μ.

Let

R := \underline{lim}_{k→∞} (β(k))^{− \frac{1}{k}}

the radius of convergence of z ↦ ∑_{k=1}^{∞} β(k)z^{k}.

Note that h⃗_{μ} ∈ ℓ^{p}(N, β⃗) if and only if ∑_{k=1}^{∞} β(k)|μ|^{k} < ∞. Thus P σ(L) equals either R^{\frac{1}{p}} D (if ∑_{k=1}^{∞} β(k)R^{k} = ∞) or R^{\frac{1}{p}} \overline{D} (if ∑_{k=1}^{∞} β(k)R^{k} < ∞).

Lemma 2.1. Suppose that the open set Ω ⊆ R^{\frac{1}{p}} D has an accumulation point. Then

A_{μ} := span {h⃗_{μ} | μ ∈ Ω}

is dense in ℓ^{p}(N, β⃗).

Proof. Suppose ϕ⃗ ∈ A_{μ}^{⊥}. Then

g(μ) :=< ϕ⃗, h⃗_{μ} >:= ∑_{k=1}^{∞} ϕ_{k}μ^{k} (μ ∈ R^{\frac{1}{p}} D)

is an analytic function in R^{\frac{1}{p}} D. Since g(μ) = 0 for all μ ∈ Ω, a set with an accumulation point, g := 0, so that ϕ⃗ = 0, as desired. □

Lemma 2.2. Suppose f is holomorphic and nonconstant in a neighborhood of ∥L∥\overline{D} and R > 0.

(1) If f(R^{\frac{1}{p}} D) ∩ ∂D is nonempty, then f(L) is chaotic.

(2) If f(L) is chaotic, then f(P σ(L)) ∩ ∂D is infinite.

(3) If f(L) generates a chaotic semigroup, then f(P σ(L)) ∩ iR is infinite.

Proof. (1). Let

Ω_{1} := \{μ ∈ R^{\frac{1}{p}} D | |f(μ)| > 1\}, Ω_{2} := \{μ ∈ R^{\frac{1}{p}} D | |f(μ)| < 1\},
Ω_{3} := \{μ ∈ R^{\frac{1}{p}} D | f(μ) ∈ exp(2πiQ)\}.

Suppose z_{0} ∈ f(R^{\frac{1}{p}} D)∩∂D. Since f is holomorphic, f(R^{\frac{1}{p}} D) is an open set. It follows that an open ball containing z_{0} is contained in f(R^{\frac{1}{p}} D). Thus Ω_{1} and Ω_{2} are open, nonempty subsets of R^{\frac{1}{p}} D, and \{μ ∈ R^{\frac{1}{p}} D | |f(μ)| = 1\} includes f^{−1} of an interval on the unit circle. This implies that, for j = 1, 2, 3, Ω_{j} has an accumulation point.

For j = 1, 2, 3, let

Y_{j} := span {h⃗_{μ} | μ ∈ Ω_{j}}.

By Lemma 2.1, each of Y_{1}, Y_{2} and Y_{3} is dense in X.

Let A := f(L), and define Z := (f(L))^{−1} on Y_{1}; that is,

Z (∑_{k=1}^{N} α_{k}h⃗_{μ_{k}}) := ∑_{k=1}^{N} α_{k}(f(μ_{k}))^{−1}h⃗_{μ_{k}},

for μ_{k} ∈ Ω_{1}, α_{k} ∈ C.

It is clear that we have the hypotheses of Lemma 1.1, so that the hypercyclicity of A follows.

To see that Y_{3} is contained in the set of all periodic points of f(L), suppose, for α_{k} ∈ C, n_{k}, m_{k} ∈ Z, f(μ_{k}) = exp(2πi\frac{n_{k}}{m_{k}}),

y := ∑_{k=1}^{N} α_{k}h⃗_{μ_{k}};

then, if M := ∏_{k=1}^{N} m_{k},

[f(L)]^{M} y = y,

as desired.

(2). Let P be the set of all periods for f(L); that is, n ∈ P if and only if there exists nontrivial x ∈ X such that n is the first positive integer such that [f(L)]^{n} x = x.

We claim that P must be infinite. To see this, suppose for the sake of contradiction that P = {n_{j}}_{j=1}^{N}. Let M ≡ ∏_{j=1}^{N} n_{j}; then [f(L)]^{M} x = x for all periodic points x; since the set of all periodic points is dense, this holds for all x ∈ X. This contradicts the hypercyclicity of f(L), proving the claim.

Fix n ∈ P and let x be as in the definition of P. Then

[(z ↦ (f(z))^{n}) (L)] x = x,

thus 1 ∈ P σ ([(z ↦ (f(z))^{n}) (L)]). By point-spectral mapping theorems ([9, Theorem 5.12.2]), there exists μ ∈ P σ(L) such that [f(μ)]^{n} = 1, so that

f(μ) = exp(\frac{2πik}{n})

for some integer k, relatively prime with respect to n, between 0 and n.

Since exp(\frac{2πik}{n}) ∈ f(P σ(L)) ∩ ∂D, for any n ∈ P, and P is infinite, this proves (2).

(3). Let P be the set of all periodic points of {e^{tf(L)}}_{t≥0}; that is, t ∈ P if and only if there exists nontrivial x such that t is the first positive number such that e^{tf(L)}x = x.

As in the proof of (2), if P were finite, then there would exist T > 0 such that

e^{T f(L)}x = x

for all x ∈ X. This would imply that all orbits of {e^{tf(L)}}_{t≥0} are compact, hence the hypercyclicity of {e^{tf(L)}}_{t≥0} would be violated. Thus P is infinite.

As in the proof of (2), \{\frac{2πi}{t} | t ∈ P\} is contained in f(P σ(L)) ∩ iR. This concludes the proof. □

A surprising result is that the existence of a single nontrivial periodic point can generate in many cases the chaoticity of the operator. This phenomenon is observed in a particular case by Grosse-Erdmann [7,8] and Martinez-Gimenez and Peris [10]. For some results we will impose on the weight vector β⃗, the following hypothesis:

(H1)  R > 0,  and  ∑_{k=1}^{∞} β(k)R^{k} diverges.

Theorem 2.3. Suppose f is holomorphic and nonconstant in a neighborhood of ∥L∥\overline{D}. Under assumption (H1) the following are equivalent.

(a) f(L) is chaotic.

(b) f(R^{\frac{1}{p}} D) ∩ ∂D is nonempty.

(c) f(L) has a nontrivial periodic point.

Proof. (a) ⟹ (c) is clear from the definition of chaos. (c) ⟹ (b). By the same proof as the second half of the proof of Lemma 2.2(2), f(P σ(L)) ∩ ∂D is nonempty. Since (H1) implies P σ(L) = R^{\frac{1}{p}} D, (b) follows. (b) ⟹ (a) is Lemma 2.2(1). □

Remark 2.4. Although (b) ⟹(a) ⟹ (c) of Theorem 2.3 always holds, none of the equivalences are true if we suppress (H1). The following two counterexamples are due to Alfredo Peris. Take β(k) ≡ \frac{1}{k^{2}} (k ∈ N). If we suppress the assumption (H1), by [10, Theorem 2.4], L is chaotic, so that (a) of Theorem 2.3 holds, with f(z) ≡ z, but f(R^{\frac{1}{p}}) = f(D) = D, hence (b) of Theorem 1.2 fails.

Now take f(z) ≡ \frac{1}{2}(1 + z), so that f(L) = \frac{1}{2}(I + L). Since Lh⃗_{1} = h⃗_{1} and f(1) = 1, if we suppress the assumption (H1), h⃗_{1} is a nontrivial periodic point for f(L). Thus f(L) satisfies (c) of Theorem 1.2. But since P σ(L) = \overline{D} and f(\overline{D}) ∩ ∂D = \{1\}, Lemma 2.2(2) implies that f(L) is not chaotic; that is, f(L) does not satisfy (a) of Theorem 1.2.

Now we will present many special cases with more readily verifiable equivalences. We will begin by seeing how Theorem 2.3 looks on the unweighted sequence space ℓ^{p}(N).

Corollary 2.5. The following are equivalent, for f holomorphic and nonconstant in a neighborhood of \overline{D}.

(a) f(L) on ℓ^{p}(N) is chaotic.

(b) f(\overline{D}) ∩ ∂D is nonempty.

(c) f(L) on ℓ^{p}(N) has a nontrivial periodic point.

Remark 2.6. In [13] it is asserted that f(L) on ℓ^{1}(N) is chaotic whenever σ(f(L)) ∩ ∂D is nonempty. This is not quite correct, as can be seen by choosing f(z) := z: f(L) = L is not chaotic on ℓ^{1}(N), but σ(f(L)) = σ(L) = \overline{D}.

For many functions f, (a) and (b) of Theorem 2.3 remain equivalent, without the restriction on β⃗.

Lemma 2.7. Suppose f is holomorphic and nonconstant in a neighborhood of ∥L∥\overline{D}, R > 0 and f(R^{\frac{1}{p}} ∂D) ∩ ∂D is finite. Then the following are equivalent.

(a) f(L) is chaotic.

(b) f(R^{\frac{1}{p}} D) ∩ ∂D is nonempty.

Proof. (a) ⟹ (b). By Lemma 2.2(2), f(R^{\frac{1}{p}} \overline{D}) ∩ ∂D is infinite. Since f(R^{\frac{1}{p}} ∂D) ∩ ∂D is finite, (b) follows.

(b) ⟹(a) is Lemma 2.2(1). □

Theorem 2.8. Suppose Q is a polynomial other than a complex multiple of z ↦ z^{n}, for some nonnegative integer n and R > 0. Then the following are equivalent.

(a) Q(L) is chaotic.

(b) Q(R^{\frac{1}{p}} D) ∩ ∂D is nonempty.

Proof. By Lemma 2.7, it is sufficient to show that, if Q(R^{\frac{1}{p}} ∂D) ∩ ∂D is infinite, then Q(z) = αz^{n}, for some n ∈ N, α ∈ C.

To this end, suppose Q is an n^{th} degree polynomial,

Q(z) ≡ ∑_{k=0}^{n} α_{k}z^{k}.

If Q(R^{\frac{1}{p}} e^{iθ}) ∈ Q(R^{\frac{1}{p}} ∂D) ∩ ∂D, then

1 = [Q(R^{\frac{1}{p}} e^{iθ})] \overline{[Q(R^{\frac{1}{p}} e^{iθ})]} = ∑_{k,j=0}^{n} R^{\frac{k+j}{p}} α_{k}\overline{α_{j}}e^{i(k−j)θ},

so that

0 = [∑_{k=0}^{n} |α_{k}|^{2}R^{\frac{2k}{p}} − 1] + ∑_{k,j=0,k≠j}^{n} R^{\frac{k+j}{p}} α_{k}\overline{α_{j}}e^{i(k−j)θ}.  (2.9)

If Q(R^{\frac{1}{p}} ∂D) ∩ ∂D is infinite, then (2.9) holds for infinitely many points R^{\frac{1}{p}} e^{iθ} ∈ R^{\frac{1}{p}} ∂D. Since a nonconstant trigonometric polynomial has only finitely many zeroes and R > 0, this implies that α_{k}\overline{α_{j}} = 0 whenever k ≠ j, 0 ≤ k, j ≤ n. This is saying that α_{k} = 0 for 0 ≤ k < n, as desired. □

Remark 2.10. Theorem 2.8 and Lemma 2.7, combined with [10, Theorem 2.4], imply that, for Q a polynomial, Q(L) is chaotic if and only if Q(P σ(L)) ∩ ∂D is infinite.

Affine functions of L may be of particular interest (see [14]).

Corollary 2.11. Suppose a and b are nonzero complex numbers. Then the following are equivalent.

(a) a + bL is chaotic.

(b) |b|R^{\frac{1}{p}} > |1 − |a||.

Proof. By Theorem 2.8, we must show that (b) holds if and only if (|a| + |b|R^{\frac{1}{p}} D) ∩ ∂D is nontrivial. By rotation, we may assume both a and b are nonnegative real numbers. Then (a+|b|R^{\frac{1}{p}} D) intersects ∂D if and only if there exists r, −1 < r < 1, so that 1 = a+r|b|R^{\frac{1}{p}}; by solving for |b|R^{\frac{1}{p}}, it is clear that this is equivalent to (b). □

Remarks 2.12. When the assumption (H1) holds, Theorem 2.3 implies that (a) and (b) of Corollary 2.11 are equivalent to

(c) (a + bL) has a nontrivial periodic point; and

(d) ∑_{k=1}^{∞} (\frac{1−|a|}{|b|})^{pk} β(k) < ∞.

When a = 0, the equivalence of (a), (c) and (d) holds without the extra hypothesis on β⃗, at least when b = 1 (see [10, Theorem 2.4]). This remark illustrates how unstable chaos is under perturbations. For β(k) ≡ 1(k ∈ N), (a + L) is chaotic if and only if (a + L) has a nontrivial periodic point if and only if 0 < |a| < 2. For β(k) ≡ \frac{1}{k^{2}} (k ∈ N), (a + L) is chaotic if and only if |a| < 2 and (a + L) has a nontrivial periodic point if and only if |a| ≤ 2.

It also seems of interest that for any nonzero complex b and any β⃗ (bL ± 1) is chaotic. The special case of a, b real, b > −a ≥ 1 appears in [14].

Now we turn to functions of L generating a chaotic semigroup.

Theorem 2.13. Assume (H1). The following are equivalent, for g holomorphic and constant on a neighborhood of ∥L∥\overline{D}.

(a) g(L) generates a chaotic semigroup.

(b) g(R^{\frac{1}{p}} D) ∩ iR is nonempty.

(c) e^{g(L)} has a nontrivial periodic point.

(d) {e^{tg(L)}}_{t≥0} has a nontrivial periodic point.

(e) e^{g(L)} is chaotic.

Proof. Let f(z) := e^{g(z)}. By Theorem 2.3, since f(L) = e^{g(L)}, (b) is equivalent to (c) and (e).

It is clear that (e) implies (a) and (a) implies (d).

(d) ⟹ (b). As in the proof of Theorem 2.3, (d) implies that there exists μ ∈ R^{\frac{1}{p}} D such that

e^{tg(μ)} = 1,

so that g(μ) ∈ iR. □

Theorem 2.14. Suppose Q is a polynomial. Then the following are equivalent. (a) Q(L) generates a chaotic semigroup.

(b) Q(R^{\frac{1}{p}} D) ∩ iR is nonempty.

(c) e^{Q(L)} is chaotic.

Proof. Let f(z) ≡ e^{Q(z)}. Then (b) is equivalent to f(R^{\frac{1}{p}} D) ∩ ∂D being nonempty.

As with the proof of Theorem 2.8, the equivalence of (b) and (c) will follow once we show that f satisfies the hypotheses of Lemma 2.7; this is equivalent to showing that Q(R^{\frac{1}{p}} ∂D) ∩ iR is finite. To this end, suppose z = x + iy ∈ R^{\frac{1}{p}} ∂D (x, y ∈ R) and Q(z) ∈ iR. Then Re(Q(z)) = 0, while x^{2} + y^{2} = R^{\frac{2}{p}} and the fact that Re(Q(z)) contains only even powers of y imply that Re(Q(z)) is a polynomial in x, hence Re(Q(z)) has only finitely many zeroes for z ∈ R^{\frac{1}{p}} ∂D, as desired.

This proves the equivalence of (b) and (c). (a) ⟹ (b) follows from Lemma 2.2(3) and the fact, just shown, that Q(R^{\frac{1}{p}} ∂D) ∩ iR is finite, since P σ(L) ⊆ R^{\frac{1}{p}} \overline{D}. (c) ⟹ (a) is obvious from the definition of chaos □

Remarks 2.15. When Q(z) ≡ a + bz (a, b ∈ C), so that Q(L) = a + bL, it is not hard to see that (b) of Theorem 2.14 is equivalent to

|Re(a)| < |b|R^{\frac{1}{p}} .

Thus Theorem 2.14 generalizes [3, Theorem 5.3]. Special cases also appear in [12], where it is shown that L generates a chaotic semigroup on ℓ^{2}(N), and in [14], where it is shown that (a + bL) generates a chaotic semigroup when b > −a > 0.

It is interesting that bL generates a chaotic semigroup, for any complex b.

The equivalence of (a) and (d) in Theorem 2.13 fails in general, even for first-degree polynomials as in [3, Theorem 5.3]. Take f(z) ≡ z − 1. If β(k) ≡ \frac{1}{k^{2}} (k ∈ N), then \{e^{tf(L)}\} has the periodic point h⃗_{1}, but since f(P σ(L)) ∩ iR = {0}, Lemma 2.2(3) implies that {e^{tf(L)}}_{t≥0} is not chaotic. If β(k) ≡ 1, then {e^{tf(L)}}_{t≥0} no longer has any periodic points.

Here is another statement about the abundance of chaos, when any chaos at all exists.

Corollary 2.16. The following are equivalent, for 1 ≤ p < ∞, r > ∥L∥.

(a) There exists f holomorphic on rD such that f(L) is chaotic.

(b) R > 0.

(c) P σ(L) is nonempty.

(d) For any nonconstant f holomorphic on rD, there exists a constant c > 0 so that cf(L) is chaotic.

Proof. The equivalence of (b) and (c) is clear from the fact that h⃗_{μ} is an eigenvector if and only if ∑_{k=1}^{∞} β(k)|μ|^{kp} < ∞. The remaining equivalences are a consequence of Lemma 2.2(1). □

Remark 2.17. A statement similar to (d), except that “chaotic” is replaced with “hypercyclic,” for generalized backward shifts, appears in [5, Theorem 3.6].

3. DISCRETE WEIGHTED BILATERAL SHIFTS. It is not hard to show that by straightforward modifications of Section 2, we can obtain the above results for a weighted bilateral shifts.

Let us denote by ℓ^{p}(Z, β⃗) (1 ≤ p < ∞) the Banach space X of complex-valued sequences x⃗ := {x(k)}_{k=−∞}^{∞} such that

∥x⃗∥^{p} := ∑_{k=−∞}^{∞} |x(k)|^{p}β(k)

is finite. Here the sequence β⃗ satisfies

sup_{k∈Z} \frac{β(k)}{β(k + 1)} < ∞.

We will denote by ℓ^{p}(Z) ℓ^{p}(Z, β⃗) with β(k) := 1.

Denote by L the left-shift on ℓ^{p}(Z, β⃗)

(Lx⃗) (k) := x(k + 1) (k ∈ Z).

For a, b extended real nonnegative numbers, b > a, denote by A(a, b) the annulus

A(a, b) := {z ∈ C | a < |z| < b}.

If b ≤ a, A(a, b) will be the empty set.

12 RALPH DELAUBENFELS AND HASSAN EMAMIRAD

As in Section 2, f(L) is defined by the Riesz functional calculus when f is holomorphic in a neighborhood of σ(L), and we denote by

h⃗_{μ} := (..., μ^{−2}, μ^{−1}, 1, μ, μ^{2}, ...);

that is, h⃗_{μ}(k) := μ^{k}(k ∈ Z); the eigenvector for L corresponding to the eigenvalue μ. Denote by

R_{1} := \underline{lim}_{k→∞} (β(k))^{− \frac{1}{k}} , and R_{2} := \overline{lim}_{k→∞} (β(−k))^{\frac{1}{k}} .

Note that h⃗_{μ} ∈ ℓ^{p}(Z, β⃗) if and only if ∑_{k=−∞}^{∞} β(k)|μ|^{k} < ∞. Thus P σ(L) ⊆ \overline{A(R_{2}, R_{1})} and equals A(R_{2}, R_{1}) if ∑_{k=−∞}^{∞} β(k)R_{i}^{k} = ∞ for i = 1, 2.

The following lemma can be obtained just by mimicking the proof of Lemma 2.2. Lemma 3.1. Suppose f is holomorphic and nonconstant in a neighborhood of \overline{P σ(L)} and 0 < R_{2} < R_{1}.

(1) If f (A(R2^{\frac{1}{p}} , R1^{\frac{1}{p}})) ∩ ∂D is nonempty, then f(L) is chaotic;

(2) If f(L) is chaotic, then f (P σ(L)) ∩ ∂D is infinite;

(c) If f(L) generates a chaotic semigroup, then f (P σ(L)) ∩ iR is infinite.

In this context the hypothesis (H1) of the previous section is modified as follows

(H2)  0 < R_{2} < R_{1},  and  ∑_{k=−∞}^{∞} β(k)R_{i}^{k} diverges for i = 1, 2.

Under this assumption the analogue of Theorem 2.3 is the following. Theorem 3.2. Suppose (H2) holds. Then the following are equivalent, for f holomorphic and nonconstant in a neighborhood of σ(L).

(a) f(L) is chaotic.

(b) f (A(R2^{\frac{1}{p}} , R1^{\frac{1}{p}})) ∩ ∂D is nonempty.

(c) f(L) has a nontrivial periodic point.

Remarks 3.3. (i) Corollary 2.11 is modified as follows.

Suppose a and b are complex numbers. Then the following are equivalent.

(a) (a + bL) is chaotic.

(b) R_{2} < (\frac{|1−|a||}{|b|})^{p} < R_{1}.

If we impose the assumption (H2) the above assertions are equivalent to

(c) (a + bL) has a nontrivial periodic point.

(d) ∑_{k=−∞}^{∞} (\frac{|1−|a||}{|b|})^{pk} β(k) < ∞.

(ii) Analogously to Theorem 2.13, we get:

Assume (H2) holds. The following are equivalent, for g holomorphic and nonconstant on a neighborhood of σ(L).

(a) g(L) generates a chaotic semigroup.

(b) g (A(R2^{\frac{1}{p}} , R1^{\frac{1}{p}})) ∩ iR is nonempty.

(c) e^{g(L)} has a nontrivial periodic point.

(d) {e^{tg(L)}}_{t≥0} has a nontrivial periodic point.

(e) e^{g(L)} is chaotic.

(iii) Here is the analogue of Theorem 2.14.

Suppose Q is a polynomial. Then the following are equivalent.

(a) Q(L) generates a chaotic semigroup.

(b) Q (A(R2^{\frac{1}{p}} , R1^{\frac{1}{p}})) ∩ iR is nonempty.

(c) e^{Q(L)} is chaotic.

(iv) Analogously to Corollary 2.16, we get:

If 0 < R_{2} < R_{1}, then for any nonconstant f holomorphic on σ(L), there exists c > 0 such that cf(L) is chaotic.

4. CONTINUOUS WEIGHTED UNILATERAL SHIFTS.

Throughout this section, we will denote by L^{p}([0, ∞), ρ) the Banach space X of complexvalued functions f on [0, ∞) such that

∥f∥^{p} ≡ ∫_{0}^{∞} |f(s)|^{p} ρ(s) ds

is finite. Here ρ : [0, ∞) → (0, ∞) is measurable and satisfies

sup_{s≥0} \frac{ρ(s)}{ρ(s + t)} < ∞,  (4.0)

for all t ≥ 0.

Let B be \frac{d}{ds}, the generator of

(e^{tB}f) (s) ≡ f(s + t) (s, t ≥ 0)

on L^{p}([0, ∞), ρ). Note that (4.0) is precisely what is needed to make {e^{tB}}_{t≥0} a strongly continuous semigroup.

Let LHP denote the open left half plane {z ∈ C | Re(z) < 0}. The continuous analogue of the radius of convergence of a power series will be

ω ≡ sup{μ ∈ R | ∫_{0}^{∞} e^{μs}ρ(s) ds < ∞}.

If the set we are trying to take a supremum of is empty, we will say that ω = −∞. Denote by

h_{μ}(s) ≡ e^{μs} (s ≥ 0)

the eigenvector for B corresponding to the eigenvalue μ.

Note that P σ(B) is equal either to \frac{ω}{p} + LHP if ∫_{0}^{∞} e^{ωs}ρ(s)ds = ∞ or its closure if not. We adopt the custom that

(∞ + LHP) ≡ C and (−∞ + LHP) ≡ the empty set .

Here is the continuous analogue of Lemma 2.1.

Lemma 4.1. Suppose Ω ⊆ \frac{ω}{p} + LHP. If Ω has an accumulation point, then H_{Ω} := {h_{μ} | μ ∈ Ω} is dense in L^{p}([0, ∞), ρ).

Proof. Suppose ϕ annihilates H_{Ω}. Define

g(μ) ≡ ∫_{0}^{∞} e^{μs}ϕ(s) ds.

Then g is analytic on \frac{ω}{p} + LHP, and since ϕ annihilates H_{Ω}, it follows that g(μ) = 0 in Ω. Since Ω has an accumulation point, g ≡ 0, so that by uniqueness of Laplace transform, ϕ ≡ 0. □

Lemma 4.2. Suppose g ↦ g(B) is a linear map from a vector space A of complex-valued functions holomorphic on a neighborhood of σ(B) into B(L^{p}([0, ∞), ρ)) such that

P σ(g(B)) ⊆ g(P σ(B)) ∪ {0}, ∀g ∈ A.  (4.3)

Then for f ∈ A we have the following.

(1) If f(\frac{ω}{p} + LHP) ∩ ∂D is nonempty, then f(B) is chaotic.

(2) If f(B) is chaotic, then f(P σ(B)) ∩ ∂D is infinite.

Proof. The proof of (1) is the same as the proof of Lemma 2.2(1), with \frac{ω}{p} +LHP replacing

R^{\frac{1}{p}} D and Lemma 4.1 replacing Lemma 2.1. The proof of (2) is the same as the proof of Lemma 2.2(2), with (4.3) replacing [9, Theorem 5.12.2]. □

The continuous analogue of the hypothesis (H1) on the weight function ρ will be

(H3)  ω < ∞  and  ∫_{0}^{∞} e^{ωs}ρ(s)ds  diverges

As with Theorem 2.3 one can prove that

Theorem 4.4. Under the assumptions of Lemma 4.2 together with (H3) for f ∈ A, the following are equivalent

(a) f(B) is chaotic.

(b) f(\frac{ω}{p} + LHP) ∩ ∂D is nonempty.

(c) f(B) has a nontrivial periodic point.

Proof. (c) ⟹ (b). By the same proof as Lemma 4.2(2), f(P σ(L)) ∩ ∂D is nonempty. Since (H3) implies P σ(L) = \frac{ω}{p} + LHP, (b) follows. (b) ⟹ (a) is Lemma 4.2(1). □

We will dispense with a detailed discussion of which g satisfy the hypotheses of Lemma 4.2; in a future paper [2] it will be shown that (4.3) holds when g(z) → 0 for |z| → ∞ in σ(B). Our interest in this paper is g equal to exponentials of polynomials, so that we may address semigroups generated by differential operators.

For Q a polynomial, define the differential operator Q(B) as follows. If

Q(z) ≡ ∑_{k=0}^{n} a_{k}z^{k},

for some complex a_{0}, a_{1}, ..., a_{n}, with a_{n} ≠ 0, define

Q(B) ≡ ∑_{k=0}^{n} a_{k}B^{k}, D(Q(B)) ≡ D(B^{n}).  (4.5)

Theorem 4.6. Suppose that the weight function ρ satisfies the assumption (H3) and Q is a polynomial such that Q(B) generates a strongly continuous semigroup {T (t)}_{t≥0}. Then the following are equivalent.

(a) {T (t)}_{t≥0} is chaotic.

(b) Q(\frac{ω}{p} + LHP) ∩ iR is nonempty.

(c) {T (t)}_{t≥0} has a nontrivial periodic point.

(d) T (1) is chaotic.

(e) T (1) has a nontrivial periodic point.

Proof. For any t ≥ 0, let f_{t}(z) ≡ e^{tQ(z)}, and let A be the span of {f_{t} | t ≥ 0}, with

f_{t}(B) ≡ T (t) (t ≥ 0).

To see that f_{t} satisfies (4.3), note first that, since {T (t)}_{t≥0} is a strongly continuous semigroup generated by Q(B),

P σ(f_{t}(B)) ≡ P σ(T (t)) ⊆ e^{tP σ(Q(B))} ∪ {0}.

([11, Theorem 2.2.4]) Since Q is a polynomial,

P σ(Q(B)) = Q(P σ(B)).

The equivalence of (d) and (e) now follows from Theorem 4.4. Theorem 4.4 now also implies the equivalence of (d) and (b), since (b) is equivalent to

e^{Q( \frac{ω}{p} +LHP)} ∩ ∂D

being nonempty.

Clearly (d) ⟹ (a) ⟹ (c). Since we have shown that f_{t} satisfies (4.3), the same argument as in the proof of Theorem 2.3(c) ⟹ (b) shows that (c) implies that there exists μ ∈ \frac{ω}{p} + LHP such that

e^{tQ(μ)} = 1,

which implies (b). This completes the proof. □

Corollary 4.7. Under the assumptions of Lemma 4.2 and (H3) the following are equivalent, for a, b ∈ C, b > 0, if we define the strongly continuous semigroup {T (t)}_{t≥0} on L^{p}([0, ∞), ρ) as follows:

(T (t)f) (s) ≡ e^{at}f(s + bt) (s, t ≥ 0).

(a) {T (t)}_{t≥0} is a chaotic semigroup.

(b) ∫_{0}^{∞} e^{− \frac{p}{b} [Re(a)]s}ρ(s) ds < ∞.

(c) Re(a) + \frac{bω}{p} > 0.

(d) There exists t > 0 and nontrivial f ∈ L^{p}([0, ∞), ρ) so that

f(s + bt) = e^{−at}f(s) ∀s ≥ 0.

(e) T (1) has a nontrivial periodic point.

(f) T (1) is chaotic.

Proof. Apply Theorem 4.6 with Q(z) ≡ a + bz.

(a) ⟺ (c). By Theorem 4.6, (a) is equivalent to

[(Re(a) + \frac{bω}{p} + LHP ) ∩ iR] = [a + b (\frac{ω}{p} + LHP )] ∩ iR

being nonempty. This is equivalent to (c).

It is clear from the definition of ω that (c) is equivalent to (b). The remaining equivalences are a special case of Theorem 4.6. □

Corollary 4.8. Under the assumptions of Lemma 4.2 and (H3) the following are equivalent, if

(T (t)f) (s) ≡ f(s + t) (s, t ≥ 0).

(a) {T (t)}_{t≥0} is a chaotic strongly continuous semigroup.

(b) ∫_{0}^{∞} ρ(s) ds < ∞.

Remark 4.9. It is interesting to compare Corollary 4.8 to [3, Theorem 4.7], where it is shown that \{e^{tB}\}_{t≥0} is hypercyclic if and only if \underline{lim}_{s→∞}ρ(s) = 0. For example, if ρ(s) ≡ \frac{1}{s}, then \{e^{tB}\}_{t≥0} is hypercyclic but not chaotic. Corollary 4.10. The following are equivalent, for 1 ≤ p < ∞.

(a) There exists a polynomial Q such that Q(B) generates a chaotic semigroup.

(b) ω > −∞.

(c) There exists real μ such that ∫_{0}^{∞} e^{μs}ρ(s) ds < ∞.

(d) ρ is exponentially bounded.

(e) P σ(B) is nonempty.

(f) Whenever Q is a polynomial such that Q(B) generates a strongly continuous semigroup, then there exists a real constant c such that (Q(B) + c) generates a chaotic semigroup {e^{t(Q(B)+c)}}_{t≥0} = {e^{tc}e^{tQ(B)}}_{t≥0}.

5. CONTINUOUS WEIGHTED BILATERAL SHIFTS It is straightforward to modify the results and proofs in Section 4. Throughout this section, we will denote by L^{p}(R, ρ) the Banach space X of complex-

valued functions f on R such that

∥f∥^{p} ≡ ∫_{−∞}^{∞} |f(s)|^{p} ρ(s) ds

is finite. Here ρ : R → (0, ∞) is measurable and satisfies

sup_{s∈R} \frac{ρ(s)}{ρ(s + t)} < ∞,

for all t ≥ 0.

Let B be \frac{d}{ds}, the generator of the strongly continuous semigroup

(e^{tB}f) (s) ≡ f(s + t) (s ∈ R, t ≥ 0)

on L^{p}(R, ρ).

For b > a extended real numbers, denote by V_{a,b} the vertical strip

V_{a,b} ≡ {z | a < Re(z) < b}.

If b ≤ a, V_{a,b} will be the empty set.

Define

ω_{1} ≡ sup{μ ∈ R | ∫_{0}^{∞} e^{μs}ρ(s) ds < ∞};

ω_{2} ≡ inf{μ ∈ R | ∫_{−∞}^{0} e^{μs}ρ(s) ds < ∞}.

As in the previous section, denote by

h_{μ}(s) ≡ e^{μs} (s ∈ R)

the eigenvector for B corresponding to the eigenvalue μ.

Note that for ω_{1} > ω_{2}

P σ(B) ⊂ \frac{1}{p}\overline{V_{ω_{2},ω_{1}}}.

In order to have

P σ(B) = \frac{1}{p}V_{ω_{2},ω_{1}},

we will sometimes need the hypotheses analogous to (H3):

(H4)  ω_{1} > ω_{2} > −∞,  ∫_{0}^{∞} e^{ω_{1}s}ρ(s)ds and ∫_{−∞}^{0} e^{ω_{2}s}ρ(s)ds  divergent.

For Q a polynomial, define Q(B) as in (4.5).

Theorem 5.1. Assume (H4) and suppose Q is a polynomial such that Q(B) generates a strongly continuous semigroup {T (t)}_{t≥0}. Then the following are equivalent.

(a) {T (t)}_{t≥0} is chaotic.

(b) Q(\frac{1}{p}V_{ω_{2},ω_{1}}) ∩ iR is nonempty.

(c) {T (t)}_{t≥0} has a nontrivial periodic point.

(d) T (1) is chaotic.

(e) T (1) has a nontrivial periodic point.

Corollary 5.2. Assume (H4). The following are equivalent, for a, b ∈ C, b > 0, if we define the strongly continous semigroup {T (t)}_{t≥0} on L^{p}(R, ρ) as follows:

(T (t)f) (s) ≡ e^{at}f(s + bt) (t ≥ 0, s ∈ R).

(a) {T (t)}_{t≥0} is a chaotic semigroup.

(b) ∫_{−∞}^{∞} e^{− \frac{p}{b} [Re(a)]s}ρ(s) ds < ∞.

(c) \frac{bω_{2}}{p} < −Re(a) < \frac{bω_{1}}{p}.

(d) There exists t > 0 and nontrivial f ∈ L^{p}(R, ρ) so that

f(s + bt) = e^{−at}f(s) ∀s ∈ R.

(e) T (1) has a nontrivial periodic point.

(f) T (1) is chaotic.

Corollary 5.3. Assume (H4). The following are equivalent, if

(T (t)f) (s) ≡ f(s + t) (t ≥ 0, s ∈ R).

(a) {T (t)}_{t≥0} is a chaotic strongly continuous semigroup.

(b) ∫_{−∞}^{∞} ρ(s) ds < ∞.

Remark 5.4. In [3, Theorem 4.8], it is shown that {e^{tB}}_{t≥0} is hypercyclic if and only if for each s ∈ R there is a sequence {t_{j}}_{j=1}^{∞} of positive numbers such that

lim_{j→∞} ρ(s + t_{j}) = lim_{j→∞} ρ(−t_{j} + s) = 0.

As in Remark 4.9, it is easy to construct weight functions ρ such that {e^{tB}}_{t≥0} is hypercyclic but not chaotic.

Analogously to Corollary 4.10, we are guaranteed many chaotic semigroups under weak hypotheses on the point spectrum of B. Indeed, Lemma 4.2 is valid if we replace \frac{ω}{p} +LHP by \frac{1}{p}V_{ω_{2},ω_{1}} and this result implies the following Corollaries.

Corollary 5.5. Suppose ω_{1} > ω_{2} and Q is a polynomial such that Q(B) generates a strongly continuous semigroup. Then there exists c ∈ R such that (Q(B) + c) generates a chaotic semigroup {e^{t(Q(B)+c)}}_{t≥0} = {e^{tc}e^{tQ(B)}}_{t≥0}.

Theorem 5.1 and Corollary 5.5 are of more interest than Theorem 4.6 and Corollary 4.10 because there are many more polynomials Q for which Q(B) generates a strongly continuous semigroup. The heat semigroup is only one example, but may be of particular interest.

Corollary 5.6. Suppose ω_{1} > ω_{2} and

sup_{s∈R} \frac{ρ(s)}{ρ(s − t)} < ∞, ∀t ≥ 0.  (*)

Then B^{2} generates a chaotic strongly continuous semigroup.

Proof. Hypothesis (*) guarantees that {e^{tB}}_{t≥0} extends to a strongly continous group. This implies ([6, 2.8.7 and 2.8.11]) that B^{2} generates a strongly continuous semigroup.

By the proof of Theorem 5.1 and the analogue of Lemma 4.2(1), it is sufficient to show that

{z^{2} |z ∈ V_{ω_{2},ω_{1}}}

intersects the imaginary axis. But this is clear when we write z = u + iv, with u and v real and note that the lines |v| = |u| are mapped to the imaginary axis by z ↦ z^{2}. □

Acknowedgement We are extremely indebted to Alfredo Peris who read the first version of the paper with care for detail and made us aware of the necessity of the assumption (H1) in some of our results by giving the counterexamples cited in Remark 2.4. The authors appreciate also the helpful discussions with V. Protopopescu.

References

[1] J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly 99 (1992), 332–334.

[2] R. deLaubenfels, Spectral mapping theorems for operators with polynomially bounded resolvent, preprint.

[3] W. Desch, W. Schappacher, and G. F. Webb, Hypercyclic and chaotic semigroup of linear operators, Ergodic Theory Dyn. Syst. 17 (1997), 793–819.

[4] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, New York, 1989.

[5] G. Godefroy and J. H. Shapiro, Operators with dense, invariant cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229–269.

[6] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, 1985.

[7] K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999), 345–381.

[8] K.-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. (to appear).

[9] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, vol. 31, Amer. Math. Soc. Coll. Publ., Providence, R. I., 1957.

[10] F. Martinez-Gimenez and A. Peris, Hypercyclic and chaotic backward shift operators on Köthe echelon spaces, Preprint.

[11] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, Berlin, 1983.

[12] C. R. MacCluer, Chaos in linear distributed systems, J. Dyn. Systems, Measurement, and Control 114 (1992), 322–324.

[13] V. Protopopescu, Chaotic behaviour of linear orbits, Proceedings of Conference “Mathematical models of non-linear excitation, transport, dynamics, control in condensed systems and other mediums (1996).

[14] V. Protopopescu and Y. Y. Azmy, Topological chaos for a class of linear models, Math. Models and Methods in Appl. Sci. 2 (1992), 79–90.

ERRATA, to “Chaos for functions of discrete and continuous weighted shift operators,” by R. deLaubenfels and H. Emamirad, Ergodic Theory and Dynamical Systems 21 (2001), 1411–1427.

1. The condition on the weight function ρ, in Sections IV and V, should be

\frac{ρ(s)}{ρ(s + t)} ≤ Me^{ωt},

for some M ≥ 1, ω ∈ R, for all t ≥ 0, and, in Section IV, all s ≥ 0, and, in Section V, all s ∈ R.

2. The proof of Lemma 2.2(2) and (3) is incorrect. Here are correct proofs.

Proof of Lemma 2.2(2)

By [9, Theorem 5.12.2], it is sufficient to show the following.

Proposition. If B is a chaotic (bounded) operator, then P σ(B) ∩ ∂D is infinite.

Proof: Suppose C ≡ P σ(B) ∩ ∂D is finite. Define a polynomial

p(z) ≡ ∏_{λ∈C}(λ − z).

Suppose x is a periodic point of B. We claim p(B)x = 0.

To show the claim, say that B^{n}x = x. Then

0 = ∏_{k=1}^{n}(e^{\frac{2πik}{n}} − B)x = [  ∏_{exp( \frac{2πik}{n})∉C}(e^{\frac{2πik}{n}} − B)] (  ∏_{exp( \frac{2πik}{n})∈C}(e^{\frac{2πik}{n}} − B)) x,

thus, applying more factors (λ − B), λ ∈ C, if necessary, we have

[ ∏_{exp( \frac{2πik}{n})∉C}(e^{\frac{2πik}{n}} − B)] (p(B)x) = 0.

If p(B)x were not trivial, this would show that there exists k such that e^{\frac{2πik}{n}} ∈ P σ(B) − C, contradicting the definition of C. This proves the claim.

If B were chaotic, then the set of all periodic points would be dense, thus the claim would imply that

p(B) ≡ 0.  (*)

However this would make it impossible for B to be hypercyclic, since (*) implies that there exist complex λ_{j} such that, if N is the degree of p,

B^{N} = ∑_{j=0}^{N−1} λ_{j}B^{j},

which implies that, for any x ∈ X,

{B^{n}x}_{n=0}^{∞} ⊆ span {x, Bx, B^{2}x, …, B^{N−1}x},

which cannot be dense.

This shows that C finite implies B cannot be chaotic, as desired. □

This actually shows that, if B is chaotic, then P σ(B) ∩ {e^{2πiθ} | θ is rational } is infinite.

Proof of Lemma 2.2(3)

By [9, Theorem 5.12.2], it is sufficient to show the following.

Proposition. If B is an operator that generates a chaotic semigroup, then P σ(B) ∩ iR is infinite.

Proof: Suppose C ≡ P σ(B) ∩ iR is finite.

First, we claim that, if x is a periodic point of {e^{tB}}_{t≥0}, then x is a finite sum of eigenvectors for B corresponding to eigenvalues from C. More specifically, for T > 0, k ∈ Z, denote by

y_{k,T,x} ≡ [ \frac{1}{T} ∫_{0}^{T} e^{−2πi \frac{k}{T} r}e^{rB}x dr] .

Then an integration by parts shows that, if e^{T B}x = x, then

By_{k,T,x} = (2πi \frac{k}{T} ) y_{k,T,x} (k ∈ Z).

We claim that, if e^{T B}x = x, then

x = ∑_{k∈Z,2πi \frac{k}{T} ∈C} y_{k,T,x}.  (*)

To prove the claim, note that, for any ϕ ∈ X^{∗}, since s ↦ ϕ(e^{sB}x) has period T, we have

ϕ(e^{sB}x) = ∑_{k∈Z} [ \frac{1}{T} ∫_{0}^{T} e^{−2πi \frac{k}{T} r}ϕ(e^{rB}x) dr] e^{2πi \frac{k}{T} s} = ∑_{k∈Z} ϕ (y_{k,T,x})e^{2πi \frac{k}{T} s} (s ≥ 0),  (**)

with the series converging in L^{2}([0, T]). Since C is supposed finite, y_{k,T,x} = 0 for all but finitely many k ∈ Z, thus the sum in (**) is actually finite, with

e^{sB}x = ∑_{k∈Z,2πi \frac{k}{T} ∈C} y_{k,T,x}e^{2πi \frac{k}{T} s} (s ≥ 0).  (***)

In particular, letting s = 0 in (***) gives us (*).

Next, we claim that (*) implies that there are only finitely many periods for {e^{tB}}_{t≥0}. To see this, note that, if

x = ∑_{j=1}^{m} h_{α_{j}} ,  (****)

where h_{λ} is an eigenvector for B corresponding to the eigenvalue λ, then, since

e^{tB}x = ∑_{j=1}^{m} e^{tα_{j}} h_{α_{j}} ,

e^{tB}x = x if and only if tα_{j} ∈ 2πiZ, for j = 1, 2, …, m. Thus the period for x as in (****) is purely a function of {α_{j}}_{j=1}^{m}; there exists T (α_{1}, α_{2}, …, α_{m}) defined to be the smallest T such that T α_{j} ∈ 2πiZ, for j = 1, 2, …, m, if it exists; define T (α_{1}, α_{2}, …, α_{m}) to be infinity otherwise.

This shows that the set of all periods of {e^{tB}}_{t≥0} equals {T (α_{1}, α_{2}, …, α_{n}) | α_{j} ∈ C, for j = 1, 2, …, n, and T (α_{1}, α_{2}, …, α_{n}) < ∞}. Since C is finite, this set is finite, proving the second claim.

Finally, we now show that the second claim implies that {e^{tB}}_{t≥0} cannot be hypercyclic. Denote by {T_{j}}_{j=1}^{N} the periods of {e^{tB}}_{t≥0}, and for j = 1, 2, …, N, let W_{j} be the set of all x of period T_{j}. Then

X = \overline{∪_{j=1}^{N}W_{j}} = ∪_{j=1}^{N}\overline{W_{j}},

since the union is finite. Thus for any x ∈ X there exists j such that x ∈ \overline{W_{j}}, which implies that e^{T_{j} B}x = x. Then {e^{tB}x}_{t≥0} = {e^{tB}x}_{0≤t≤T_{j}} , which is compact, hence cannot be dense. As in 2.2(2), we have shown that C finite implies that {e^{tB}}_{t≥0} is not chaotic, as desired. □

We are indebted to Andreas Weber, of Karlsruhe, for pointing out the mistake in the proof of Lemma 2.2(2) and (3).