Appeared in J. Approx. Theory 105, (2000), 176–187

LINEAR CHAOS AND APPROXIMATION

R. deLaubenfels, H. Emamirad, and V. Protopopescu

Abstract. Let {T_{n}(t)}_{n=1}^{∞} be a sequence of strongly continuous linear semigroups on Banach spaces X_{n} converging in the sense of Kato to a semigroup T (t) on the Banach space X. We discuss under what conditions the chaoticity of T_{n}(t) is inherited by T (t). We apply our results to a discrete parabolic equation.

1. Introduction

During the last twenty years or so, several definitions of ”chaos” have been proposed (see [Dev], [K–S],[L–Y], and [Wig]). One of the most used among these definitions can be formulated as follows [Dev]:

Definition 1.1. Let X be a metric space. A continuous map f : X ↦ X is said to be chaotic on X, if

(1) f is topologically transitive;

(2) the periodic points of f are dense in X;

(3) f has sensitive dependence on initial conditions.

Condition (1) means that for all non-empty open subsets U and V of X there exists a natural number n such that f^{n}(U) ∩ V ≠ ∅, while a point x is called periodic if there exists some n ∈ ℕ, n > 1 such that f^{n}(x) = x. For the notion of sensitive dependence on initial conditions we refer to [Dev], Definition 8.2, p. 49. In the sequel we shall dispense with condition (3). Indeed, it has been proven in [Ba-al] that if f is transitive and has dense periodic points then f has sensitive dependence on initial conditions. One of the most appealing properties of Definition 1.1 is that it extends rather naturally to linear

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 47D05, Secondary 41A65, 47A58.

Key words and phrases. Hypercyclic and chaotic semigroups, approximation in the sense of Kato, convection-diffusion equation.

systems([McC], [P-A], [D-S-W]). More precisely, let X be an infinite-dimensional separable Banach space and T a linear bounded operator on X. The operator T is called hypercyclic if there exists x^{∗} ∈ X such that its orbit under the operator T, {T^{n}x^{∗} | n ∈ ℕ}, is dense in X. The operator T is hypercyclic if and only if it is topologically transitive (see for instance [D-S-W, Theorem 2.2]). Thus, a hypercyclic operator T is chaotic if and only if the set of X_{per} of the periodic points of T, i.e.

X_{per} := {x ∈ X | T^{n}x = x, for some n ∈ ℕ}

is dense in X. Let us now restate Definition 1.1 in the context of linear semigroups defined on a Banach space.

Definition 1.2. A C_{0}-semigroup {T (t)}_{t≥0} on the Banach space X is called chaotic provided that T (t) satisfies the following two conditions:

(H) (Hypercyclicity) There exists x^{∗} ∈ X such that {T (t)x^{∗} | t ≥ 0} is dense in X. (P) The set X_{per} of the periodic points of {T (t)}_{t≥0} given by X_{per} := {x ∈ X | ∃ t >

0 : T (t)x = x} is dense in X.

Our objective in this paper is to investigate the chaoticity of the approximating semigroups. In order to analyze this question, we define the approximation in the sense of Kato, which is one of the most general frameworks in approximation theory for linear semigroups (see [Kat], Chapter IX, Section 4). Let (X_{n}, ∥.∥_{n}) be a sequence of Banach spaces such that on each X_{n} there is defined a C_{0}-semigroup T_{n}(t). Assume that T_{n}(t) converges in the sense of Kato to a C_{0}-semigroup T (t), i.e.

lim_{n→∞} sup_{t∈[0,T]} ∥T_{n}(t)P_{n}f − P_{n}T (t)f∥_{n} = 0 for any f ∈ X  (1.1)

for some approximating operator P_{n} (the precise definition is given in Section 3).

To approximate a chaotic semigroup T (t) by a sequence of C_{0}-semigroups {T_{n}(t)}, one should first settle the following questions: (Q1) If {T_{n}(t)} is a sequence of linear chaotic C_{0}-semigroups which converges to T (t) in

the sense of Kato, can we assert that T (t) is chaotic? (Q2) Assuming that T (t) is chaotic and {T_{n}(t)} converges to T (t) in the sense of Kato,

under which conditions can we assert that at least one of T_{n}(t) is chaotic? The relevance of these questions can be seen in the two examples of the next section.

The first example is constructed to show that in general the statement:

(S1)  T_{n}(t) is chaotic ∀n ∈ ℕ ⟹ T (t) is chaotic,

is not true, which gives a negative anwer to question (Q1). The converse statement:

(S2)  T (t) is chaotic ⟹ T_{n}(t) is chaotic ∀n ∈ ℕ,

is not true either. But we can show that (S2) is true for those n’s such that instead of (1.1) we have

P_{n}T (t)f = T_{n}(t)P_{n}f  ∀f ∈ X.  (1.2)

We apply the above result to the chaotic semigroup generated by a parabolic equation which will be introduced in Example 2.4 of the following section. We do not expect that e^{tA_{h}} ⟶^{K} e^{tA} (as is mentioned in [Kat], even for the heat semigroup we can not expect such convergence). In the Section 4 we construct explicitly the approximating spaces and we prove that for some specific mesh sizes the discretized problem becomes chaotic. We mention that this feature has already been observed and studied in the context of nonlinear chaos (see [T-W-P] ).

2. Two examples

In this section we give two examples of linear chaotic semigroups.

Example 2.1 [P-A]. Let X := ℓ^{1} and A := −αI + βB, where B is the backward shift in ℓ^{1}, i.e. if f = (f_{1}, f_{2}, ⋯) ∈ ℓ^{1}, then (Bf)_{n} = f_{n+1}. Then the linear evolution problem in X

\{ \begin{matrix}\frac{df}{dt} = −αIf + βBf := Af, \\ f(0) = f_{0} ∈ ℓ^{1}, \end{matrix}

generates a chaotic semigroup T (t) = e^{tA}, provided that β > α ≥ 0.

For the reader’s convenience we briefly sketch the proof by using a Lemma which gives a sufficient condition that a C_{0}-semigroup be hypercyclic. For the original proof see [P-A].

Lemma 2.2 (see [D-S-W, Theorem 2.3]). Let {T (t)}_{t≥0} be a C_{0}-semigroup on the Banach space X. Define

X_{0} := {x ∈ X | lim_{t→∞} T (t)x = 0}

and X_{∞} := {x ∈ X | for any ϵ > 0, there exist u ∈ X and t > 0 such that ∥u∥ < ϵ and ∥T (t)u−x∥ < ϵ}. If X_{0} and X_{∞} are both dense in X, then {T (t)}_{t≥0} is hypercyclic.

Thus, in order that {T (t)}_{t≥0} be a chaotic semigroup in X, it suffices that X_{0}, X_{∞} and X_{per} be simultaneously dense in X.

Proof of Example 2.1. The spectrum of A is σ(A) = {−α + βμ | |μ| ≤ 1}. Indeed for any |μ| < 1, the vector

h_{μ} = {μ^{n}}_{n∈N}

satisfies the eigenvalue equation Ah = γh with eigenvalue γ = −α + βμ and the points with |μ| = 1 belong to the spectrum by the closure property.

We assume that 0 ≤ \frac{α}{β} < 1.

(1) \underline{X_{0} is dense in X.} Indeed, for 0 ≤ μ < \frac{α}{β}, one has e^{tA}h_{μ} = e^{−αt+βμt}h_{μ} → 0 as t → ∞. Hence h_{μ} and any finite linear combination of these vectors belongs to X_{0}. Since this set of vectors is dense in X, X_{0} is also dense in X.

(2) \underline{X_{∞} is dense in X.} Since e^{tA} is a C_{0}-group, for 1 > μ > \frac{α}{β}, one has e^{−tA}h_{μ} = e^{αt−βμt}h_{μ} → 0 as t → ∞. Any such h_{μ} ∈ X_{∞}, since for any ϵ > 0 one can choose t large enough such that for g_{μ} = e^{−tA}h_{μ}, ∥g_{μ}∥ < ϵ and ∥e^{tA}g_{μ} − h_{μ}∥ = 0. The same argument shows that X_{∞} is dense in X.

(3) \underline{X_{per} is dense in X.} Indeed, since β > α, the spectrum of the operator A, σ(A), contains a nonempty segment of the imaginary axis. On that segment, the complex numbers κ = i\frac{m}{n}, (m, n) ∈ ℤ^{2} form a dense set. Each such κ is an eigenvalue of

A with the eigenvector h_{\frac{κ+α}{β}} = (\frac{κ+α}{β}, (\frac{κ+α}{β})^{2}, ⋯). Each h_{\frac{κ+α}{β}} ∈ X_{per} and linear

combinations of such vectors are dense in ℓ^{1}. Indeed, if they were not dense, there would exist a non-identically zero functional Φ = (ϕ_{1}, ϕ_{2}, ...) ∈ ℓ^{∞} such that ⟨Φ, h_{\frac{κ+α}{β}} ⟩ = 0 for any h_{\frac{κ+α}{β}} constructed as above. This would imply that the

analytic function represented by the series ϕ_{1} \frac{κ+α}{β} + ϕ_{2}(\frac{κ+α}{β})^{2} + ϕ_{3}(\frac{κ+α}{β})^{3} + ⋯ is zero on a set of points κ with an accumulation point, which implies it is identically zero, in contradiction with the assumption that Φ ≠ 0.

□

Lemma 2.3. If in the above example α = β, then the generating C_{0}-semigroup e^{tA} is not chaotic.

Proof. For α = β, −αI + βB has no purely imaginary eigenvalues and the uniform boundedness

∥T (t)∥ = ∥e^{αt(B−1)}∥ = e^{−αt}∥e^{tαB}∥ ≤ e^{−αt}e^{∥tαB∥} = 1,

contradicts the hypercyclicity of the semigroup, since it is now clear that the orbit of any vector will be contained in the unit ball of X. Thus T (t) cannot be chaotic. □

Example 2.4 [D-S-W, Example 4.12]. The linear evolution problem is a convection– diffusion type equation of the form:

\{ \begin{matrix}f_{t} = af_{xx} + bf_{x} + cf := Af, \\ f(0, t) = 0 for t ≥ 0, \\ f(x, 0) = f_{0}(x) for x ≥ 0, \end{matrix}

with some f_{0} ∈ L^{1}(ℝ^{+}, ℂ). In [D-S-W], this problem is considered in the Hilbert space L^{2}(ℝ_{+}, ℂ) but the same proof is valid in the Banach space L^{1}(ℝ^{+}, ℂ). So the operator A,with domain {f ∈ W^{2,1}([0, ∞)) | f(0) = 0}, generates an analytic chaotic semigroup T (t) = e^{tA} in L^{1}(ℝ^{+}, ℂ), provided that a, b, c > 0 and c < \frac{b^{2}}{2a} < 1.

If in the above example we replace df/dx by its finite difference approximation [δf]_{j} = (f_{j} − f_{j−1})/h_{n} and d^{2}f/dx^{2} by

[δ^{2}f]_{j} = \frac{f_{j+1} − 2f_{j} + f_{j−1}}{h_{n}^{2}} ,

then the generator A of Example 2.4 becomes

[Af]_{j} = a\frac{f_{j+1} − 2f_{j} + f_{j−1}}{h_{n}^{2}} + b\frac{f_{j} − f_{j−1}}{h_{n}} + cf_{j}.

By taking h_{n} = a/b one gets [Af]_{j} = (−\frac{b^{2}}{a} + c) f_{j} + (\frac{b^{2}}{a} ) f_{j+1} and this is the generator

of Example 2.1, namely Af := −αI + βB with α = \frac{b^{2}}{a} − c and β = b^{2}/a. Since c < b^{2}/2a, this implies 0 < α < β, which is a sufficient condition for e^{tA} in Example 2.1 to be chaotic. By disregarding the fact that the underlying spaces are different in these examples, the condition h_{n} = a/b answers the question (Q2). In Section 4 we construct the precise spaces to answer this question.

3. Approximation in the sense of Kato

Definition 3.1. We say that a sequence of Banach spaces {(X_{n}, ∥.∥_{n}) : n = 1, 2, ⋯ } converges to a Banach space (X, ∥.∥) in the sense of Kato and we write

X_{n} ⟶^{K} X

if for any n there is a linear operator P_{n} ∈ L(X, X_{n}) (called an approximating operator) satisfying the following two conditions: (K1) lim_{n→∞} ∥P_{n}f∥_{n} = ∥f∥ for any f ∈ X; (K2) for any f_{n} ∈ X_{n}, there exists f^{(n)} ∈ X such that f_{n} = P_{n}f^{(n)} with ∥f^{(n)}∥ ≤

C∥f_{n}∥_{n} (C is independent of n).

Definition 3.2. Let X_{n} ⟶^{K} X, B_{n} ∈ L(X_{n}) and B ∈ L(X). We say that B_{n} converges to B in the sense of Kato and we write B_{n} ⟶^{K} B if lim_{n→∞} ∥B_{n}P_{n}f − P_{n}Bf∥_{n} = 0 for any f ∈ X. Let A_{n} and A be the generators of the C_{0}–semigroups {T_{n}(t)}_{t≥0} ⊆ L(X_{n}) and {T (t)}_{t≥0} ⊆ L(X), respectively. Consider the following three conditions:

(A) (Consistency). There is a complex number λ contained in the resolvent sets ⋂_{n∈ℕ} ρ(A_{n}) and ρ(A), respectively, such that

(λ − A_{n})^{−1} ⟶^{K} (λ − A)^{−1}.

(B) (Stability). There exists a positive constant M and a real number ω such that

∥T_{n}(t)∥ ≤ Me^{ωt}, for any t ≥ 0 and for any n ∈ ℕ.

(C) (Convergence). For any finite T > 0

T_{n}(t) ⟶^{K} T (t)

uniformly on [0, T], i.e.

lim_{n→∞} sup_{t∈[0,T]} ∥T_{n}(t)P_{n}f − P_{n}T (t)f∥_{n} = 0 for any f ∈ X.  (3.1)

In [Ush] one can retrieve the standard version of the Lax equivalence theorem which says that the conditions (A) and (B) hold if and only if (C) holds.

The following Theorem gives a negative answer to question (Q1).

Theorem 3.3. There exists a sequence of chaotic semigroups which converges in the sense of Kato to a non-chaotic semigroup.

Proof. By using the semigroup of Example 2.1, one can construct such a sequence. Let X := ℓ^{1} and A := −αI + βB. From Lemma 2.3 the C_{0}-semigroup T (t) = e^{tA} is chaotic for β > α > 0 and non-chaotic for β = α > 0. Hence, if {β_{n}} is a sequence such that β_{n} > α and β_{n} → α, then the sequence of chaotic semigroups {T_{n}(t)}_{t≥0} converges uniformly to the semigroup {T (t)}_{t≥0}. By taking X_{n} := X and P_{n} := I the identity operator, the sequence {T_{n}(t)}_{t≥0} converges also in the sense of Kato to {T (t)}_{t≥0}, which is not chaotic. □ Theorem 3.4. Let {(X_{n}, ∥.∥_{n}) | n = 1, 2, ⋯ } be a sequence of Banach spaces such that X_{n} ⟶^{K} X. Suppose {T (t)}_{t≥0} is a chaotic semigroup on (X, ∥.∥) and on each (X_{n}, ∥.∥_{n}) one defines a C_{0}-semigroup {T_{n}(t)}_{t≥0}. Now, if for some n ∈ ℕ one has

P_{n}T (t)f = T_{n}(t)P_{n}f  for any f ∈ X and for any t ≥ 0,  (3.2)

then {T_{n}(t)}_{t≥0} is also chaotic.

Proof. From the hypercyclicity of {T (t)}_{t≥0} it follows that there exists an f^{∗} ∈ X such that {T (t)f^{∗} | t ≥ 0} is dense in X. For the hypercyclicity of {T_{n}(t)}_{t≥0}, take g_{n} ∈ X_{n}; from (K2) there exists g^{(n)} ∈ X such that g_{n} = P_{n}g^{(n)}. Then for any ϵ > 0 there exists t > 0 such that ∥g^{(n)} − T (t)f^{∗}∥ < ϵ. The assumption (K1) implies the uniform boundedness of {P_{n}}, hence for f_{n}^{∗} := P_{n}f^{∗} we have

∥g_{n} − T_{n}(t)f_{n}^{∗}∥_{n} = ∥g_{n} − T_{n}(t)P_{n}f^{∗}∥_{n} = ∥P_{n}(g^{(n)} − T (t)f^{∗})∥_{n}
≤ M∥g^{(n)} − T (t)f^{∗}∥ ≤ Mϵ.

To prove (P) for T_{n}, take g_{n} ∈ X_{n} and denote g^{(n)} ∈ X such that g_{n} = P_{n}g^{(n)}. For ϵ > 0 we take f^{(p)} ∈ X_{per} such that ∥g^{(n)} − f^{(p)}∥ < ϵ, then for f_{n}^{(p)} = P_{n}f^{(p)} we have

∥g_{n} − f_{n}^{(p)}∥_{n} = ∥P_{n}(g^{(n)} − f^{(p)})∥_{n} ≤ Mϵ.

To finish the proof it suffices to see that f_{n}^{(p)} is a periodic vector for T_{n}. Indeed,

T_{n}(t)f_{n}^{(p)} = T_{n}(t)P_{n}f^{(p)} = P_{n}T (t)f^{(p)} = P_{n}f^{(p)} = f_{n}^{(p)}.
□

To prove the converse we have to impose extra conditions on P_{n}. Let us denote by (X_{n})_{per} the set of all periodic vectors of T_{n} in X_{n} and for any constant C > 0 let us define

C(f_{n}) := {f^{(n)} ∈ X | P_{n}f^{(n)} = f_{n} with ∥f^{(n)}∥ ≤ C∥f_{n}∥_{n}}.

Theorem 3.5. Suppose that (3.2) holds for some n ∈ ℕ. Suppose {T_{n}(t)}_{t≥0} is chaotic and P_{n} satisfies (X_{n})_{per} ⊆ P_{n} (X_{per}). If there exists a constant C such that for every f ∈ X and ϵ > 0 there is an f^{(n)} ∈ C(P_{n}f) with ∥f − f^{(n)}∥ < ϵ, then {T (t)}_{t≥0} is also chaotic.

Proof. Let us prove that condition (H) holds for {T (t)}_{t≥0}. Since {T_{n}(t)}_{t≥0} is hypercyclic, there exists f_{n}^{∗} ∈ X_{n} such that the orbit {T_{n}(t)f_{n}^{∗} | t ≥ 0} is dense in X_{n}. Let g ∈ X and g_{n} = P_{n}g. For any ϵ > 0 there exists t > 0 such that ∥g_{n} − T_{n}(t)f_{n}^{∗}∥_{n} < ϵ. According to (K2), let f_{∗}^{(n)} ∈ X be such that f_{n}^{∗} = P_{n}f_{∗}^{(n)}. Then for h = g − T (t)f_{∗}^{(n)} there exists h^{(n)} ∈ C(P_{n}h), with ∥h−h^{(n)}∥ < ϵ and P_{n}h^{(n)} = P_{n}h = g_{n} −P_{n}T (t)f_{∗}^{(n)}. As a consequence of (3.2), we obtain that g_{n} − P_{n}T (t)f_{∗}^{(n)} = g_{n} − T_{n}(t)f_{n}^{∗}. Thus the inequality

∥g − T (t)f_{∗}^{(n)}∥ ≤ ∥h − h^{(n)}∥ + ∥h^{(n)}∥
≤ ϵ + C∥g_{n} − T_{n}(t)f_{n}^{∗}∥_{n}
≤ (1 + C)ϵ  (3.3)

implies the hypercyclicity of T (t). To prove (P) for T (t), take g ∈ X and g_{n} = P_{n}g. For ϵ > 0 we choose f_{n}^{(p)} ∈ (X_{n})_{per} such that ∥g_{n} − f_{n}^{(p)}∥_{n} < ϵ. The bijectivity of P_{n} on X_{per} implies that we may choose f^{(p)} ∈ X_{per} such that f_{n}^{(p)} = P_{n}f^{(p)}. By taking h = g − f^{(p)} and h^{(n)} ∈ C(P_{n}h), an inequality similar to (3.3) implies the Theorem. □

4. Chaoticity of the discrete parabolic equation

In this section we construct approximating operators for X := L^{1}([0, ∞)) in the sense of Kato. To this end, we define for each n ∈ ℕ the step size h_{n} such that h_{n} → 0 as n → ∞. The corresponding space would be the space of step functions generated by the characteristic functions χ_{j}^{n} := χ_{[(j−1)h_{n},jh_{n})} of the intervals [(j − 1)h_{n}, jh_{n}). That is, any function f_{n} ∈ X_{n} can be expressed by

f_{n} = ∑_{j=1}^{∞} a_{j}^{n}χ_{j}^{n}.  (4.1)

Each X_{n} is endowed with the norm induced by X. Hence, if f_{n} ∈ X_{n} is given by (4.1), then ∥f_{n}∥ := h_{n} ∑_{j=1}^{∞} |a_{j}^{n}|. Now we define P_{n} as a map from X into X_{n} by

f_{n} = P_{n}f = ∑_{j=1}^{∞} f(jh_{n})χ_{j}^{n}.

Riemann’s sum theorem implies that

lim_{n→∞} ∥P_{n}f∥_{n} = lim_{h_{n}→0} h_{n} ∑_{j=1}^{∞} |f(jh_{n})| = ∥f∥.

Hence we have (K1); for (K2) it suffices to remark that the injection of X_{n} into X is isometric. It follows that the operator P_{n} is an approximating operator for X.

Now let us consider f ∈ X, f_{n} := P_{n}f and define the operator A_{n} by

A_{n}f_{n} := β ∑_{j=1}^{∞} f((j + 1)h_{n})χ_{j}^{n} − α ∑_{j=1}^{∞} f(jh_{n})χ_{j}^{n},

with D(A_{n}) = X_{n}.

Lemma 4.1. For β > α ≥ 0 the operator A_{n} generates a chaotic semigroup, e^{tA_{n}}, on X_{n}.

Proof. One can repeat the proof of Example 2.1, by taking h_{μ} = ∑_{j=1}^{∞} μ^{j}χ_{j}^{n} as the eigenfunction of A_{n}. □

Here we want to define an operator A such that

P_{n}Af = A_{n}P_{n}f  for any f ∈ D(A).  (4.2)

Actually the operator A is not exactly aD^{2} +bD+cI of Example 2.4, but a discrete version thereof, namely

A_{h} := a ((\frac{1}{h}χ_{h/2}) ∗ D)^{2} + b (\frac{1}{h}χ_{h} ∗ D) + cI,  (4.3)

where χ_{h} and χ_{h/2} are the characteristic functions of [0, h) and [−\frac{h}{2}, \frac{h}{2}).

Lemma 4.2. For h = h_{n} = a/b, α = (\frac{b^{2}}{a} − c), and β = \frac{b^{2}}{a}, we have

P_{n}A_{h}f = A_{n}P_{n}f.  (4.4)

Proof. We can write

[\frac{1}{h}χ_{h} ∗ D]f(x) = \frac{1}{h} ∫_{0}^{h} f′(x − y)dy = [\frac{f(x) − f(x − h)}{h}]  (4.5)

and for h = h_{n}

P_{n}[\frac{1}{h}χ_{h} ∗ D]f(x) = \frac{1}{h} ∑_{j=1}^{∞}[f(jh) − f((j − 1)h)]χ_{j}^{n}(x).

In the same manner

[\frac{1}{h}χ_{\frac{h}{2}} ∗ D]^{2}f(x) = \frac{f(x + h) − 2f(x) + f(x − h)}{h^{2}}  (4.6)

and

P_{n}[(\frac{1}{h}χ_{h/2} ∗ D)^{2}]f(x) = \frac{1}{h^{2}} ∑_{j=1}^{∞}[f((j + 1)h) − 2f(jh) + f((j − 1)h)]χ_{j}^{n}(x).

Thus, if A_{h} := a ((\frac{1}{h}χ_{h/2}) ∗ D)^{2} + b (\frac{1}{h}χ_{h} ∗ D) + cI, we obtain (4.4) provided that h = a/b, α = (\frac{b^{2}}{a} − c) and β = \frac{b^{2}}{a}. □

Lemma 4.3. Suppose that e^{tA} and e^{tA_{n}} are the C_{0}-semigroups generated by A and A_{n}, which are related by relation (4.2). Then

P_{n}e^{tA}f = e^{tA_{n}}P_{n}f  for any f ∈ X.  (4.7)

Proof. It is sufficient to differentiate, at least for f ∈ D(A):

\frac{d}{dt} [P_{n}e^{tA}f] = P_{n} \frac{d}{dt} [e^{tA}f] = P_{n}Ae^{tA}f = A_{n} [P_{n}e^{tA}f] ,

for all t ≥ 0, and identify

[P_{n}e^{tA}f] |_{t=0} = P_{n}f,

so that [P_{n}e^{tA}f] and e^{tA_{n}}P_{n}f are both the mild solutions of the Cauchy problem

\{ \begin{matrix}\frac{d}{dt}u = A_{n}u; \\ u(0) = P_{n}f. \end{matrix}

Since D(A) is dense, (4.7) is true for all f ∈ X. □

From (4.5) and (4.6) it follows that A_{h} is a bounded operator on L^{1}([0, ∞)). Consequently it generates a uniformly continuous semigroup e^{tA_{h}}. Furthermore, if f is a step function, then A_{h}f is also a step function hence the subspace X_{n} is invariant under A_{h} and e^{tA_{h}}. Then we can prove:

Theorem 4.4. For h = a/b, e^{tA_{h}} is a chaotic semigroup on X_{n}.

Proof. For the proof we will use Theorem 3.5. Lemma 4.3 implies (3.2) and Lemma 4.1 asserts that e^{tA_{n}} is chaotic. To see that P_{n} is a bijective mapping between X_{per} and (X_{n})_{per}, one takes f ∈ X_{per} and writes e^{tA_{n}}P_{n}f = P_{n}e^{tA_{h}}f = P_{n}f. Thus, P_{n}f ∈ (X_{n})_{per}. Conversely, if f_{n} ∈ (X_{n})_{per} by considering f_{n} as an element of X (since X_{n} is a subset of X) we have

P_{n}e^{tA_{h}}f_{n} = e^{tA_{n}}P_{n}f_{n} = e^{tA_{n}}f_{n} = f_{n},

since P_{n} acts as the identity on X_{n}. On the other hand, f_{n} ∈ X_{n} implies e^{tA_{h}}f_{n} ∈ X_{n}, thus P_{n}e^{tA_{h}}f_{n} = e^{tA_{h}}f_{n} and consequently e^{tA_{h}}f_{n} = f_{n}.

Finally, for given f ∈ X_{n}, any function g ∈ X_{n} which satisfies |g| ≤ |P_{n}f| and interpolates f on the points {(jh, f(jh)), j = 1, 2, ⋯ } belongs to C(P_{n}f) with C = 1. Thus for any ϵ > 0 we can find an interpolating function f^{(n)} with the additional condition ∥f^{(n)} − f∥ < ϵ. This concludes the theorem. □

Acknowledgements

We are extremely indebted to the referees who read the paper with unusual attention and care for detail. They made numerous valuable suggestions to improve the general style and the occasional casualness of the first version of the manuscript.

V. P. was supported in part by the U. S. Department of Energy, Office of Basic Energy Sciences, under contract no. DE-AC05-96OR22464 with Lockheed Martin Energy Research Corporation.

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