On reducibility of linear quasiperiodic systems with bounded solutions ^{1}

Victor I. Tkachenko

Institute of Mathematics National Academy of Sciences of Ukraine Tereshchenkivs’ka str. 3, Kiev, Ukraine e-mail: vitk@imath.kiev.ua

Abstract

It is proved that nonreducible systems form a dense G_{δ} subset in the space of systems of linear differential equations with quasiperiodic skew-symmetric matrices and fix frequency module. There exists an open set of nonreducible systems in this space.

AMS Subject Classification: 34C27, 34A30

1. Introduction.

We consider a linear quasiperiodic system of differential equations

\frac{dx}{dt} = A(φ · t)x,  (1)

where x ∈ ℝ^{n}, φ ∈ 𝕋_{m}, 𝕋_{m} = ℝ^{m}/2πℤ^{m} is an m-dimensional torus, A(φ) is a continuous function 𝕋_{m} → o(n), o(n) is the set of nth-order skew-symmetric matrices, φ · t denotes an irrational twist flow on the torus 𝕋_{m}

φ · t = ωt + φ, φ ∈ 𝕋_{m}.  (2)

ω = (ω_{1}, ..., ω_{m}) is a constant vector with rationally independent coordinates.

Fix a flow (2) on the torus 𝕋_{m} and consider a set of all quasiperiodic systems (1) with continuous skew-symmetric matrices A(φ). The distance between two systems (1) is introduced in terms of uniform norm of matrixfunctions A(φ) on the torus. The aim of this paper is to prove that C^{0}-generic subsets of systems (1) are not reducible to systems with constant matrices.

1This paper is in final form and no version of it will be submitted for publication elsewhere.

There exists an open subset of nonreducible systems in the space of systems (1).

We note that near system (1) on 𝕋_{m} × ℝ^{3} with constant coefficients the reducible and uniquely ergodic (end, hence, nonreducible) analytic systems are mixed: analytic nonreducible systems form a dense G_{δ} set but there is a dense set of reducible systems (see [2], [3], [5]).

2. Main results.

For x = (x_{1}, …, x_{n}) ∈ ℝ^{n} we define the norm ∥x∥ = (∑_{j=1}^{n} x_{j}^{2})^{1/2}. The corresponding norm ∥A∥ for n-dimensional matrix A is defined as follows: ∥A∥ = sup{∥Ax∥, x ∈ ℝ^{n}, ∥x∥ = 1}. Thus ∥Ax∥ = 1 for A ∈ SO(n), where SO(n) is the set of all orthogonal matrices of dimension n with determinant equal to 1.

Let C(𝕋_{m}, o(n)) be the space of continuous functions on the torus 𝕋_{m} with values in the group o(n). In the space C(𝕋_{m}, o(n)), we introduce the ordinary norm

∥a(φ)∥_{0} = sup_{φ∈𝕋_{m}} ∥a(φ)∥.

Let Φ(t, φ), Φ(0, φ) = I(I is the identity matrix) be the fundamental solution for system (1). It forms a cocycle

Φ(t_{1} + t_{2}, φ) = Φ(t_{2}, φ · t_{1})Φ(t_{1}, φ).  (3)

If A(φ) ∈ o(n) then Φ(t, φ) ∈ SO(n).

Define a quasiperiodic skew-product flow on 𝕋_{m} × SO(n) as follows

(φ, X) · t = (φ · t, Φ(t, φ)X), t ∈ ℝ,  (4)

where (φ, X) ∈ 𝕋_{m} × SO(n).

Let X = cls{(φ_{0}, I) · t : t ∈ ℝ} ⊆ 𝕋_{m} × SO(n) be the closure of the trajectory with the initial point (φ_{0}, I) of the flow (4). The set X is minimal and distal. Let π be the projector onto the first component, i.e., π : X → 𝕋_{m}. As shown in [1], π^{−1}(φ_{0}) forms a compact group. Denote it by G. For all φ ∈ 𝕋_{m}, the preimage π^{−1}(φ) is the uniform space of the group G ⊆ SO(n).

Definition 1 System (1) is said to be reducible if there is a linear change of variables x = P (φ)y that transforms (1) to a system with a constant matrix where P (φ) is a continuous map P : 𝕋_{m} → SO(n) the map t → P (φ · t) : ℝ → SO(n) is continuously differentiable and φ → (d/dt)P (φ · t)|_{t=0} : 𝕋_{m} → SO(n) is continuous.

The flow (4) preserves the product Haar measure μ × ν on 𝕋_{m} × SO(n). It preserves distances both in the 𝕋_{m}-direction and in the SO(n)-direction but it is not isometry. In the reducible case this measure is not ergodic and there are invariant measures supported on each invariant torus.

Theorem 1 The functions A(φ) corresponding to systems (1) whose trajectories (φ, I) · t is dense in 𝕋_{m} × SO(n) form a dense G_{δ} subset of the space C(𝕋_{m}, o(n)). These systems have unique invariant measure and are not reducible.

The proof of theorem is preceded by the two lemmas.

Lemma 1 [10] Suppose that a continuous function A(φ) : 𝕋_{m} → SO(n) satisfies

sup_{φ∈𝕋_{m}} ∥A(φ) − I∥ ≤ ε ≤ \frac{1}{2}.  (5)

Then there exists a real continuous logarithm of the function A(φ) defined on the torus T_{m} such that

sup_{φ∈𝕋_{m}} ∥ ln A(φ)∥ ≤ \frac{4\sqrt{2}ε}{1 − 2ε}.  (6)

Lemma 2 Assume that the mapping F (t, φ) : [0, 1] × 𝕋_{m} → SO(n) is continuous in t and φ, continuously differentiable with respect to t, and such that F (0, φ) = I and F (1, φ) = a(φ). Then, for ε ≤ α < 1/2 and b(φ) ∈ C(𝕋_{m}, SO(n)), such that ∥a(φ) − b(φ)∥_{0} < ε there exists a mapping G(t, φ) : [0, 1] × 𝕋_{m} → SO(n) continuous in t and φ, continuously differentiable with respect to t, and such that G(0, φ) = I, G(1, φ) = b(φ), and

∥G(t, φ) − F (t, φ)∥_{0} < ε,  ∥\frac{∂G(t, φ)}{∂t} − \frac{∂F (t, φ)}{∂t}∥_{0}< Kε,  (7)

where K > 0 is a constant depending only on the function F (t, φ).

Proof. We have

∥a^{∗}(φ)b(φ) − I∥_{0} ≤ ∥a^{∗}(φ)∥_{0} · ∥a(φ) − b(φ)∥_{0} < ε.

For sufficiently small ε > 0, there exists a logarithm

ln(a^{∗}(φ)b(φ)) = \frac{1}{2πi} ∫_{Γ}(λI − a^{∗}(φ)b(φ))^{−1} ln λdλ,  (8)

continuous on the torus 𝕋_{m}; here, Γ is the boundary of a simply connected domain in the complex plane which contains the closure of the set of eigenvalues of the matrices a^{∗}(φ)b(φ), φ ∈ 𝕋_{m}, and does not contain zero. By lemma 1, we get

∥ ln(a^{∗}(φ)b(φ))∥_{0} ≤ \frac{4\sqrt{2}ε}{1 − 2ε}.

The function H(t, φ) = exp[t ln(a^{∗}(φ)b(φ))] realizes the homotopy of a^{∗}(φ)b(φ) to the identity matrix and

∥\frac{∂H(t, φ)}{∂t}∥_{0}≤ ∥H(φ)∥_{0} · ∥ ln(a^{∗}(φ)b(φ))∥_{0} ≤ \frac{4\sqrt{2}ε}{1 − 2ε}.

The required function G(t, φ) has the form G(t, φ) = F (t, φ)H(t, φ). Taking the last inequality into account, we arrive at estimates (7). Indeed,

∥G − F ∥_{0} ≤ ∥H − I∥_{0} < ε,

∥\frac{∂G}{∂t} − \frac{∂F}{∂t} ∥_{0}≤ ∥\frac{∂F}{∂t} (H − I)∥_{0}+∥F \frac{∂H}{∂t} ∥_{0}≤ ε(∥\frac{∂F}{∂t} ∥_{0}+8)= Kε.

The lemma is proved.

Proof of theorem 1. We consider the torus 𝕋_{m} as a product 𝕋_{m} = 𝕋_{m−1} × 𝕋_{1} of the (m − 1)-dimensional torus 𝕋_{m−1} and of the circle 𝕋_{1}. Then φ = (ψ, ξ), ψ ∈ 𝕋_{m−1}, ξ ∈ 𝕋_{1}, and Φ(t, φ) = Φ(t, ψ, ξ).

For an open nonempty subset W of the set 𝕋_{m−1} × SO(n), we consider a set

E(W) = {A(φ) : A(φ) ∈ C(𝕋_{m}, o(n)), O′(φ_{0}, A) ∩ W ≠ ⊘},

where O′(φ_{0}, A) is a trajectory of system (1) in 𝕋_{m} × SO(n) which passes through the point (φ_{0}, I) = (ψ_{0}, ξ_{0}, I).

Using continuous dependence of solutions for (1) on parameters, we get that E(W) is open in C(𝕋_{m}, o(n)).

Let us prove that this set is dense. We fix an arbitrary number ε > 0 and a continuous function A(φ) : 𝕋_{m} → o(n).

Equation (1) is associated with the linear discrete system

y_{n+1} = a(ψ · n)y_{n}, n ∈ ℤ,  (9)

where ψ ∈ 𝕋_{m−1}, a(ψ) = Φ_{A}(2π/ω_{m}, ψ, ξ_{0}), ψ · n = ω̄n + ψ, ω̄ = (ω_{1}, ..., ω_{m−1}).

Analogously to Lemma 2, [4], we prove that in uniform on 𝕋_{m−1} topology the ε-neighborhood of the function a(ψ) contains a function a_{1}(ψ) ∈ C(𝕋_{m−1}, SO(n)) such that a_{1}(ψ) ∈ E′(W), where

E′(W) = {a(ψ) : a(ψ) ∈ C(𝕋_{m−1}, SO(n)), O′(ψ_{0}, a) ∩ W ≠ ⊘},

where O′(φ_{0}, a)) is a trajectory of system (9) in 𝕋_{m−1} × SO(n) which passes through the point (ψ_{0}, I). For sufficiently small ε, the function a_{1}(ψ) is homotopic to a(ψ), and, hence, to the identity matrix.

Let us show that a_{1}(ψ) is associated with quasiperiodic system

\frac{dx}{dt} = A_{1}(φ · t)x,  (10)

such that the difference between the matrices A(φ) and A_{1}(φ) is small. For φ = (ψ, ξ_{0}), t ∈ [0, 2π/ω_{m}], the matrix function Φ_{A}(t, ψ, ξ_{0}) defines a homotopy of a(ψ) = Φ_{A}(2π/ω_{m}, ψ, ξ_{0}) to the identity matrix. Suppose that the function F (t, ψ) : [0, 2π/ω_{m}]×𝕋_{m−1} → SO(n) defines the homotopy of a_{1}(φ) to the identity matrix. By virtue of Lemma 2, the function F (t, ψ) can be chosen so that

∥F (t, ψ) − Φ_{A}(t, ψ)∥_{0} < ε, ∥\frac{∂F (t, ψ)}{∂t} − \frac{∂Φ_{A}(t, ψ, ξ_{0})}{∂t}∥_{0}< Kε

where K > 0 is a constant depending only on the right-hand side of system (1). The function F (t, ψ) is the solution of system (10) for φ = (ψ, ξ_{0}), t ∈ [0, 2π/ω_{m}]. For φ = (ψ, ξ_{0}), t ∈ ℝ, the solution of system (10) is given by the cocycle formula

Φ_{A_{1}}(t, φ) = Φ_{A_{1}}(t − [tω_{m}/2π], φ · [tω_{m}/2π])Φ_{A_{1}}([tω_{m}/2π], φ),

where [t] is the integer part of the number t. The map φ · [tω_{m}/2π] don’t change the m-coordinate of the point φ ∈ 𝕋_{m}. Therefore,

Φ_{A_{1}}(t − [tω_{m}/2π], φ · [tω_{m}/2π]) = F (t − [tω_{m}/2π], ψ · [tω_{m}/2π])

for φ = (ψ, ξ_{0}) and Φ_{A_{1}}([tω_{m}/2π], φ) can be found if we know the solution of the discrete equation x_{n+1} = a_{1}(ψ · n)x_{n}. For other values of φ ∈ 𝕋_{m}, the function Φ_{A_{1}}(t, φ) are determined by using the property of cocycle (3).

The matrix A_{1}(φ) in system (10) can be found by formula

A_{1}(φ) = \frac{∂F (t, ψ)}{∂t}F^{∗}(t, ψ),

where φ = (ψ, ξ_{0}), t ∈ [0, 2π/ω_{m}]. Note that, for each point φ ∈ 𝕋_{m}, there exists a mapping φ = (ψ, ξ_{0}) · t, t ∈ [0, 2π/ω_{m}]. Taking into account inequalities (7), we get

∥A(φ) − A_{1}(φ)∥_{0} < 3Kε.

Thus, the systems whose solutions pass through the set W form an open everywhere dense subset of the set of all systems (1).

For any δ > 0, there exists a bounded covering of the compact set 𝕋_{m−1} × SO(n) by open sets W_{i}^{0}, i = 1, ..., n_{0}, such that the diameter of each set W_{i}^{0} is less than δ. Then the set ⋂_{i=1}^{n_{0}} E(W_{i}^{0}) is open and dense in C(𝕋_{m}, o(n)). The elements B ∈ ⋂_{i=1}^{n_{0}} E(W_{i}^{0}) satisfy the condition that the trajectory of corresponding equation ẋ = B(φ · t)x passes through all balls with diameter 2δ in the space 𝕋_{m−1} × SO(n).

Consider a covering of the set 𝕋_{m−1} × SO(n) by open sets W_{i}^{1} with diameter δ/2. Assume that, for any W_{i}^{1} there exists a set W_{j}^{0} such that W_{i}^{1} ⊂ W_{j}^{0}, i = 1, ..., n_{1}.

By analogy, we consider the covering of the set 𝕋_{m−1} × SO(n) by the sets W_{i}^{j} with diameter δ/2^{j}. The elements A(φ) of the set E_{j} = ⋂_{i=1}^{n_{j}} E(W_{i}^{j}) the trajectory of corresponding equation ẋ = A(φ · t)x passes through all balls with diameter δ/2^{j−1} in the space 𝕋_{m−1} × SO(n). The sets E_{j} are open and dense in C(𝕋_{m}, o(n)) by construction.

The set F = ⋂_{i=0}^{∞} E_{i} is dense G_{δ} in C(𝕋_{m}, o(n)). The elements A(φ) ∈ F correspond to Eqn. (1) whose trajectories are dense in 𝕋_{m−1} × SO(n). For these equations, the set π^{−1}(φ) coincides with SO(n).

Therefore, the systems with G = SO(n) form a dense G_{δ} subset in the set of systems (1).

Now we prove that systems (1) with G = SO(n) are unique ergodic. We use ideas of [2].

Let f(φ, X) be a continuous function on 𝕋_{m} × SO(n) and let

g(φ, X) = lim_{T →∞} \frac{1}{T} ∫_{0}^{T} f(φ · t, Φ(t, φ)X)dt.  (11)

Since the product Haar measure μ × ν is invariant under the flow (4), the limit (11) exists for a. e. (φ, X) and is measurable. Since the flow preserves distances in the SO(n) direction, the function g(φ, X) exists for a.e. φ ∈ 𝕋_{m} and all X ∈ SO(n) and X → g(φ, X) is equicontinuous. Hence, for a.e. φ ∈ 𝕋_{m} and for all t ∈ ℝ and X ∈ SO(n) :

g(φ, X) = g(φ · t, Φ(t, φ)X).

It is easily seen that if φ · t_{n} → φ for all φ ∈ 𝕋_{m} then

g(φ · t_{n}, X) → g(φ, X)

for some subsequence {n = n_{i}}, for a.e. φ ∈ 𝕋_{m} and all X ∈ SO(n).

Let φ ∈ 𝕋_{m} be a point for which the function g(φ, X) exists. Since G = SO(n), for any B ∈ SO(n), there exists a sequence t_{n} such that φ · t_{n} → φ and Φ(t_{n}, φ) → B as n → ∞. Hence, for some subsequence {n = n_{i}},

g(φ, X) = lim_{n→∞} g(φ · t_{n}, Φ(t_{n}, φ)X) = g(φ, BX).

Therefore, g(φ, X) is independent of X for a.e. φ ∈ 𝕋_{m} and invariant under φ → φ · t.

Similarly to [2], p. 18, we prove that the system under consideration has a unique invariant Borel measure. This system is nonreducible.

Theorem 2 There exists a system (1) such that all the systems from some neighborhood of it (in topology C(𝕋_{m}, o(n))) have no nontrivial almost periodic solutions and, hence, they are not reducible.

Proof We consider system (1) in 𝕋_{2} × ℝ^{3}. Let φ = (θ, ψ) ∈ 𝕋_{2}. We consider the fundamental solution Φ(t, θ, ψ) with following properties:

i) Φ(t, θ, 0) = G(g(t), θ) for all θ ∈ 𝕋_{1} and t ∈ [0, 2π], where g : [0, 2π] → [0, 2π] is a continuously differentiable function which is zero near 0 and 2π near 2π. The function G(τ, θ) : [0, 2π] × 𝕋_{1} → SO(3) is continuously differentiable and

G(0, θ) = I, G(2π, θ) = ( \begin{matrix}cos θ & sin θ & 0 \\ − sin θ & cos θ & 0 \\ 0 & 0 & 1 \end{matrix} ) ,  (12)

ii) for all values of t, θ, and φ, the function Φ(t, θ, 0) is extended by cocycle formula (3).

The function Φ(t, θ, ψ) satisfies the following system of differential equations

\frac{dθ}{dt} = ω, \frac{dψ}{dt} = 1, \frac{dx}{dt} = B(θ, ψ)x,  (13)

where

B(θ, ψ) = \frac{∂Φ(0, θ, ψ)}{∂t}

and ω is irrational number.

System (13) has two invariant bundles γ_{1} and γ_{2} defined by projectors

P_{1} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} ) , P_{2} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} )  (14)

on the circle (θ, 0) of torus 𝕋_{2}. By construction, the bundle γ_{1} is nontrivial.

Let P (θ) = {p_{ij}(θ)}_{i,j=1}^{3} be another projector on the circle (θ, 0) defining an invariant bundle of the system (13). Rewrite the projector P (θ) in the form P (θ) = {p_{ij}(θ)}_{i,j=1}^{2}, where p_{11}, p_{12}, p_{21}, and p_{22} are 2 × 2, 2 × 1, 1 × 2, and 1 × 1 matrices, respectively. Then

G(2π, θ)P (θ) = P (θ + 2πω)G(2π, θ),  (15)

hence

( \begin{matrix}p_{11}(θ + 2πω) & p_{12}(θ + 2πω) \\ p_{21}(θ + 2πω) & p_{22}(θ + 2πω) \end{matrix} ) = ( \begin{matrix}O_{2}(θ)p_{11}(θ)O_{2}^{∗}(θ) & O_{2}(θ)p_{12}(θ) \\ p_{21}(θ)O_{2}^{∗}(θ) & p_{22}(θ) \end{matrix} ) ,

where

O_{2}(θ) = ( \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ) .

Since P (θ) is 2π-periodic, vector-functions p_{12}(θ) and p_{21}(θ) are identically zero. Otherwise, the equation

p(θ + 2πω) = O_{2}(θ)p(θ), p ∈ ℝ^{2},  (16)

has a nontrivial 2π-periodic solution p(θ). For periodic function p(θ), the sequence p_{k} = p(θ + 2πωk), k ∈ ℤ, is almost periodic (see [7], p. 198). By (16), p_{k} has the following explicit form

p_{k} = O_{2}(kθ + k(k + 1)ω/2)P (θ).

By definition of almost periodic sequence, for ε > 0, there exists a positive integer q = q(ε) such that ∥p_{k+q} − p_{k}∥ < ε for all k ∈ ℤ. Hence

∥O_{2}(qθ + \frac{q(q − 1)ω}{2} + \frac{(2q − 1)kω}{2} )−I∥=  (17)

= ∥p_{k}^{−1}(p_{k+q} − p_{k})∥ ≤ ε.

The set

\{\frac{(2q − 1)ωk}{2} (mod2π), k ∈ ℤ\}

is dense on the circle 𝕋_{1}, therefore we can select k_{0} such that

\frac{3π}{4} ≤ (qθ + \frac{q(q − 1)ω}{2} + \frac{k_{0}(2q − 1)ω}{2} )(mod2π) ≤ \frac{5π}{4} .

Then

∥O_{2}(qθ + \frac{q(q − 1)ω}{2} + \frac{k_{0}(2q − 1)ω}{2} )−I∥> 1.

Choosing ε < 1 we have contradiction. Therefore, system (13) has only invariant bundles γ_{1} and γ_{2}.

Fix ε > 0. We consider another system in 𝕋_{2} × ℝ^{3}

\frac{dθ}{dt} = ω, \frac{dψ}{dt} = 1, \frac{dx}{dt} = B̃(θ, ψ)x,  (18)

which fundamental solution Φ̃(t, θ, ψ) satisfies condition

∥Φ̃(t, θ, ψ) − Φ(t, θ, ψ)∥ ≤ ε  (19)

for t ∈ [0, 2π], (θ, ψ) ∈ 𝕋_{2}. Then

Φ̃(2π, θ, 0) = ( \begin{matrix}cos θ & sin θ & 0 \\ − sin θ & cos θ & 0 \\ 0 & 0 & 1 \end{matrix} ) (I + U),  (20)

where ∥U∥ ≤ ε.

We consider a projector P̃(θ) on the circle (θ, 0) defining an invariant bundle γ̃ of system (18). Denote P̃(θ) = {p̃_{ij}(θ)}_{i,j=1}^{2}, where p̃_{11}, p̃_{12}, p̃_{21}, and p̃_{22} are 2 × 2, 2 × 1, 1 × 2, and 1 × 1 matrices, respectively.

The projector P̃(θ) satisfies the equation

P̃(θ + 2πω) = Φ̃(t, θ, 0)P̃(θ)Φ̃(t, θ, 0).  (21)

Using (19), we get

∥P̃(θ + ω) − G(2π, θ)P̃(θ)G^{∗}(2π, θ)∥ < 2pε,  (22)

where p = sup ∥P̃(θ)∥. Hence

∥ ( \begin{matrix}p̃_{11}(θ + 2πω) & p̃_{12}(θ + 2πω) \\ p̃_{21}(θ + 2πω) & p̃_{22}(θ + 2πω) \end{matrix} ) −

− ( \begin{matrix}O_{2}(θ)p̃_{11}(θ)O_{2}^{∗}(θ) & O_{2}(θ)p̃_{12}(θ) \\ p̃_{21}(θ)O_{2}^{∗}(θ) & p̃_{22}(θ) \end{matrix} ) ∥< 2pε,  (23)

There exists ε_{0} > 0 such that the bundle γ̃ is homotopic to the bundle γ_{i} if ∥P̃(θ) − P_{i}∥ < ε_{0}, i = 1, 2.

It can be shown that inequality ∥p(θ + 2πω) − O_{2}(θ)p(θ)∥ ≤ ε for continuous 2π-periodic vector-function p(θ) implies ∥p(θ)∥ ≤ a_{0}ε, θ ∈ [0, 2π], where constant a_{0} is independent from ε.

Therefore, by (23), we get ∥p_{12}(θ)∥ < 2pa_{0}ε, ∥p_{21}(θ)∥ < 2pa_{0}ε. Hence, for ε > 0 satisfying inequality 2pa_{0}ε < ε_{0}, the bundle γ̃ is homotopic to the bundle γ_{1} or to the bundle γ_{2} in accordance with rankP̃(θ) = 2 or rankP̃(θ) = 1 for all θ ∈ [0, 2π]. If P (θ) defines an invariant bundle then I − P (θ) defines an invariant bundle too. Therefore, if system (18) has an invariant bundle, it has an invariant bundle of rank 2. By proved above, this bundle is homotopic to the nontrivial bundle γ_{1} and, hence, is nontrivial.

If system (18) has almost periodic fundamental solution and, hence, it is reducible then the space 𝕋_{2}×ℝ^{3} is the Whitney sum of three one-dimensional trivial invariant bundles over the torus 𝕋_{2} or the Whitney sum of onedimensional and two-dimensional trivial invariant bundles over the torus 𝕋_{2} [6], [9]. Therefore, system (18) is not reducible if ε < ε_{0}/2pa_{0}. The theorem is proved.

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