Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 45, pp. 1–15. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.45 OSELEDETS DECOMPOSITION ON SUB SEMIFLOWS MAREK KRYSPIN Abstract. The existence of the Oseledets decomposition on continuously em- bedded subspaces of Banach spaces is proved in this paper. Natural assump- tions facilitating such transfer of the Oseledets decomposition are presented, notably conditions often met by dynamical systems generated by differential equations. Introduction The primary objective of this work is to delineate natural assumptions facilitat- ing the transfer of the Oseledets decomposition from a Banach space into another Banach space continuously embedded in the former. In general, the Oseledets-type decomposition implies the partitioning/splitting (in the form of a direct sum) of the fiber space, within which a dynamical system operates, into finite-dimensional sub- spaces, potentially an infinite many of them, each corresponding to a specific Lya- punov exponent. This process gives rise to a hierarchy of subspaces often referred to as the Oseledets filtration or flags. Lyapunov exponents hold paramount signifi- cance in the realm of systems dynamics, as they dictate the exponential asymptotic growth rate along trajectories. For results, see, e.g., [7, 9, 10, 11, 12, 16, 17, 26]. There are numerous papers that concentrate on the subject of dynamical systems generated by differential equations and various phase space decompositions; for instance [1, 4, 5, 6, 7, 8, 14, 20, 21, 22, 18, 19]. In many cases of differential equations there is no “natural” phase space. Nonetheless, it appears that there is a scarcity of research papers addressing the topics of regularization and the transfer of Oseledets-type decomposition. As examples, we can mention here, first, ordinary or partial differential equations with delay, as considered in [15, 23, 24], and, second, advection diffusion equations and others, as investigated in [2]. The paper [13] is also worth mentioning, where the authors discuss the possibility of transferring the Oseledets decomposition to dense subspaces of the fiber space with separable dual. The presuppositions of this paper, meaning the requirements set for the solv- ing operators and the fiber space, are often considerably weaker than the specific properties of these (for instance, compactness or separable dual are not necessarily required). On the other hand, such relaxation of assumptions may lead to new results. To be precise, decompositions in subspaces (in particular, subspaces with a 2020 Mathematics Subject Classification. 37H15, 34K06. Key words and phrases. Oseledets decomposition; random dynamical systems; random delay differential systems. ©2024. This work is licensed under a CC BY 4.0 license. Submitted July 4, 2024. Published August 15, 2024. 1 2 M. KRYSPIN EJDE-2024/45 finer/stronger topology are of particular interest here) of Banach spaces (in terms of scales and continuous embeddings) frequently encountered in real-world problems related to differential equations, may arise from this approach. To the author’s knowledge, there are no known theorems allowing the transfer of Oseledets decom- positions to subspaces without assuming the previously mentioned separability of the fiber space and/or its dual. To give a flavor of our results, we formulate now some specializations of our main results. It is well-known that a linear ordinary differential equation with delay generates a dynamical system that admits the Oseledets splitting in the fiber space Lp([−1, 0],RN )⊕RN . This follows from nice properties of fiber spaces, such as separability and reflexivity. It is also known that in practice, such equations possess a regularisation property that leads to continuous solutions. Pullback technique allows us to transfer the Oseledets decomposition to more regular spaces (with finer topology). 1. Preliminaries and definitions In this section, we will present definitions of measurable dynamical systems, measurable linear skew-product semidynamical systems, and the Oseledets decom- position. 1.1. Measurable dynamical systems. We write R+ for [0,∞). For a metric space S by B(S) we denote the σ-algebra of Borel subsets of S. A probability space is a triple (Ω,F,P), where Ω is a set, F is a σ-algebra of subsets of Ω, and P is a probability measure defined for all F ∈ F. We always assume that the measure P is complete. A measurable dynamical system on the probability space (Ω,F,P) is a (B(R)⊗ F,F)-measurable mapping θ : R× Ω → Ω such that • θ(0, ω) = ω for any ω ∈ Ω, • θ(t+ s, w) = θ(t, θ(s, ω)) for any ω ∈ Ω and t, s ∈ R+. We write θ(t, ω) as θtω. Also, we usually denote measurable dynamical systems by ((Ω,F,P), (θt)t∈R) or simply by (θt)t∈R. A metric dynamical system is a measurable dynamical system ((Ω,F,P), (θt)t∈R) such that for each t ∈ R the mapping θt : Ω → Ω is P-preserving (i.e., P(θ−1 t (F )) = P(F ) for any F ∈ F and t ∈ R). A subset Ω′ ⊂ Ω is invariant if θt(Ω ′) = Ω′ for all t ∈ R, and the metric dynamical system is said to be ergodic if for any invariant subset F ∈ F, either P(F ) = 1 or P(F ) = 0. Throughout the paper we will assume that P is ergodic. 1.2. Measurable linear skew-product semidynamical systems. By a mea- surable linear skew-product semidynamical system or semiflow, Φ = ((Uω(t))ω∈Ω,t∈R+ , (θt)t∈R) onX covering a metric dynamical system (θt)t∈R we understand a (B(R+)⊗ F⊗B(X),B(X))-measurable mapping [R+ × Ω×X ∋ (t, ω, u) 7→ Uω(t)u ∈ X] satisfying Uω(0) = IdX for each ω ∈ Ω, Uθsω(t) ◦ Uω(s) = Uω(t+ s) for each ω ∈ Ω and t, s ∈ R+, [X ∋ u 7→ Uω(t)u ∈ X] ∈ L(X) for each ω ∈ Ω and t ∈ R+. (1.1) EJDE-2024/45 OSELEDETS DECOMPOSITION 3 Equation (1.1) is called the cocycle property. By the positive semiorbit passing through (ω, u) ∈ Ω × X we understand a (B([0,∞)),B(X))-measurable mapping[ [0,∞) ∋ t 7→ Uω(t)u ∈ X ] . A negative semiorbit passing through (ω, u) ∈ Ω × X is a (B((−∞, 0]),B(X))- measurable mapping ũ : (−∞, 0] → X such that • ũ(0) = u; • ũ(s+ t) = Uθsω(t)ũ(s) for each s ≤ 0, t ≥ 0 such that s+ t ≤ 0. By a full or entire orbit passing through (ω, u) ∈ Ω ×X we understand a (B(R), B(X))-measurable mapping ũ : R → X such that • ũ(0) = u; • ũ(s+ t) = Uθsωũ(s) for each s ∈ R and t ≥ 0. From now on, we will focus on separable Banach spaces. Furthermore, based on the results from [17, Lemma 5.6 and Corollary 7.3], we will refrain from discussing measurability in the Grassmanian sense in favor of an equivalent definition of a mea- surable basis. Let Ω0 ∈ F. A family {E(ω)}ω∈Ω0 of l-dimensional vector subspaces of X is measurable if there are (F,B(X))-measurable functions v1, . . . , vl : Ω0 → X such that (v1(ω), . . . , vl(ω)) forms a basis of E(ω) for each ω ∈ Ω0. Let {E(ω)}ω∈Ω0 be a family of l-dimensional vector subspaces of X, and let {F (ω)}ω∈Ω0 be a family of l-codimensional closed vector subspaces of X, such that E(ω) ⊕ F (ω) = X for all ω ∈ Ω0. We define the family of projections associated with the decomposition E(ω)⊕F (ω) = X as {P (ω)}ω∈Ω0 , where P (ω) is the linear projection of X onto F (ω) along E(ω), for each ω ∈ Ω0. The family of projections associated with the decomposition E(ω)⊕ F (ω) = X is called strongly measurable if for each u ∈ X the mapping [Ω0 ∋ ω 7→ P (ω)u ∈ X] is (F,B(X))-measurable. We say that the decomposition E(ω) ⊕ F (ω) = X, with {E(ω)}ω∈Ω0 finite- dimensional, is invariant if Ω0 is invariant, Uω(t)E(ω) = E(θtω) and Uω(t)F (ω) ⊂ F (θtω), for each t ∈ R+. A strongly measurable family of projections associated with the invariant de- composition E(ω)⊕ F (ω) = X is referred to as tempered if lim t→±∞ ln ∥P (θtω)∥ t = 0 P-a.e. on Ω0. 2. Oseledets filtration and decomposition From now on we assume that for a given semiflow (E1) the functions[ Ω ∋ ω 7→ sup 0≤s≤1 ln+ ∥Uω(s)∥ ∈ R+ ] ∈ L1(Ω,F,P),[ Ω ∋ ω 7→ sup 0≤s≤1 ln+ ∥Uθsω(1− s)∥ ∈ R+ ] ∈ L1(Ω,F,P). Then it follows from the Kingman subadditive ergodic theorem that there exists λtop ∈ [−∞,∞) such that lim t→∞ ln ∥Uω(t)∥ t = λtop for P-a.e. ω ∈ Ω, which is referred to as the top Lyapunov exponent of Φ. 4 M. KRYSPIN EJDE-2024/45 (E2) λtop > −∞. Definition 2.1 (Oseledets decomposition). Φ admits an Oseledets decomposition if there exists an invariant subset Ω0 ⊂ Ω, P(Ω0) = 1, with the property that one of the following mutually exclusive cases, (O1) or (O2), holds: (O1) There are k real numbers λ1 = λtop > · · · > λk, called the Lyapunov exponents for Φ, k measurable families {E1(ω)}ω∈Ω0 , . . . , {Ek(ω)}ω∈Ω0 of finite dimensional vector subspaces, and a family {F∞(ω)}ω∈Ω0 of closed vector subspaces of finite codimension such that (i) for j = 1, . . . , k, any ω ∈ Ω0 and t ≥ 0 Uω(t)Ej(ω) = Ej(θtω) & Uω(t)F∞(ω) ⊂ F∞(θtω); (ii) E1(ω)⊕ · · · ⊕ Ek(ω)⊕ F∞(ω) = X for any ω ∈ Ω0; we write Fj(ω) := k⊕ m=j+1 Em(ω)⊕ F∞(ω) for j = 0, . . . , k. In particular, Fj(ω) = Ej+1(ω)⊕ Fj+1(ω) for j = 0, 1, . . . , k − 2; (iii) for j = 1, . . . , k, the families of projections associated with the decom- position ( j⊕ n=1 En(ω) ) ⊕ Fj(ω) = X is strongly measurable and tempered; (iv) for j = 1, . . . , k , any ω ∈ Ω0 and any nonzero u ∈ Ej(ω) lim t→∞ ln ∥Uω(t)u∥ t = λj ; (v) for j = 1, . . . , k and any ω ∈ Ω0, a nonzero u ∈ Fj−1(ω) belongs to Ej(ω) if and only if there exists a negative semiorbit ũ : (−∞, 0] → X passing through (ω, u) such that lim s→−∞ ln ∥ũ(s)∥ s = λj ; (vi) for any ω ∈ Ω0 lim t→∞ ln ∥Uω(t) ∣∣ F∞(ω) ∥ t = −∞. In this case, {F1(ω)}ω∈Ω0 , . . . , {Fk−1(ω)}ω∈Ω0 , {F∞(ω)}ω∈Ω0 is called the Oseledets filtration for Φ. (O2) There is a decreasing sequence of real numbers λ1 = λtop > · · · > λj > λj+1 > · · · with limit −∞, called the Lyapunov exponents for Φ, countably many measurable families {Ej(ω)}ω∈Ω0 , j ∈ N, of finite dimensional vector subspaces, and countably many families {Fj(ω)}ω∈Ω0 , j ∈ N, of closed vector subspaces of finite codimensions, called the Oseledets filtration for Φ, such that (i) for j ∈ N, any ω ∈ Ω0 and t ≥ 0: Uω(t)Ej(ω) = Ej(θtω) & Uω(t)Fj(ω) ⊂ Fj(θtω); (ii) for j ∈ N and any ω ∈ Ω0: E1(ω)⊕ . . .⊕ Ej(ω)⊕ Fj(ω) = X & Fj(ω) = Ej+1(ω)⊕ Fj+1(ω); EJDE-2024/45 OSELEDETS DECOMPOSITION 5 (iii) for j ∈ N, the families of projections associated with the decomposi- tions ( j⊕ n=1 En(ω) ) ⊕ Fj(ω) = X are strongly measurable and tempered; (iv) for j ∈ N, any ω ∈ Ω0 and any nonzero u ∈ Ej(ω) lim t→∞ ln ∥Uω(t)u∥ t = λj ; (v) for j ∈ N and any ω ∈ Ω0, a nonzero u ∈ Fj−1(ω) belongs to Ej(ω) if and only if there exists a negative semiorbit ũ : (−∞, 0] → X passing through (ω, u) such that lim s→−∞ ln ∥ũ(s)∥ s = λj ; (vi) for j ∈ N and any ω ∈ Ω0 lim t→∞ ln ∥Uω(t) ∣∣ Fj(ω) ∥ t = λj+1; 3. Sub-semiflows Let Φ(1) = ((U (1) ω (t))ω∈Ω,t∈R+ , (θt)t∈R) and Φ(2) = ((U (2) ω (t))ω∈Ω,t∈R+ , (θt)t∈R) be measurable linear skew-product semidynamical systems on Banach spaces (X1, ∥· ∥1) and (X2, ∥·∥2) respectively. The operator norm on L(Xm, Xn), m,n = 1, 2, will be denoted by ∥ · ∥m,n. By Ls(Xm, Xn) we understood the space of linear bounded operators equipped with strong operator topology. We assume the following: (A1) There is an injective bounded linear map i : X2 → X1. (A2) For any ω ∈ Ω and t ≥ 0 the equality U (1) ω (t) ◦ i = i ◦ U (2) ω (t) holds. (A3) For any ω ∈ Ω there is a map U (1,2) ω (1) ∈ L(X1, X2) such that U (1) ω (1) = i ◦ U (1,2) ω (1) and U (2) ω (1) = U (1,2) ω (1) ◦ i. Moreover,[ Ω ∋ ω 7→ sup 0≤s≤1 ln+ ∥U (1,2) θsω (1)∥ ∈ R+ ] ∈ L1(Ω,F,P). (A4) The operator is U (1,2) ω (1) is (F,B(Ls(X1, X2)))-measurable i.e., for all u ∈ X1 the mapping[ Ω ∋ ω 7→ U (1,2) ω (1)u ∈ X2 ] is (F,B(X2))-measurable. (A5) X1 and X2 are separable. (A6) For any A ∈ B(X2) there exists B ∈ B(X1) such that A = i−1(B). It should be noted that under the assumption (A3) the operator norm ∥U (1,2) θtω (1)∥1,2 along P-a.e. ω ∈ Ω trajectories is sub-exponential, that is, lim t→±∞ ln+ ∥U (1,2) θtω (1)∥1,2 t = 0, for all P-a.e. ω ∈ Ω. For t → ∞ it is a consequence of the following more general lemma presented below. For t → −∞ we can repeat argument for inverse ergodic flow (θ−t)t∈R. 6 M. KRYSPIN EJDE-2024/45 Lemma 3.1. Let (Ω,F,P, (θt)t∈R) be an ergodic dynamical system and let the func- tion f : Ω → R+ such[ Ω ∋ ω 7→ sup 0≤s≤1 f(θsω) ∈ R+ ] ∈ L1(Ω,F,P) be given, then f(θtω)/t → 0 as t → ∞ for P-a.e. ω ∈ Ω. Proof. From [25, Theorem 5] it follows that there is a time t0 ∈ (0, 1] such that θt0 is an ergodic transformation. Moreover, 0 ≤ t0 f(θtω) t ≤ 1 n sup 0≤s≤1 f(θs+t0nω) = 1 n n∑ k=0 sup 0≤s≤1 f(θs+t0kω)− 1 n n−1∑ k=0 sup 0≤s≤1 f(θs+t0kω). Furthermore, the application of Birkhoff Ergodic Theorem allows us to write that for P-a.e. ω ∈ Ω, lim n→∞ ( 1 n n∑ k=0 sup 0≤s≤1 f(θs+t0kω)− 1 n n−1∑ k=0 sup 0≤s≤1 f(θs+t0kω) ) = ∫ Ω sup 0≤s≤1 f(θsω) dP(ω)− ∫ Ω sup 0≤s≤1 f(θsω) dP(ω) = 0, this completes the proof. □ Later we evaluate limits of ln ∥U (1,2) θtω (1)∥1,2/t as t → ±∞. Convergence to zero is not necessarily true. However, the use of the inequality ln ∥U (1,2) θtω (1)∥1,2 ≤ ln+ ∥U (1,2) θtω (1)∥1,2 will be sufficient in order to obtain desired results. Lemma 3.2. For a separable Banach space X1 the mapping[ Ls(X1, X2)×X1 ∋ (T, x) 7→ Tx ∈ X2 ] (B(Ls(X1, X2))⊗B(X1),B(X2))-measurable. The above lemma is comptible with [11, Lemma A.6 (2)] and [3, Lemma. 6.4.2(i)]. Lemma 3.3. Assume (A1), (A2) and (A3). For any ω ∈ Ω and t ≥ 1 the equality U (1,2) θt−1ω (1) ◦ U (1) ω (t− 1) = U (2) θ1ω (t− 1) ◦ U (1,2) ω (1) holds. Proof. Fix ω and t. We are going to use the assumptions (A2) and (A3). Moreover, since i is injective it suffices to observe that i ◦ U (1,2) θt−1ω (1) ◦ U (1) ω (t− 1) = U (1) θt−1ω (1) ◦ U (1) ω (t− 1) = U (1) θ1ω (t− 1) ◦ U (1) ω (1) = U (1) θ1ω (t− 1) ◦ i ◦ U (1,2) ω (1) = i ◦ U (2) θ1ω (t− 1) ◦ U (1,2) ω (1). □ EJDE-2024/45 OSELEDETS DECOMPOSITION 7 The above observation allows us to extend the definition of the operator U (1,2) ω (1) to all t ≥ 1 as U (1,2) ω (t) : = U (1,2) θt−1ω (1) ◦ U (1) ω (t− 1) = U (2) θ1ω (t− 1) ◦ U (1,2) ω (1) ∈ L(X2, X1). Before we investigate how to transfer Oseledets decomposition from one measur- able linear skew-product semidynamical systems Φ(1) onto another semiflow Φ(2), we will introduce a useful Claim. Claim 3.4. Assume (A1), (A2), (A3) and let Ω0 ∈ F be such that P(Ω0) = 1. For any family of subspaces {W (ω)}ω∈Ω0 of X1 such that the equality U (1) ω (t)W (ω) = W (θtω) holds for all t ≥ 0 and all ω ∈ Ω0, there exists a family of subspaces {V (ω)}ω∈Ω0 of X2 such (i) iV (ω) = W (ω) for any ω ∈ Ω0, (ii) U (2) ω (t)V (ω) = V (θtω) for all t ≥ 0 and ω ∈ Ω0. Proof. For ω ∈ Ω0 let V (ω) := U (1,2) θ−1ω (1)W (θ−1ω). Therefore, iV (ω) = i ◦ U (1,2) θ−1ω (1)W (θ−1ω) = U (1) θ−1ω (1)W (θ−1ω) = W (ω), and the part (i) is done. Furthermore, for fixed t ≥ 0 we have i ◦ U (2) ω (t)V (ω) = i ◦ U (2) ω (t) ◦ U (1,2) θ−1ω (1)W (θ−1ω) = U (1) ω (t) ◦ i ◦ U (1,2) θ−1ω (1)W (θ−1ω) = U (1) ω (t) ◦ U (1) θ−1ω (1)W (θ−1ω) = U (1) θt−1ω (1) ◦ U (1) θ−1ω (t)W (θ−1ω) = U (1) θt−1ω (1)W (θt−1ω) = i ◦ U (1,2) θt−1ω (1)W (θt−1ω) = i ◦ U (1,2) θ−1θtω (1)W (θ−1θtω) = iV (θtω), which completes the proof. □ It should be noted that V (ω) can be defined as i−1(W (ω)). In such a case, firstly, we can observe that by invariance of W (ω) and the assumption (A3) we have W (θtω) = U (1) θt−1ω (1)W (θt−1ω) = i ◦ U (1,2) ω (1)W (θt−1ω) ⊂ iX2. Hence, by proceeding as in the proof of Claim 3.4 we can show invariance of V (ω). Indeed, i ◦ U (2) ω (t)V (ω) = i ◦ U (2) ω (t)i−1(W (ω)) = U (1) ω (t) ◦ i(i−1(W (ω))) = U (1) ω (t)(W (ω) ∩ iX2) = U (1) ω (t)W (ω) = W (θtω) = W (θtω) ∩ iX2 = i(i−1W (θtω)). 8 M. KRYSPIN EJDE-2024/45 Assume that Φ(1) admits an Oseledets decomposition. By taking in Claim 3.4 {E(1) j (ω)}ω∈Ω0,j=1,...,k or {E(1) j (ω)}ω∈Ω0,j∈N for {W (ω)} we obtain families {E(2) j (ω)}ω∈Ω0,j=1,...,k or {E(2) j (ω)}ω∈Ω0,j∈N of invariant, finite-dimensional vector subspaces of X2. Fix j. Let l be the dimension of E (1) j . For ω ∈ Ω0 denote by G(ω) the linear isomorphism from E (1) j (ω) onto Rl given by G(ω)u := (α1, . . . , αl) where u = α1 v1(ω) + . . .+ αl vl(ω) is written in the basis. The family {Ûω(1)}ω∈Ω0 of linear automorphisms of Rl defined by Ûω(1) := G(θ1ω) ◦ U (1) ω (1) ∣∣ E (1) j (ω) ◦G(ω)−1, ω ∈ Ω0, generates a two-sided discrete-time linear skew-product dynamical system Φ̂ = ((Ûω(n)), (θn)) on Ω0 × Rl, with Ûω(n) := G(θnω) ◦ U (1) ω (n) ∣∣ E (1) j (ω) ◦G(ω)−1, ω ∈ Ω0, n ∈ N. This allows us to use results in [27, Subsection 4.2.4]. Indeed, those results are formulated for discrete time, but assumption (E1) allows us to extend them to the continuous time case. As a consequence of [27, Proposition 4.11], lim t→∞ 1 t ln ∥U (1) ω (t) ∣∣ E (1) j (ω) ∥1,1 = λj , lim t→∞ 1 t ln ∥(U (1) ω (t) ∣∣ E (1) j (ω) )−1∥−1 1,1 = λj . (3.1) Based on the above, we can introduce the following lemma, which will be useful in the subsequent analysis. Lemma 3.5. For P-a.e. ω ∈ Ω and each j, there exists a function c : (1,∞) → (0,∞) such that (i) for each t ≥ 1 and u ∈ E (2) j (θtω) we have c(t)∥u∥2 ≤ ∥iu∥1, (ii) c is sub-exponential i.e. lim t→∞ ln c(t) t = 0. Proof. Let c(t) = inf {∥iu′∥1 ∥u′∥2 : u′ ∈ E (2) j (θtω) \ {0} } Hence, part (i) is trivially satisfied. For part (ii) observe that the upper bound for c is obviously ∥i∥. Therefore, we are concentrating on the sub-exponential lower limit. It is claimed that ∥(U (1) ω (t) ∣∣ E (1) j (ω) )−1∥−1 1,1 ∥U (1,2) θt−1ω (1)∥1,2 ∥U (1) ω (t− 1) ∣∣ E (1) j (ω) ∥1,1 ≤ c(t), for t ≥ 1. Indeed, fix t ≥ 1 and u′ ∈ E (2) j (θtω) \ {0}. Let u ∈ E (1) j (ω) be such that u′ = U (1,2) ω (t)u. Since ∥u′∥2 = ∥U (1,2) θt−1ω (1)U (1) ω (t− 1)u∥2 ≤ ∥U (1,2) θt−1ω (1)∥1,2 ∥U (1) ω (t− 1) ∣∣ E (1) j (ω) ∥1,1 ∥u∥1, ∥iu′∥1 = ∥U (1) ω (t)u∥1,1 ≥ ∥(U (1) ω (t) ∣∣ E (1) j (ω) )−1∥−1 1,1 ∥u∥1. EJDE-2024/45 OSELEDETS DECOMPOSITION 9 We thus have further ∥(U (1) ω (t) ∣∣ E (1) j (ω) )−1∥−1 1,1 ∥U (1,2) θt−1ω (1)∥1,2 ∥U (1) ω (t− 1) ∣∣ E (1) j (ω) ∥1,1 ≤ ∥iu′∥1 ∥u′∥2 . From this the preliminary claim is obtained, which makes it possible to write 1 t ln ∥(U (1) ω (t) ∣∣ E (1) j (ω) )−1∥−1 1,1 − 1 t ln ∥U (1) ω (t− 1) ∣∣ E (1) j (ω) ∥1,1 − 1 t ln ∥U (1,2) θt−1ω (1)∥1,2 ≤ ln c(t) t ≤ ln ∥i∥2,1 t Equation (3.1) completes the proof. □ Theorem 3.6. Assume (A1)-(A6) and moreover, that Φ(1) admits an Oseledets de- composition in case (O1) (or (O2)). Then Φ(2) admits an Oseledets decomposition in case (O1) (or (O2) respectively). Proof. We are going to show that if Φ(1) admits an Oseledets decomposition in case (O1) with parameters (Ω0, (λj)j=1,...,k, {Ej(ω)}ω∈Ω0,j=1,...,k, {F∞(ω)}ω∈Ω0 ), then Φ(2) admits an Oseledets decomposition in case (O1) with parameters (Ω0, (λj)j=1,...,k, {U (1,2) θ−1ω (1)Ej(θ−1ω)}ω∈Ω0,j=1,...,k, {i−1(F∞(ω))}ω∈Ω0 ). Before we start with cases (ii)-(ivi). Note that i−1(F∞(ω)) is a closed subspace as a continuous preimage of a closed subspace. Moreover, for each j = 1, . . . , k the family {U (1,2) θ−1ω (1)Ej(θ−1ω)}ω∈Ω0 is measurable. Indeed, the mapping[ Ej(ω) ∋ u 7→ U (1,2) θ−1ω (1)u ∈ U (1,2) θ−1ω (1)Ej(θ−1ω) ] is a bijection. It follows from the observation that U (1) ω (1) is a bijection since it preserves finite dimension i.e., dimU (1) ω (1)Ej(ω) = dimEj(θ1ω) (see (O1)(ii)). Therefore, from: the injectivity of i, the bijectivity of U (1) ω (1) and i ◦ U (1,2) ω (1) = U (1) ω (1) we can deduce that u = v whenever, i◦U (1) θ−1ω (1)u = i◦U (1) θ−1ω (1)v. Moreover, U (1,2) θ−1ω (1)Ej(θ−1ω) = span{U (1,2) θ−1ω (1)v1(θ−1ω), . . . , U (1,2) θ−1ω (1)vl(θ−1ω)}, where v1, . . . , vl : Ω0 → X are (F,B(X))-measurable functions, such that (v1(ω), . . . , vl(ω)) forms a basis of Ej(ω) for each ω ∈ Ω0. Hence, it remains to show the (F,B(X2))-measurability of the maps[ Ω0 ∋ ω 7→ U (1,2) θ−1ω (1)vj(θ−1ω) ∈ X2 ] , for j = 1, . . . , l. It can be done by looking at the composition ω 7→ (ω, ω) 7→ (U (1,2) θ−1ω (1), vj(θ−1ω)) 7→ U (1,2) θ−1ω (1)vj(θ−1ω). The second mapping is (F ⊗ F,B(Ls(X1, X2)) ⊗ B(X1))-measurable. By the as- sumption (A5) and Lemma 3.2, the pairing is (B(Ls(X1, X2)) ⊗ B(X1),B(X2))- measurable. (O1)(ii) can be done via Claim 3.4, indeed U (2) ω (t)U (1,2) θ−1ω (1)Ej(θ−1ω) = U (1,2) θt−1ω (1)Ej(θt−1ω). 10 M. KRYSPIN EJDE-2024/45 It remains to show U (2) ω (t)i−1(F∞(ω)) ⊂ i−1(F∞(θtω)), equivalently i ◦ U (2) ω (t)(i−1(F∞(ω))) = U (1) ω (t) ◦ i(i−1(F∞(ω))) = U (1) ω (t)(F∞(ω) ∩ iX2) ⊂ U (1) ω (t)F∞(ω) ∩ U (1) ω (t)iX2 = U (1) ω (t)F∞(ω) ∩ iU (2) ω (t)X2 ⊂ F∞(θtω) ∩ iX2 = i(i−1(F∞(θtω))). Hence, the decomposition is invariant. (O1)(iii) To show k⊕ j=1 U (1,2) θ−1ω (1)Ej(θ−1ω)⊕ i−1(F∞(ω)) = X2 it suffices to show that the spaces U (1,2) θ−1ω (1)Ej(θ−1ω) (j = 1, . . . , k) and i−1(F∞(ω)) are linearly independent and k∑ j=1 U (1,2) θ−1ω (1)Ej(θ−1ω) + i−1(F∞(ω)) = X2. Therefore, fix e (2) j ∈ U (1,2) θ−1ω (1)Ej(θ−1ω) for each j = 1, . . . , k and f (2) ∈ i−1(F∞(ω)) such that e (2) 1 + e (2) 2 + · · ·+ e (2) k + f (2) = 0. Since 0 = i ( k∑ j=1 e (2) j + f (2) ) = k∑ j=1 ie (2) j + if (2), and by linear independence of the subspaces Ej(ω) (j = 1, . . . , k) and F∞(ω) we have ie (2) 1 = · · · = ie (2) k = if (2) = 0, so e (2) 1 = · · · = e (2) k = f (2) = 0. It remains to show the sets equality. The inclusion (⊂) is obvious. Therefore, fix x ∈ X2. Hence, ix ∈ X1 there exists a (unique) representation ix = e (1) 1 + e (1) 2 + · · · + e (1) k + f (1) for some e (1) j ∈ Ej(ω) and f (1) ∈ F∞(ω). However, from the first part of the Claim 3.4(i) we can conclude that for any e (1) j ∈ Ej(ω) there exists e (2) j ∈ U (1,2) θ−1ω (1)Ej(θ−1ω) such that e (1) j = ie (2) j . We can find f (2) ∈ F∞(ω) such that if (2) = f (1), namely f (2) = x− e (2) 1 − · · · − e (2) k . So, the inclusion (⊃) is done. (O1)(iiii) Fix l = 1, . . . , k and u ∈ X2. In order to show that the family of projections associated with the decomposition( l⊕ j=1 U (1,2) θ−1ω (1)Ej(θ−1ω) ) ⊕ ( k⊕ j=l+1 U (1,2) θ−1ω (1)Ej(θ−1ω)⊕ i−1(F∞(ω)) ) = X2 (3.2) along the first component is strongly measurable it suffices to see that for fixed set A ∈ B(X2) we have (P (2)(·)u)−1(A) = (P (2)(·)u)−1(i−1(B)) = (i ◦ P (2)(·)u)−1(B) = (P (1)(·)iu)−1(B) ∈ F, EJDE-2024/45 OSELEDETS DECOMPOSITION 11 where P (2)(ω) is the projection along U (1,2) θ−1ω (1)E1(θ−1ω)⊕ · · · ⊕U (1,2) θ−1ω (1)El(θ−1ω) and set B ∈ B(X1) is the set from the assumption (A6) such that A = i−1(B). It remains to show that the family of projections associated with the decomposi- tion (3.2) is tempered. The definition refers to projections onto the finite codimen- sional closed vector subspaces F (ω). However, it is a well-known fact that it is sufficient to equivalently demonstrate the temperedness of projections onto finite- dimensional vector subspaces (i.e. first component of (3.2), see [15, Remark 2.1]). We start by demonstrating that the projections onto U (1,2) θ−1ω (1)Ej(θ−1ω) for any j = 1, . . . , k are tempered. Let us fix j. Subsequently, in the inductive process, we will show that temperedness holds universally. Therefore, we let P̃ (2) j (ω) denote the complementary part of P (2) j (ω), namely the projection onto U (1,2) θ−1ω (1)Ej(θ−1ω). Using the results from Lemma 3.5, we can conclude that for any u ∈ X2 we have c(t)∥P̃ (2) j (θtω)u∥2 ≤ ∥iP̃ (2)(θtω)u∥1 ≤ ∥i∥2,1 ∥P̃ (2) j (θtω)u∥2. In fact, we have 1 ≤ ∥P̃ (2) j (θtω)∥2,2 = sup {∥P̃ (2) j (θtω)u∥2 ∥u∥2 : u ∈ X2 \ {0} } ≤ sup {∥P̃ (1) j (θtω)iu∥1 c(t)∥iu∥1 · ∥iu∥1 ∥u∥2 : u ∈ X2 \ {0} } ≤ ∥i∥2,1 c(t) ∥P̃ (1) j (θtω)∥1,1. Hence, ln ∥P̃ (2)(θtω)∥2,2/t → 0 as t → ∞. Furthermore, in the spirit of Tanny’s the- orem, when t → −∞, we can obtain ln ∥P̃ (2)(θtω)∥2,2/t → 0 as well, see [11, Lemma C2]. In conclusion, let us observe that generally for the projection P̃ (2) 1,...,l(θtω) onto the first component of (3.2) we have 0 ≤ ln ∥P̃ (2) 1,...,l(θtω)∥2,2 t ≤ ln l +maxj ln ∥P̃ (2) j (θtω)∥2,2 t . Hence, ln ∥P̃ (2) 1,...,l(θtω)∥2,2/t → 0 as t → ∞ and again by [11, Lemma C2] we can obtain this result for t → −∞. (O1)(iiv) fix j = 1 . . . k, ω ∈ Ω0 and u ∈ U (1,2) θ−1ω (1)Ej(θ−1ω). Hence, from Claim 3.4(i) iu ∈ Ej(ω). Therefore, by (O1)(iiv) for Φ(1), ln ∥U (1) ω (t)iu∥1/t → λj . Moreover, we have two inequalities, first one from the assumption (A2): ∥U (1) ω (t) ◦ iu∥1 = ∥i ◦ U (2) ω (t)u∥1 ≤ ∥i∥2,1∥U (2) ω (t)u∥2 and the second one, ∥U (2) ω (t)u∥2 ≤ ∥U (1,2) θt−1ω (1)∥1,2 ∥U (1) ω (t−1)◦ iu∥1, from the fact that U (2) ω (t)u = U (1,2) θt−1ω (1)◦U (1) ω (t− 1) ◦ iu. Therefore, 1 t ln ∥U (1) ω (t) ◦ iu∥1 − 1 t ln ∥i∥2,1 ≤ 1 t ln ∥U (2) ω (t)u∥2 ≤ 1 t ln ∥U (1,2) θt−1ω (1)∥1,2 + 1 t ln ∥U (1) ω (t− 1) ◦ iu∥1. Thus, we have proven (O1)(iiv) for Φ(2). 12 M. KRYSPIN EJDE-2024/45 (O1)(iv) Fix ω and u(2) ∈ k⊕ m=j U (1,2) θ−1ω (1)Em(θ−1ω)⊕ i−1(F∞(ω)). To begin with, let us note that Claim 3.4 allows for demonstrating only the equiv- alence between the existence of negative semiorbits (in X1 and X2) on which the appropriate Lyapunov exponent is achieved, since the membership of u(2) in the subspace U (1,2) θ−1ω (1)Ej(θ−1ω) is equivalent to the membership of iu(2) in the subspace Ej(ω). Assume that there exists a negative semiorbit ũ(2)(s) in X2 passing through (ω, u(2)). We commence by demonstrating that within the space X1, there exists a negative semiorbit passing through (ω, iu(2)), such that the appropriate Lyapunov exponent is attainable. Namely, we define ũ(1) : (−∞, 0] → X1 by ũ(1)(s) = iũ(2)(s); such Lyapunov exponent in X2 is (by assumption) λj . Upon examination, it be- comes evident that the thus-defined ũ(1) indeed is a negative semi-orbit. Indeed ũ(1)(0) = iu(2) and ũ(1)(s+ t) = iũ(2)(s+ t) = iU (2) θsω (t)ũ(2)(s) = U (1) θsω (t)iũ(2)(s) = U (1) θsω (t)ũ(1)(s). Moreover, ũ(1) is (B((−∞, 0]),B(X1))-measurable as a continuous composition. Furthermore, ln ∥ũ(1)(s)∥1/s → λj as s → −∞ since ∥ũ(2)(s+ 1)∥2 = ∥U (1,2) θsω (1)iũ(2)(s)∥2 ≤ ∥U (1,2) θsω (1)∥1,2∥iũ(2)(s)∥1 = ∥U (1,2) θsω (1)∥1,2∥ũ(1)(s)∥1 ≤ ∥U (1,2) θsω (1)∥1,2∥i∥2,1∥ũ(2)(s)∥2. Conversely, if ũ(1) is a negative semiorbit on X1, then ũ(2) defined as ũ(2)(s) = U (1,2) θs−1ω (1)ũ(1)(s− 1) serves as a negative semiorbit on X2. Clearly, iũ(2)(0) = iU (1,2) θ−1ω (1)ũ(1)(−1) = U (1) θ−1ω (1)ũ(1)(−1) = ũ(1)(0) = u(1) = iu(2), and for any s ≤ 0, t ≥ 0 such that s+ t ≤ 0, we have ũ(2)(s+ t) = U (1,2) θs−1+tω (1)ũ(1)(s− 1 + t) = U (1,2) θs−1+tω (1) ◦ U (1) θs−1ω (t)ũ(1)(s− 1) = U (2) θsω (t) ◦ U (1,2) θs−1ω (1)ũ(1)(s− 1) = U (2) θsω (t)ũ(2)(s). Moreover, ln ∥ũ(2)(s)∥2/s → λj as s → −∞ since ∥ũ(1)(s)∥1 ≤ ∥i∥2,1∥ũ(2)(s)∥2 ≤ ∥i∥2,1∥U (1,2) θs−1ω (1)∥1,2∥ũ(1)(s− 1)∥1. It remains to show the (B((−∞, 0]),B(X2))-measurability of the map[ (−∞, 0] ∋ s 7→ U (1,2) θs−1ω (1)ũ(1)(s− 1) ∈ X2 ] . EJDE-2024/45 OSELEDETS DECOMPOSITION 13 For this purpose, let us observe that it can be rewritten as the measurable compo- sition s 7→ (θs−1ω, s) 7→ (U (1,2) θ−1ω (1), ũ(1)(s− 1)) 7→ U (1,2) θs−1ω (1)ũ(1)(s− 1). The first mapping is (B((−∞, 0]),F⊗B((−∞, 0]))-measurable, the second mapping is (F⊗B((−∞, 0]),B(Ls(X1, X2))⊗B(X1))-measurable. By the assumption (A5) and Lemma 3.2, the pairing is (B(Ls(X1, X2))⊗B(X1),B(X2))-measurable. (O1)(ivi) Firstly, we claim that U (2) ω (t) = U (1,2) θt−1ω (1) ◦ U (1) ω (t− 1) ◦ i. With i(i−1(F∞(ω))) ⊂ F∞(ω) we can see that the inequality ∥U (2) ω (t) ∣∣ i−1(F∞(ω)) ∥2,2 ≤ ∥i∥2,1 ∥U (1,2) θt−1ω (1)∥1,2 ∥U (1) ω (t− 1) ∣∣ i(i−1(F∞(ω))) ∥1,1 ≤ ∥i∥2,1 ∥U (1,2) θt−1ω (1)∥1,2 ∥U (1) ω (t− 1) ∣∣ F∞(ω) ∥1,1 holds for any ω ∈ Ω and t ≥ 1. So, 1 t ln ∥U (2) ω (t) ∣∣ i−1(F∞(ω)) ∥2,2 ≤ 1 t ln ∥i∥2,1 + 1 t ln ∥U (1,2) θt−1ω (1)∥1,2 + 1 t ln ∥U (1) ω (t− 1) ∣∣ F∞(ω) ∥1,1, with under assumptions (A1) and (A3) suffices to finish the proof of this part. Furthermore, we have concluded the first and, as it turns out, the main part of the proof. The remaining part (O2) pertains to the scenario in which there is a countable number of Lyapunov exponents. Nevertheless, it is easy to notice that practically the entire proof is replicated word for word in this situation. However, for the sake of completeness, let us note the fact that if Φ(1) admits Oseledets decomposition in the case (O2) with parameters (Ω0, (λj)j∈N, {Ej(ω)}ω∈Ω0,j∈N, {Fj(ω)}ω∈Ω0,j∈N) then Φ(2) also admits Oseledets decomposition in the case (O2), with parameters (Ω0, (λj)j∈N, {U (1,2) θ−1ω (1)Ej(θ−1ω)}ω∈Ω0,j∈N, {i−1(Fj(ω))}ω∈Ω0,j∈N). □ Assumption discussion In this section, we aim to discuss the scenario where assumption (A6) is met. Indeed, we will show something more, namely that this assumption is fulfilled when, in particular, X1 = Lp([−1, 0]) (1 < p < ∞) and X2 = C([−1, 0]), and that measurability of function factors follows from it. Lemma 3.7. Let i : C([−1, 0]) → Lp([−1, 0]) be the continuous injection defined by i(f) = f . Moreover, let w(1) : Ω → Lp([−1, 0]) be a (F,B(Lp([−1, 0])))-measurable function with the factor w(2) : Ω → C([−1, 0]), i.e. w(1) = i ◦ w(2). Then w(2) : Ω → C([−1, 0]) is (F,B(C([−1, 0])))-measurable. Proof. The proof is based on observation that every closed ball, specifically the unit ball, in C([−1, 0]) corresponds to a preimage of i of some Lp-closed subset F ⊂ Lp([−1, 0]), i.e. B̄C([−1,0])(0, 1) = i−1(F ) for some Lp-closed F . The set F := {i(f) ∈ Lp([−1, 0]) : −1 ≤ f(x) ≤ 1 for a.e. x ∈ [−1, 0]} proves to be a witness, as Lp([−1, 0]) \ F is Lp-open. Indeed, for g ∈ Lp([−1, 0]) \ F there exists 14 M. KRYSPIN EJDE-2024/45 a Lebesgue-measurable set M ⊂ [−1, 0] such that its Lebesgue measure λ(M) > 0 and ϵ > 0 such that g(x) ≥ 1 + ϵ for all x ∈ M . Therefore, for f ∈ F we have ∥g − f∥Lp ≥ ϵλ1/p(M) so BLp(−1,0)(g, ϵλ 1/p(M)) ⊂ Lp([−1, 0]) \ F . It remains to show i−1(F ) = B̄C([−1,0])(0, 1). However, it can be easily done via definition of preimage since, conditions −1 ≤ f(x) ≤ 1 for a.e. x and −1 ≤ f(x) ≤ 1 for all x are equivalent for continuous functions. As a final step, we demonstrate that the map w(2) is (F,B(C([−1, 0])))-measurable. Let D be an open subset of C([−1, 0]) in the sense of the C([−1, 0]) topology. Since C([−1, 0]) is separable, it has a countable base, and D can be expressed as a count- able union of open balls. Moreover, open balls can be written as countable union of closed ones. As a consequence the set D is a countable union of preimages of Lp-closed sets. The observation that (w(2))−1[D] = (w(2))−1 ( ∪n∈N i−1(Fn) ) = (i ◦ w(2))−1 ( ∪n∈N Fn ) = (w(1))−1 ( ∪n∈N Fn ) ∈ F completes the proof. □ Acknowledgments. This research was supported by the National Science Centre, Poland (NCN) under grant Sonata Bis NCN 2020/38/E/ST1/00153. The author is grateful to Janusz Mierczyński for his inspiring conversations. References [1] A. Bátkai, S. Piazzera; Semigroups for Delay Equations, Res. Notes Math., 10, A K Peters, Wellesley, MA, 2005. MR 2181405 [2] A. Blumenthal, S. Punshon-Smith; On the norm equivalence of Lyapunov exponents for regularizing linear evolution equations, Arch. Ration. Mech. 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