Takens theorem for random dynamical systems

In this paper, we study random dynamical systems with partial hyperbolic fixed points and prove the smooth conjugacy theorems of Takens type based on their Lyapunov 
exponents.


1.
Introduction. This paper is a continuation of our previous one [21] on smooth linearization for random dynamical systems. In [21] we proved the smooth conjugacy theorems of Sternberg type for random dynamical systems based on their Lyapunov exponents, where we assumed that there is no zero Lyapunov exponent.
1 n log − ρ(θ n ω) = 0, P − a.s.), such that each sup is tempered from above (i.e., lim n→±∞ 1 n log + C i (θ n ω) = 0, P − a.s.). The size of a tempered ball may decrease as ω varies, but these changes along each orbit θ n ω are at a subexponential rate. The upper bound of C i (ω) may grow to infinity as ω varies. But along each orbit θ n ω, it may increase only at a subexponential rate. This nonuniform behavior is one of the intrinsic features of random dynamical systems.
Two local tempered random diffeomorphisms φ and ψ with fixed point x = 0 are C k locally conjugate for 1 ≤ k ≤ ∞ if there exists a C k random diffeomorphism h(ω, x) defined on a tempered ball V (ω) with h(ω, 0) = 0 such that h(θω, φ(ω, x)) = ψ(ω, h(ω, x)) for x ∈ V (ω), a. s. ω ∈ Ω This conjugacy relationship implies that h carries orbits of φ to orbits of ψ, when the orbits stay in the corresponding domains.
For the remainder of the paper, we assume that θ n is ergodic. Otherwise, we restrict our study to each ergodic component.
In the next section, we will see that there is a linear random transformation which changes the linear part A(ω) of φ to a block diagonal: and preserve the Lyapunov spectrum.
1 A large class of random dynamical systems may be converted into this case. See [1], page 310 Our main result can be summarized as Theorem. (Takens Theorem for Random Diffeomorphisms) For each integer k > 0, there exists a positive integer N 0 = N 0 (k) such that (i) If φ and ψ are C N0 locally tempered random diffeomorphisms and have a common center manifold , then φ and ψ are C k locally conjugate. (ii) If φ is C N0 tempered and the nonzero Lyapunov exponents satisfy where (λ h , τ ) := λi =0 λ i τ i , then φ is locally C k conjugate to a RDS whose time one map has the form If, in addition to the condition (2), the nonzero Lyapunov exponents also satisfy then φ is locally C k conjugate to a RDS whose time one map has the form Remark. (i) When φ is a deterministic diffeomorphism, this result was proved by Takens [32]. (ii) When θ n is not ergodic, the quantities such as the dimensions of the stable and unstable Oseledets subspaces may vary as ω changes. Ω may be decomposed into a union of countably many disjoint θ-invariant measurable setsΩ i such that on eachΩ i , these quantities are constant. The construction of conjugacy is carried out independently over each invariant set Ω i . Patching them together gives the conjugacy over the whole Ω.
The study of smooth conjugacy of deterministic dynamical systems to their normal forms has a long and rich history. Analytic linearization was first studied by Poincaré and Birkhoff, later by Siegel [40], Arnold [3], Moser [28], Zehnder [44], Brjuno [8], and others. There is an extensive literature on C k smooth linearization that was initiated by Sternberg [41,42]. Some classical results may be found in Nagumo and Isé [30] , Chen [10], Hartman [16,17], and Nelson [29]. For more delicate conditions such that the nonlinear system admits a C k -smooth linearization we refer to Sell [37,38], Beleskii [6,7], ElBialy [13], Zhang and Zhang [45]. A result which preserves the geometric structure of the original equation was recently obtained by Banyaga, de la Llave, and Wayne [5]. A C 0 linearization (A Hartman-Grobman Theorem) associated with the local structural stability may be found in Grobman [14], Hartman [16], Pugh [31], and Kirchgraber and Palmer [33].
For random dynamical systems, Wanner [43] proved a Hartman-Grobman theorem, also see [1] and [12] for topological conjugacy. A structural stability theorem for deterministic dynamical systems under random perturbations was obtained by P-D. Liu [24]. In [2], also see [1], Arnold and Xu gave a theorem on formal linearization of random diffeomorphisms. In [21], we established theorems of Poincaré type for random dynamical systems. In [22], we established theorems of Sternberg type for random dynamical systems. A theorem of Siegel type for random dynamical systems was presented in [23].
In Section 2, we introduce basic concepts on random dynamical systems, the Multiplicative Ergodic Theorem, and some basic lemmas; In Section 3, we prove our main results.
2. Random dynamical systems and center manifolds. In this section, we first review some of the basic concepts and results on random dynamical systems including the Multiplicative Ergodic Theorem, which are taken from [1]. We also introduce basic notations and state the assumptions on the systems. Then, we review the theorems on center-unstable manifolds, center-stable manifolds, and center manifolds for random dynamical systems, which we borrowed from [15].
Note that F (ω, 0) = 0 and D x F (ω, 0) = 0. In addition, Φ(n, ω) is generated by A(ω): Let {x n } n∈Z be an orbit of φ(n, ω, x) with initial value x 0 , i.e., x n := φ(n, ω, x). Then, {x n } n∈Z satisfies the equation Conversely, a solution of the above equation is also an orbit of φ(n, ω, x 0 ). The next concept is of fundamental importance in the study of random dynamical systems.  Moreover, we recall that a multifunction W = {W (ω)} ω∈Ω of nonempty closed
Theorem 2.1. (Multiplicative Ergodic Theorem) Let Φ be a linear random dynamical system over the metric dynamical system (Ω, F, P, (θ n ) n∈Z ). Assume that Then there exists an invariant subsetΩ ⊂ Ω of full measure such that for each ω ∈Ω the following hold: Then is ergodic, the functions p(ω), λ i (ω) and d i (ω) are constant onΩ. (4): For each ω ∈Ω, there exists a splitting equivalently, Here λ i (ω) and E i (ω) are so-called Lyapunov exponents and Oseledets spaces, respectively. In the remainder of this paper, we denoteΩ by Ω and assume that all statements are true for ω ∈ Ω.
We divide the Lyapunov exponents into three groups based their signs. Let and denote with corresponding projections and d c (ω) are measurable functions from Ω to {1, . . . , d} and P s (ω), P u (ω) and P c (ω) are measurable projections. When θ is not ergodic, Ω can be decomposed into a union of l disjoint θ-invariant measurable sets where on each Ω i , d u (ω), d s (ω) and d c (ω) are constant. One can build the centerunstable manifold, the center-stable manifold and the center manifold over Ω i , then patch them together to get dimension-varying invariant manifolds on the whole Ω.
Next, we assume that the fixed point is partially hyperbolic.
The following lemma is on the nonuniform partial hyperbolicity of the system, which is a consequence of the Multiplicative Ergodic Theorem 2.1.
Here β(ω) is chosen to be smaller than the absolute values of all non-zero Lyapunov exponents. For example, one may choose As ω varies, β(ω) may be arbitrarily small and K(ω) may be arbitrarily large. However, along each orbit θ n ω, β(ω) is a constant and K(ω) can increase only at a subexponential rate. When θ is ergodic, both α and β are constants.
Thus, the unstable, stable and center Oseledets subspaces for Ψ arẽ Then R d has an orthogonal decomposition: We still use P u (ω), P s (ω), and P c (ω) to denote the corresponding projections. The following lemma is a consequence of Lemmas 2.1 and 2.2. Lemma 2.3. Assume that Hypotheses 1 and 2 hold. There exists a θ-invariant random variables β : Ω → (0, ∞) such that for each θ-invariant random variables α(ω) > 0 satisfying α(ω) < β(ω)/2N there is a tempered random variable K(ω) : Set P cu (ω) = P c (ω) + P u (ω), P cs (ω) = P c (ω) + P s (ω), We call E cu (ω) center-unstable Oseledets subspace and E cs (ω) center-stable Oseledets subspace. On Ω i , by Lemma 2.2, R d has an invariant splitting R d = E cu ⊕E s (R d = E cs ⊕ E u ) independent of ω. Surely, the projection operators P cu , P s (P cs , P u ) are also independent of ω on Ω since θ is ergodic. For each x ∈ R d , we write it as x = x cu + x s for some x cu ∈ E cu and x s ∈ E s , x = x cs + x u for some x cs ∈ E cs and x u ∈ E u .

2.3.
Random center manifolds. In this subsection, we state the random centerunstable, center-stable, and center manifolds, which we take from [15].
For the nonlinear term f (1, ω, x) we assume that where theB k are tempered from above and the L k (R d , R d ) are the Banach space of all k-linear maps from R d to R d with the norm · L k (R d ,R d ) .
Then f (1, ω, ·) is extended to the outside of U (ω). By simple calculation, we have the following lemma.
for all 2 ≤ k ≤ N and ω ∈ Ω, where each B k (ω) is random variable tempered from above.
By choosing sufficiently small tempered radius ρ(ω), one has that ψ(1, ω, {x n } n∈Z is an orbit of φ(n, ω, x 0 ) if and only if Then by Lemma 2.4(1) we obtain that for every sequence x n ∈ B(0, ρ(θ n ω)), x n = φ(n, ω, x 0 ) if and only if From now on, we consider modified equation (7). To simplify the notation, the modified random dynamical system is still denoted by φ(n, ω, x).
The intersection of the center-unstable manifold and the center-stable manifold gives the center manifold as follows.
Theorem 2.4. (Center Manifold Theorem) Assume that Hypotheses 1-3 hold. Then for a sufficiently small tempered radius ρ(ω), φ(n, ω, x) has a C N center manifold which is given by where h c : E c × Ω → E u ⊕ E s satisfies the following: } is a local center-stable manifold for the original system (4).

Smooth conjugacy.
In this section, we prove our main results. First, we introduce the following concept.
The next theorem gives a smooth conjugacy in a jet class.
Theorem 3.2. Let φ be a random dynamical system satisfying Hypothesis 1-3. Then for any integer k > 0, there exists a θ-invariant measurable integer function N 0 = N 0 (α, β, k) such that if φ and ψ are C N (N ≥ N 0 ) locally tempered and have a common center manifold W c (ω) with φ − ψ = O(|x| N ) for ω ∈ Ω, x ∈ W c (ω), then φ and ψ are C k locally conjugate.
The proof of this theorem is based on the following lemmas and proposition. Some of them are directly taken from our previous work [22]. We first applying the center-unstable manifold theorem (Theorem 2.2) and the center-stable manifold theorem (Theorem 2.3) to φ. Then, φ has a C N local center-unstable manifold We may identify E u , E c and E s with R du , R dc and R ds , respectively. Next, we use the center-stable, center-unstable and the common center manifolds as new axes to rewrite φ. Consider the coordinate random transformation defined on B r(ω) (0) It also follows from Theorem 2.2 and Theorem 2.3 that this transformation is a C N random diffeomorphism. By this transformation, the random diffeomorphism φ(1, ω) is locally conjugate to a C N random diffeomorphism where A(ω) = Dφ(1, ω) is written as diag(A s (ω), A c (ω), A u (ω)), F 1 (ω, x) satisfies Hypothesis 3, and F 1 (ω, 0) = 0, DF 1 (ω, 0) = 0. Furthermore, which implies that y = 0 is the center-stable manifold, z = 0 is the center-unstable manifold and y = 0, z = 0 is the common center manifold for the new random dynamical system.
In the following, we will decompose R into two parts, one part is dominated by a power of y while another is dominated by a power of z. Lemma 3.3. The function R can be written as R = R 1 + R 2 , where R i (ω, x) are measurable and R i (ω, ·) are C [N/2] tempered functions satisfying Π s R 2 (ω, 0, w, z) = 0, Π u R 1 (ω, y, w, 0) = 0, Proof. Consider the Taylor expansion of R of order [N/2] with respect to the variable y: [N/2] ) and a i is a homogeneous polynomial of variable y of degree i whose coefficients are the measurable functions of (ω, w, z) and C [N/2] with respect to the variable (w, z). Since jet N y=0,z=0 R = 0, we have ω, y, w, z). Then (10) holds. Other properties of R 1 and R 2 follows from the properties of F 1 and F 2 . This completes the proof of the lemma.

Lemma 3.4. [22]
For any given integer k > 0, there exist a constant M k (ω) = M k (α, β) and a random variable d k (ω) tempered from above such that The next lemma states that one may reduce the problem of C k conjugacy (8) to a problem of solving a linear recurrent functional equation. This idea is based on the so-called homotopy method which was used in [18] and [19] for finitely smooth normal forms of deterministic dynamical systems. We extend it in [22] further so that it can be applied to random dynamical systems where the partial hyperbolicity is nonuniform.
Next, we construct formal solutions of linear functional equation (15) in terms of an infinite series. We will prove later that they are convergent for R being R 1 and R 2 .
For simplicity, we set (f * R)(x) := (Df R)•f −1 (x) for a diffeomorphism f : R d → R d and a vector field R in R d . Notice that f * (g * R) = (f • g) * R. Then, we have two formal solutions to equation (15). and are formal solutions of (15).
We are now ready to show the convergence of the formal series solutions. Let ω, x)), thenR 1 andR 2 are C [N/2] (resp. C ∞ ) tempered and satisfy where functionsC 0 is tempered from above. The next lemma gives two smooth solutions of (15) with R =R 1 andR 2 .
This proposition together with Lemma 3.3 and the random tranformation (9) gives Theorem 3.2.
Denote by H n,r,s the vector space of homogeneous polynomials of degree n ≥ 2 in r variables with values in R s . Lemma 3.9. Let φ and ψ be two linear cocycles in R r and R s over the metric DS (Ω, F, P, θ(t) t∈Z ) respectively. Let φ and ψ be generated by A : Ω → Gl(r, R) and B : Ω → Gl(s, R) respectively. We assume that Proof. Consider a RDS in R r × R s over θ whose time one map is Let a(ω) be the solution of the cocycle generated by the affine difference equation a n+1 = M (θ n ω, 0)a n − b(θ n ω, 0).
Then, by Theorem 5.6.5 in [1], a(ω) is tempered. Let u = x, v = y − a(ω). Then, ψ in the new coordinate has the form The origin (0, 0) is a fixed point of ψ and the linearization matrix of ψ at (0,0) is  M (ω, 0)). By center manifold theorem 2.4, there exist a local center manifold W c (ω) which is a graph of , which implies that the function h = h c + a(ω) satisfies the equation (22). Lemma 3.11. Let φ be a C N tempered RDS in R d1 ⊕ R d2 over a metric DS θ whose time one map has the form where w(ω, 0) = 0 and f is a homogeneous polynomial of variable v of degree n with values in R d1 whose coefficients are locally C N tempered functions of variable u. If the Lyapunov spectrum Sp(θ, Dw(ω, 0)) = {0} and the Lyapunov spectrum Λ = Sp(θ, A(ω, 0)) satisfies (Λ, τ ) = 0, f or |τ | = n, then φ is locally C N −1 conjugate to a C N −1 tempered RDS whose time one map has the form Proof. Let H ω (u, v) = (u + h(ω, u, v), v), where h is a homogeneous polynomial of variable v of degree n with values in R d1 whose coefficients are locally C N tempered functions of variable u. Then If we denote by h ω (u) and b(ω, u) the coefficient vectors of the polynomials h(ω, u, v) and b(ω, u, v) respectively, then the equation (24) can be written in the form By Lemma 3.9 and Lemma 3.10, equation (25) has locally a C N −1 tempered solution h ω .
Proof. Let H ω (u, v) = (u, v + h(ω, u, v)), where h is a homogeneous polynomial of variable v of degree n with values in R d2 whose coefficients are locally C N tempered functions of variable u. Then If we denote by h ω (u) and b(ω, u) the coefficient vectors of the polynomials h(ω, u, v) and b(ω, u, v) respectively, then the equation (27) can be written in the form By Lemma 3.9 and Lemma 3.10, equation (28) has locally a C N −1 tempered solution h ω .
Proof of the main theorem. The conclusion (i) follows from Lemma 3.11 and Theorem 3.2. The conclusion (ii) follows from the conclusion (i), Lemma 3.12 and Theorem 3.2.