ON INTEGRAL SEPARATION OF BOUNDED LINEAR RANDOM DIFFERENTIAL EQUATIONS

. Our aim in this paper is to investigate the openness and dense-ness for the set of integrally separated systems in the space of bounded linear random diﬀerential equations equipped with the L ∞ -metric. We show that in the general case, the set of integrally separated systems is open and dense. An exception is the case when the base space is isomorphic to the ergodic rotation ﬂow of the unit circle, in which the set of integrally separated systems is open but not dense.


Introduction. A continuous-time linear deterministic differential equation is of the formẋ
(1) The Lyapunov spectrum of (1) is the set of real part of eigenvalues of A. It is well-known that the eigenvalues depend continuously on A (see Kato [18]). Hence, the linear deterministic differential equations with simple Lyapunov spectrum, i.e. the spectrum has exact d distinct exponents, form an open (but not dense) set in the space of all systems. In the case that (1) has simple Lyapunov spectrum, the state space R d can be decomposed into a direct sum of one dimensional subspaces R d = E 1 ⊕ E 2 ⊕ · · · ⊕ E d with the property that the asymptotical behavior of solutions of (1) starting from subspaces E i are uniformly separated in the sense that there exist positive numbers K, ε such that for i < j with i, j ∈ {1, . . . , d} and t ≥ 0 A natural extension of the above analyzed uniform separation of asymptotical dynamics of autonomous systems to nonautonomous systems leads to a notion of integral separation. In [20], Millionshchikov showed that in the space of bounded piecewise-constant linear differential equations equipped with C 0 -topology, the set of systems being integrally separated is dense and open. Later, the notion of integral separation and exponential dichotomy for shifted equations give rise to an adequate spectral theory for nonautonomous linear differential equations, see Sacker & Sell [22], Palmer [21], Siegmund [23] and Kloeden & Rasmussen [19]. The objects of interest in this paper are bounded linear random differential equations, i.e. a special class of nonautonomous linear differential equations in which the base space consists of randomness and is modelled by an ergodic flow on a probability space. Our main aim is to answer the question how often bounded linear random differential equations exhibit an integral separation. Here "how often" (and later on "generic") is understood in the sense of Baire category theory for topological spaces. This question is motivated from some well-developed research areas: Firstly, we mention some research achievements on nonautonomous differential equations whose base spaces are modelled as the Kronecker flow associated with an irrationally dependent frequency on a torus. It was showed in [14,15,16] that there is a Baire residual set of frequencies with the property that for a fixed frequency in this set, the set of bounded traceless linear systems having an exponential dichotomy is C 0 -dense. An improvement of this result was obtained in Avila, Bochi & Damanik [6] which showed that for an arbitrary irrational independent frequency the set of bounded traceless linear systems having an exponential dichotomy is C 0 -dense, see also Fabbri, Johnson & Zampogni [17].
Secondly, in the discrete-time case, i.e. product of random matrices, it was proved in Cong [11] that the set of integrally separated linear cocycle is open and dense in the space of bounded linear cocycles equipped with L ∞ -metric. It is also pointed out in Cong & Doan [12] that this result cannot be extended to the space of linear cocycles satisfying integrability condition of the multiplicative ergodic theorem. In contrast to L ∞ -metric, Arbieto & Bochi [3] showed that L p -generic linear cocycles has one-point spectrum. As a consequence, the set of integrally separated linear cocycles is L p -meagre.
Concerning with continuous-time linear random differential equations, there have been a few contributions to the genericity of Lyapunov spectrum and uniform hyperbolicity. We mention here the work in Bessa [7] in which he showed that for a C 0 -generic subset of all the 2-dimensional conservative systems, either Lyapunov exponents are zero or there is a dominated splitting (an equivalent notion of integral separation). An extension of this result to arbitrary dimensional spaces was established in Bessa [8]. Recently, it has been established in Bessa & Vilarinho [9] that L p -generic linear random differential equations has one-point spectrum.
In this paper, our goal is to extend the results on genericity of integral separation from discrete-time to continuous time in L ∞ -metric. Note that coming from discrete-time setting to continuous-time setting, the main technical problem arose is the null-set property. For example, when we want to use the Millionshchikov perturbation method (see Bochi & Viana [10] and Cong [11]), we need to perturb a system along a specific orbit. For discrete-time systems, null-set property is not an issue because any countable union of null-sets will be again a null-set. However, for continuous-time systems, along any orbit the union of null-set can be no longer a null set. To get rid of this difficulty, we use Ambrose's representation theorem of ergodic flows and deal with the induced random dynamical systems. Depending on the structure of the base space in the representation theorem, we completely characterize the openness and denseness of the set of integrally separated systems.
The paper is organized as follows: In section 2, we introduce the space of linear bounded random differential equations and the spectral theory of these equations. Section 3 is devoted to our main result in this paper about denseness and openness 2. Linear bounded random differential equations. Let (Ω, F, P) be a Lebesgue probability space and (θ t ) t∈R be an ergodic flow from Ω into itself preserving the probability P (see Appendix for more details of the definition of ergodic flow). Suppose further that (θ t ) t∈R has no fixed point. The assumption of ergodicity of the flow is natural because one may easily use ergodic decomposition of Ω (see [13]) to reduce the problem to the ergodic case. Furthermore, we do not consider the fixed point of (θ t ) t∈R because this is the case of trivial dynamics and our problem reduces to the well-known problem of autonomous linear ordinary differential equations. Let It is well known that the metric space (L ∞ (Ω, R d×d ), ρ ∞ ) is complete, see e.g., Arnold & Cong [5]. For each A ∈ L ∞ (Ω, R d×d ), we consider the corresponding linear random differential equation of the following form (see Arnold [4 Let Φ A (t, ω)x denote the solution of (2) satisfying x(0) = x, i.e. Φ A (t, ω) satisfies the following integral equation where id denotes the identity matrix of size d. Thus, the linear mapping Φ A : R × Ω → R d×d is a continuous random dynamical system, i.e. Φ A is B(R) ⊗ F, B(R d×d ) measurable and the following properties hold: Lemma 2.1. Let A ∈ L ∞ (Ω, R d×d ) and define K := ess sup ω∈Ω A(ω) . Then, for any ε > 0 there exists a measurable set Ω ⊂ Ω (depending on ε) being of full measure and invariant under the flow (θ t ) t∈R , i.e. θ t Ω ⊂ Ω for all t ∈ R, such that Proof. Choose and fix an arbitrary ε > 0. Let ∈ Q >0 be an arbitrary positive rational number. Define the following set By continuity of Lebesgue integral, we have we derive that θ t M ⊂ M for all t ∈ R. Thus, by ergodicity of the flow (θ t ) t∈R , either P(M ) = 0 or P(M ) = 1 holds. We now prove that P(M ) = 1 by supposing the contrary that P(M ) = 0. Let δ := ε 2K > 0 and for each k ∈ Z we define Using analogous arguments as in the proof of measurability of M above, the set N k, is also measurable. By definition of M as in (5), where M c is the complement of M . Thus, P(M c ) = 1 implies that there exists k ∈ Z such that P(N k, ) > 0. Note that N k, is a subset of the following set N := ω ∈ Ω : By measurability of the maps (s, ω) → θ s ω and ω → A(ω), we have that the map (s, ω) → A(θ s ω) is also measurable. Thus, using Fubini's Theorem, we obtain that which together with (6) implies that From the fact that K = ess sup ω∈Ω A(ω) we derive that Since δ = ε 2K it follows that (δ + )K < (K + ε) . Thus, the preceding inequality provides a contradiction to (7). Therefore, P(M ) = 1 for all ∈ Q ≥0 . Hence, the measurable set Ω := ∈Q>0 M is of full measure and also invariant under all θ t for t ∈ R. To conclude the proof, we prove that Ω satisfies (4). For this purpose, let ω ∈ Ω and α, β ∈ R with α < β. Let (q n ) n∈N be a sequence of positive rational numbers such that lim n→∞ q n = β − α. Since ω ∈ Ω it follows that ω ∈ M qn for all n ∈ N. Therefore, Letting n → ∞ in the above inequality, we obtain that which completes the proof.
According to Lemma 2.1, replacing Ω by a set of full measure being invariant under (θ t ) t∈R , we can assume from now on that whereK > ess sup ω∈Ω A(ω) is a positive constant. In the following corollary, we establish some estimates on the norm of the solution Φ A (t, ω).

Corollary 1.
For all ω ∈ Ω and t ∈ R, we have whereK is a positive constant satisfying (8).
Proof. Choose and fix ω ∈ Ω. Let δ > 0 be arbitrary such thatKδ < 1. From (3), we obtain that which together with (8) implies that Note that the maximum max 0≤s≤δ Φ A (s, ω) exists due to continuity of the map Now let t > 0 be arbitrary. We choose n ∈ N such thatK t n < 1. By cocycle property we have Hence, using (9), we obtain Letting n → ∞ yields that Φ A (t, ω) ≤ exp(Kt). Similarly, for the case t < 0 we also obtain that Φ A (t, ω) ≤ exp(−Kt). The proof is complete.
3. Integral separation of bounded linear random differential equations. We start this section by extending the notion of integral separation of linear nonautonomous differential equations, see e.g., Adrianova [1, Definition 5.3.2], to linear random differential equations.
such that for i = 1, . . . , d the following assertions hold: Let R ⊂ L ∞ (Ω, R d×d ) be the set of bounded measurable matrices A such that the associated linear random differential equationẋ = A(θ t ω)x is integrally separated. In this section, we focus on the question: How big is the set R in the complete metric space (L ∞ (Ω, R d×d ), ρ ∞ )? The answer of this question depends on the structure of the base space (Ω, F, P, (θ t ) t∈R ). Before going to formulate this answer, we need the following preparation: Firstly, in the whole section we assume that (Ω, F, P) is a Lebesgue probability space (or Standard probability space) and (θ t ) t∈R is an ergodic flow from Ω into itself without fixed points and preserving the probability P. Hence, according to Theorem 4.2 in the Appendix, there exists a Lebesgue probability space (X, A, m) and an ergodic transformation T : X → X preserving the probability m and a measurable function f : X → R ≥0 which is bounded from 0 and ∞ such that the flow (S t ) t∈R built under a function (X, T, m, f ) is isomorphic to the flow (θ t ) t∈R , i.e. there exists a measure preserving bijective transformation H : for all ω ∈ Ω.
Since f is bounded from 0 and ∞, it follows that Finally, since (X, F, m) is a Lebesgue probability space, (X, F, m) is of one of the following cases: p i with p i is the probability of {x i } and k can be equal to ∞. Remark 1. Suppose that Case B holds. Since T is an invertible ergodic transformation on X, it follows that X has a finite atoms and from this we derive that X consists of only atoms. Hence, X is isomorphic to the set {x 1 , . . . , x k } with the probability of x i is 1 k . Consequently, the base space (Ω, F, P, (θ t ) t∈R ) is isomorphic to (S 1 , B(S 1 ), λ, (R t ) t∈R ), where S 1 := {e 2πiz : z ∈ [0, 1]} is the unit circle, λ is the standard Lebesgue probability on S 1 and R t : S 1 → S 1 is the rotation map defined by R t (e 2πiz ) = e 2πi(z+t) .

Case A.
Theorem 3.2. Suppose that Case A holds. Then, we have: As a consequence, the set of bounded linear random differential equations being integrally separated is generic.
Proof. (i) Let A ∈ L ∞ (Ω, R d×d ) and ε > 0 be arbitrary. Our aim is to construct A ε ∈ L ∞ (Ω, R d×d ) with ρ ∞ (A, A ε ) < ε such that the linear differential equatioṅ x = A ε (θ t ω)x is integrally separated. For this purpose, we define a measurable matrix-valued function M : X → R d×d by By Corollary 1, we have where c is defined as in (11). Thus, M and M −1 are essentially bounded. Let δ is a positive number satisfying that and the generated linear cocyle Φ M : Z × X → R d×d over the metric dynamical system (X, A, m, T ) is integrally separated. For any x ∈ X, we set Combining (13) and (15) yields that for m-a.e. x ∈ X Note that δ exp cK < 1 and thus R(t, x) is invertible and we have We divide the remaining of the proof of this assertion into two steps: Step 1. In this step, we construct A : Ω → R d×d such that ρ ∞ (A, A) < ε and For this purpose, we define for all (x, t) ∈ B Thus, Using (16) and (17), we obtain that which together with (14) gives that ρ ∞ (A, A) < ε. To conclude the proof of this step, let x ∈ X be arbitrary. By definition of A and (10), for any t ∈ [0, f (x)) the equationξ Consequently, By definition of R, equality (18) holds.
Step 2. In this step, we show that the linear random dynamical system Φ A (t, ω) is integrally separated. From (10) and (12), we derive that for any x ∈ X and n ∈ N Hence, from the fact that Φ M is integrally separated and f is strictly separated from 0 and ∞ we can easily deduce that the linear random dynamical system Φ A (t, ω) is also integrally separated. The proof of (i) is complete.
(ii) Let A ∈ R be arbitrary. Define a measurable matrix-valued function M : Since the linear random dynamical system Φ A is integrally separated and f is strictly separated from 0 and ∞ it follows that the discrete time linear random dynamical system Φ M generated by M over (X, A, m, T ) is also integrally separated. Note that for discrete time linear random dynamical systems, integral separation is a robust property, see e.g., Cong & Doan [12]. Thus, there exists ε < 1 such that for all M, satisfying that the generated discrete time linear random dynamical system Φ M is also integrally separated. Let A ∈ L ∞ (Ω, R d×d ) be arbitrary such that ρ ∞ (A, A) < ε, where ε < 1 is chosen later. Analog to Lemma 2.1, there existsK > 8 (independent of ε) and a set of full measure Ω being invariant under (θ t ) t∈R such that for all ω ∈ Ω, β > α we have which implies that Hence, by Corollary 1 we have Letting |t| = 1 2K and usingK > 8 yield that max Hence, using the cocycle property we obtain that for m-a.e. x ∈ X where c is given as in (11) and n is the smallest integer such that n 2K ≥ c. Choose ε := ε 2 exp(1+3cK) . Then for all A ∈ L ∞ (Ω, R d×d ) with ρ ∞ ( A, A) < ε the linear discrete-time random dynamical system generated by Φ A (f (x), H −1 (x, 0)) is integrally separated. Analog to proof of Step 2 of (i), the linear random dynamical system Φ A (t, ω) is also integrally separated. The proof is complete.

Case B.
Theorem 3.3. Suppose that Case B holds. Then, the set R is open but not generic in (L ∞ (Ω, R d×d ), ρ ∞ ).
Let ω 0 = e 2πi0 and define M := Φ A (1, ω 0 ). Thus, for any ω ∈ S 1 there exists a unique s ∈ [0, 1) such that ω = θ s ω 0 and therefore we have for all t ≥ 0 where n is the largest integer smaller or equal to t + s. Therefore, the generated linear random dynamical system Φ A is integrally separated if and only if the set of the logarithms of the moduli of eigenvalues of the matrix M has exactly d elements.
Due to the fact that eigenvalues depend continuously on the matrix, the set R is open. Also note that the multiplicity 2 of pairs of conjugate complex eigenvalues is also stable under small perturbation. Therefore, there exists an open set in (L ∞ (Ω, R d×d ), ρ ∞ ) which has no intersection with R. The proof is complete.

Appendix.
4.1. Representation of ergodic flow. Let (Ω, F, P) be a Lebesgue probability space and (θ t ) t∈R be an ergodic flow preserving the probability measure P, i.e. the map Θ : R × Ω → Ω, (t, ω) → θ t ω is (B(R) ⊗ F, F) measurable and the following properties hold: (i) θ 0 = id and θ t+s = θ t • θ s for t, s ∈ R, (ii) For all t ∈ R and U ∈ F, we have P(θ t U ) = P(U ),