On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems

In this paper, we construct an open and dense set in the space of bounded linear random dynamical systems (both discrete and continuous time) equipped with the essential sup norm such that the Lyapunov exponents depend analytically on the coefficients in this set. As a consequence, analyticity for Lyapunov exponents of bounded linear random dynamical systems is a generic property.


1.
Introduction. The fundamental results on Lyapunov exponents for random dynamical systems on finite dimensional spaces were first obtained in [13], which is now called the Oseledets Multiplicative Ergodic Theorem, see also [3].
Since the appearance of the paper [13], exploring properties of Lyapunov exponents of random dynamical systems has become one of a central task in the theory of random dynamical systems. In this task, understanding the stability of Lyapunov exponents under a small perturbation has been received a lot of interests. Now, we would like to mention some publications that are related to our work in this paper: In [14], Ruelle showed that the characteristic exponents of a random matrix product over a compact base space endowed with a probability measure depends real analytically on the data of the problem. This result was generalized in [9] to product of random linear operators having an invariant cone over a measurable base space.
Concerning random dynamical systems without assuming to share an invariant cone, in general Lyapunov exponents do not depend continuously on the coefficients, see [12]. Using a similar technique as in [12], Bochi gave a characterization of discontinuity for Lyapunov exponents of SL(2, R)-cocycles in a unpublished preprint [4]. Roughly speaking, it was showed in [4] that if a SL(2, R)-cocycle has a positive Lyapunov exponent and has not a dominated splitting (integral separation), then the upper Lyapunov exponent function is not continuous at this cocycle.
Although Lyapunov exponents are in general not continuous, it is proved in [5] that continuity of Lyapunov exponents is a generic property on the space of bounded linear cocycles. Thereafter, it is natural to investigate higher regularity property 3114 DOAN THAI SON than continuity property for Lyapunov exponents of linear random dynamical systems such as Hölder continuity, smoothness and analyticity.
In this paper, we show that analyticity of Lyapunov exponents is still a generic property on the space of bounded linear random dynamical systems. As far as we are aware, this result is neither stated nor proved elsewhere and the main ingredient in the proof of this fact is to construct a suitable cone being invariant under a long enough iteration of an integrally separated linear random dynamical system. Note that generic bounded linear random dynamical systems are integrally separated, see [5] and [6]. This fact together with results in [14,9] deduces the conclusion that generically, the top Lyapunov exponent of random dynamical systems depend analytically on the coefficients. To derive the same conclusion for other Lyapunov exponents, we work with the generated random dynamical systems on the exterior power space.
The paper is organized as follows: In Section 2, we present a basic background on Lyapunov exponents of product of random matrices and linear random differential equations. In this section, we also recall a notion of analytic functions defined in an arbitrary Banach space. Section 3 is devoted to present the main results (Theorem 4 and Theorem 5) of this paper about genericity for analyticity of Lyapunov exponents. The Appendix section is divided into two subsections. In Subsection 3.2, we recall a basic background on the exterior power of a finite dimensional vector space. A result proved in [8] about the structure of Lyapunov spectrum of induced random dynamical systems on the exterior power was also presented. Subsection 3.2 is devoted to present the work in [2] and [9] about the simplicity and analyticity of the top Lyapunov exponent of product of positive random matrices.
To conclude this section, we introduce notations which are used throughout the paper: For a probability space (Ω, F, P), let L ∞ (Ω, R d×d ) denote the space of all essentially bounded random matrix-valued maps A : Ω → R d×d . Let L ∞ (Ω, R d×d ) be endowed with the essential sup norm, i.e.
It is well known that (L ∞ (Ω, R d×d ), · ∞ ) is a Banach space.

Preliminaries.
2.1. Discrete-time bounded linear random dynamical systems. Let (X, A, m) be a Lebesgue probability space and T : X → X an invertible ergodic transformation preserving the probability m. Each random map A ∈ L ∞ (X, R d×d ) gives rise to an one-sided linear random dynamical system where id denotes the identity matrix in R d×d . By virtue of the Multiplicative Ergodic Theorem for one-sided linear random dynamical systems defined on an invertible metric dynamical system, see e.g. [10], there exist finite numbers of Lyapunov exponents λ 1 (A) > · · · > λ p(A) (A), where p(A) ∈ {1, . . . , d}, and a forward invariant decomposition with the property that is called the Lyapunov spectrum of A. For a convenience of presentation in the rest of the paper, let γ d (A) ≤ γ d−1 (A) ≤ · · · ≤ γ 1 (A) denote the Lyapunov exponents of A (counting with multiplicity), i.e. for i = 1, . . . , p(A) The Lyapunov spectrum is said to be simple if p(A) = d, i.e. all Oseledets subspaces Let GL(d) be the set of all invertible matrices in R d×d . Let L ∞ (X, GL(d)) denote the space of all A ∈ L ∞ (X, R d×d ) such that the inverse A −1 (x) exists for x ∈ X and A −1 (·) ∈ L ∞ (X, R d×d ). Note that each random map A ∈ L ∞ (X, GL(d)) generates a two-sided linear random dynamical system which is also denoted by In the following lemma, we show that L ∞ (X, GL(d)) is an open and dense subset of L ∞ (X, R d×d ). Note that this result does not depend on the norm equipped with R d and the induced norm on R d×d . Then, in what follows let R d be endowed with the standard Euclidean norm. Proof. The openness of L ∞ (X, GL(d)) in (L ∞ (X, R d×d ), · ∞ ) is obvious and we only need to prove the density of this set. Let A ∈ L ∞ (X, R d×d ) and ε > 0 be arbitrary. Let A(x) = U (x)Σ(x)V (x) be a measurable singular valued decomposition of A, where U (x) and V (x) are measurable orthogonal matrices and Then, Hence, L ∞ (X, GL(d)) is dense in (L ∞ (X, R d×d ), · ∞ ) and the proof is complete.
2.2. Bounded linear 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. Suppose further that (θ t ) t∈R has no fixed point. For each A ∈ L ∞ (Ω, R d×d ), we consider the corresponding linear random differential equation of the following forṁ Let Ψ A (t, ω)ξ denote the solution of (1) satisfying ξ(0) = ξ. The linear mapping Φ A : R × Ω → R d×d is a continuous random dynamical system, i.e. Ψ A is (B(R) ⊗ F, B(R d×d )) measurable and the following properties hold: It is well known (see e.g. [3,6]) that Ψ A (t, ω) satisfies the integrability condition of the Multiplicative Ergodic Theorem, i.e.

Analytic functions.
Let X, Y be Banach spaces. For k ∈ N, a function p : X → Y is called a continuous homogeneous polynomial of degree k if there exists a continuous k-multilinear symmetric function ϕ : . A function f : X → Y is said to be real analytic at x ∈ X if there exists δ > 0 and a sequence of continuous homogeneous polynomials ( In the following lemma, we collect some fundamental properties of analytic functions which are used in the next section.
Lemma 2. The following statements hold: (i) Let X, Y be Banach spaces. Then, the composition, the summation and the subtraction of two analytic functions from X to Y are again analytic. (ii) Let X, Y be Banach spaces. A bounded linear operator T : X → Y is analytic.
Proof. We can find a proof of (i) in e.g. [15]. Meanwhile, the statements (ii) and (iii) follow directly from the definition of analytic function.
3.1. Discrete-time linear random dynamical systems. This subsection is devoted to study discrete-time bounded linear random dynamical systems introduced in Subsection 2.1. The main result in this subsection is to show that generically Lyapunov exponents of discrete-time bounded linear random dynamical systems depends analytically on the generators. To achieve this result, we show that in the set of integrally separated bounded linear random dynamical systems the Lyapunov exponents depends analytically on the generator. This assertion together with the result on the genericity of integral separation in the space of bounded invertible random matrices (see [5]) and Lemma 1 implies that analyticity of Lyapunov exponents is a generic property.
Recall that a bounded linear random dynamical system A ∈ L ∞ (X, GL(d)) is said to be integrally separated if there exist K, α > 0 and an invariant measurable is a linear subspace of dimension 1, such that for all n ∈ N and m-a.e. x ∈ X the following inequality In what follows, we introduce and prove some fundamental properties of a specific cone in the exterior power space Λ k R d . This cone is a key object in the proof of the main result in which we show that the generated random dynamical system of any integrally separated bounded linear random dynamical system on Λ k R d after a long enough iteration and a random change of coordinate preserves this cone.
Then, the following statements hold: The dual cone C k of C k is given by Consequently, e 1 ∧ · · · ∧ e k is an interior point of both C k and C k . More, precisely, we have where Proof. The assertion (i) follows directly from the definition of C k and the space Λ k R d , see Subsection 3.2. To prove (ii), we compute the dual cone C k of C k . For this purpose, Obviously, x ∈ C k and a direct computation yields that where we use the fact that β 1...k = max 1≤i1<···<i k ≤d |β i1...i k | to obtain the preceding inequality. Therefore, x, v ≥ 0 and the assertion (ii) is verified. In the remaining part of the proof, we show (4). For this purpose, let v = 1≤i1<···<i k ≤d α i1...i k e i1 ∧ · · · ∧ e i k satisfy that e 1 ∧ · · · ∧ e k − v Λ k R d ≤ 1 3 . Thus, from Consequently, v ∈ C k and thus B(e 1 ∧ · · · ∧ e k , 1 3 ) ⊂ C k . To prove the remaining assertion in (4) Analog to the arguments yielding (5), we have which implies that α 1...k ≥ 1≤i1<···<i k ≤d,(i1,...,i k ) =(1,...,k) |α i1...i k |. Thus, v ∈ C k and the proof is complete.
Theorem 4 (Analyticity for Lyapunov exponents of generic discrete-time linear random dynamical systems). Let A ∈ L ∞ (X, GL(d)) be an integrally separated bounded linear random dynamical system. Then, for i = 1, . . . , d the map γ i (·) is analytic at A. As a consequence, analyticity of Lyapunov exponents is a generic property in the Banach space (L ∞ (X, R d×d ), · ∞ ).
be the invariant decomposition of A such that inequality (2) holds for K, α > 0 and m-a.e. x ∈ X. For i = 1, . . . , d, choose and fix a measurable unit vector u i (x) ∈ E i (x). By invariance of E i (x), we have where a i : X → R is measurable. Since A ∈ L ∞ (X, GL(d)), there exists M 1 > 0 such that and by (2) |a Define L : X → GL(d) by where e 1 , . . . , e d is the standard Euclidean orthogonal basic of R d . From boundedness of A, A −1 and (2), for i = 1, . . . , d the angle between E i (x) and ⊕ d j=1,j =i E j (x) is uniformly bounded away from zero, see e.g. [5]. Consequently, there exists M 2 > 0 such that Define a linear operator Q L : Then, Q L is invertible and from (8), we have Q L ∞ , Q −1 L ∞ ≤ M 2 2 . Therefore, to complete the proof it is sufficient to show that for i = 1, . . . , d the map γ i (·) is analytic at D ∈ L ∞ (X, GL(d)) which is defined by The proof of this fact is divided into two steps: Step 1. Choose and fix an arbitrary k ∈ {1, . . . , d}. Let N be a positive integer such that Ke αN ≥ 2. Define a random map D (N ) ∈ L ∞ (X, GL(d)) by Let Λ k D (N ) denote the generated random map of D (N ) in L ∞ (X, L(Λ k R d )). In this step, we show that Λ k D (N ) preserves the cone C k defined as in (3), i.e.
and we also show that there exists R > 0 such that Choose and fix an arbitrary x ∈ X. By linearity of D (N ) (x), to prove (10) it is sufficient to show that which together with the fact that α 1...k = max 1≤i1<···<i k ≤d |α i1...i k | implies that By definition of D (N ) and the generated random dynamical system Λ k D (N ) on Λ k R d , see Subsection 3.2, we have Using (7), we obtain that Consequently, Λ k D (N ) (x)v ∈ C k ∪ (−C k ) and (10) is proved. Furthermore, from (12) and a direct computation yields that Thus, the distance (with respect to the Hilbert metric on the cone C k ) between two vectors Λ k D (N ) (x)v and e 1 ∧ · · · ∧ e k can be estimated as follows . . a j (x)| This together with (6) and the fact that 1 √ Step 2. So far we have proved that for any k ∈ {1, . . . , d} the random linear map Λ k D (N ) preserves the cone C k . On the other hand, by Lemma 3 the cone C k satisfies conditions (C1) and (C2) of Theorem 7 in the Appendix with the measurable mappings c : X → C k and c : X → C k defined by c(x) = c (x) := e 1 ∧ · · · ∧ e k for x ∈ X.
To apply Theorem 7 to the random map Λ k D (N ) over (X, A, m, T N ), we only need to check the ergodicity of the transformation T N . In what follows, we consider two separated cases and our aim is to show that the map γ 1 (·) + · · · + γ k (·) is analytic at D, where k ∈ {1, . . . , d}.
Case 1. T N is an ergodic transformation from X into itself. In this case, using the Multiplicative Ergodic Theorem, the Lyapunov exponents of the random dynamical system generated by M ∈ L ∞ (X, R d×d ) over (X, A, m, T N ), denoted by γ Note that π is a product of bounded linear operators In view of Theorem 6, we have for all ∆ ∈ L ∞ (X, By (9) Case 2. T N is not ergodic. Since all statements above still work if N is replaced by a larger integer number. Hence, we can assume additionally that N is a prime number. Let W be a measurable set satisfying that m(W ) ∈ (0, 1) and W = T N W . We prove by induction that for k = 1, . . . , N there exists a measurable set W k such that Obviously, (14) holds for k = 1 with W 1 := W and suppose that it holds for some k ∈ {1, . . . , N − 1} with a measurable set W k . By ergodicity of T and the fact that T Then, let ∈ {1, . . . , N −1} be the smallest integer such that m W k ∩ i=1 T ik W k = 0. Hence, a measurable set W k+1 := W k ∩ −1 i=1 T ik W k is also invariant under T N and satisfies (14) for k + 1. For the measurable set W := W N we have T N W = W and the sets W , . . . , T N −1 W are pairwise disjoint. From T ( On the other hand, for any subset U ⊂ W with positive measure and m( W \ U ) > 0 we have m Consequently, m(T N U ∆U ) = 0 and therefore the restriction of the map T N on the set W is ergodic. Thus, using a similar statement as in Case 1 to the induced random dynamical systems on ( W , A, m, T N ) completes the proof.
3.2. Bounded linear random differential equations. Our aim in this subsection is to study the analyticity of Lyapunov exponents of bounded linear random differential equations over a metric dynamical system (Ω, F, P, (θ t ) t∈R ). For this purpose, we need the following preparation. Firstly, we recall a notion of flow built under a function: Definition 1 (Flow built under a function). Let (X, A, m) be a probability space and T : X → X be an invertible measurable transformation preserving the probability m. Let f : X → R ≥0 be a measurable function with X f (x) dm(x) < ∞. The flow built under the function (X, T, m, f ) is defined with the following ingredients: x ∈ X, 0 ≤ s < f (x)}. Then, B is a measurable set of the measurable space (X × R, A ⊗ B(R)). Let σ and µ denote, respectively, the restriction of the sigma algebra A ⊗ B(R) and the product measure m × λ on the measurable set B, where λ is the Lebesgue measure on R ≥0 . (ii) The flow (S t ) t∈R : For each t ∈ R ≥0 , the map S t : B → B is defined by where k is the smallest positive integer satisfying and for each t < 0, S t := (S −t ) −1 .
According to the representation theory of ergodic flow (see e.g. [1,7]), 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 Theorem 5 (Analyticity for Lyapunov exponents of generic bounded linear random differential equations). For each A ∈ L ∞ (Ω, R d×d ), let γ d (A) ≤ · · · ≤ γ 1 (A) denote the Lyapunov exponents of the linear random differential equatioṅ Then, the following statements hold: (i) Suppose that Case A holds. Then, there exists an open and dense set R ⊂ L ∞ (Ω, R d×d ) such that the functions γ 1 (·), . . . , γ d (·) are analytic at A ∈ R. (ii) Suppose that Case B holds. Then, the functions γ 1 (·), . . . , γ d (·) are analytic at all A ∈ L ∞ (Ω, R d×d ).
Proof. Define a function T : where Φ A denote the linear random dynamical system generated by (17) and H is defined as in (15). By variation of constants formula, we have for all ∆ ∈ L ∞ (Ω, R d×d ) where the operator L A : Obviously, L A is a linear operator and from the boundedness of A and (16), the operator L A is bounded. Consequently, by Lemma 2 (ii), the function T is analytic.
In the remaining part of the proof, we consider two separated cases: (i) Suppose that Case A holds. As is proved in [6, Theorem 3.2], there exists an open and dense set R ⊂ L ∞ (Ω, R d×d ) such that for all A ∈ R, the generated random dynamical system Ψ A and hence also T (A) are integrally separated. This together with analyticity of T , Theorem 4 and Lemma 2 completes the proof of this part.
(ii) Suppose that Case B holds. In this case, (Ω, F, P, (θ t ) t∈R ) can be identified with (S 1 , B(S 1 ), λ, (R t ) t∈R ). Choose and fix an arbitrary ω 0 ∈ S 1 and consider a map P : L ∞ (Ω, R d×d ) → R d×d defined by PA := Ψ A (1, ω 0 ). This is clear that the Lyapunov exponents of the linear random dynamical systems generated bẏ ξ = A(θ t ω)ξ coincides with the set of the modulus of the eigenvalues of PA. Analog to the proof of the analyticity of the map T above, the map P is also analytic. This together with the fact that the eigenvalues of matrices depend analytically on the matrices (see e.g. [11]) proves (ii). The proof is complete.
Appendix. Throughout this section, let (X, A, m) be a probability space and T : X → X is an ergodic transformation preserving the probability m.
The canonical inner product on R d induces an inner product ·, · on Λ k R d via Let {e i } i=1...,d denote the standard Euclidean orthogonal basic of R d . For each k ∈ {1, . . . , d}, we define e i1...i k := e i1 ∧ · · · ∧ e i k for 1 ≤ i 1 < · · · < i k ≤ d. Then, where L(Λ k R d ) denotes the space of linear operators from Λ k R d into itself. The generated linear random dynamical system by Λ k A on Λ k R d , which is denoted by Λ k Φ A , is called the k fold exterior of Φ A .
Positive linear random dynamical systems. A subset C ⊂ R d is called a proper cone if tv ∈ C for all t ≥ 0, v ∈ C and C ∩ (−C) = {0}. For a closed proper convex cone C, the dual cone C is defined by Recall that the Hilbert metric d C on C \ {0} is defined by d C (x, y) := log(β C (x, y)) + log(β C (y, x)) for x, y ∈ C \ {0}, where β C (x, y) := inf{t > 0 : tx − y ∈ C}. In what follows, the Hilbert metric d C will be understood in a wider sense that d C (±x, ±y) := d C (x, y) for all (x, y) ∈ C.
Theorem 7. Let (C x ) x∈X be a family of closed proper convex cones satisfying the following conditions: (C1) there is a measurable mapping c : X → R d such that c(x) = 1 and B(c(x), r) ⊂ C x with r is independent of x, (C2) there is a measurable mapping c : X → R d such that c (x) = 1 and B(c (x), r ) ⊂ C x with r is independent of x. Let A ∈ L ∞ (X, GL(d)) satisfy that A(x)C x ⊂ C T x ∪ (−C T x ) and there exists R < ∞ such that Hence, it is sufficient to prove (i) and (ii) for T A. By definition of r, the random matrix T A preserves the family of cones C x in the sense that T A(x)C x ⊂ C T x . Hence, using results in [2] yields (i). Meanwhile, the assertion (ii) is obtained by applying the result in [9]. The proof is complete.