Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 19, pp. 1–25. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOME APPLICATIONS OF LYAPUNOV REGULARITY LUÍS BARREIRA, CLAUDIA VALLS Abstract. For a non-autonomous dynamics with discrete time defined by a tempered sequence of upper-triangular matrices, we obtain lower and upper bounds for the Grobman regularity coefficients. We also give two applications of these results: we obtain an upper bound for the Grobman coefficients of the exterior powers of a tempered sequence, and we give a simple proof of Os- eledets’ multiplicative ergodic theorem for cocycles over a measure-preserving transformation without using Kingman’s subadditive ergodic theorem. 1. Introduction Our main objective is to discuss various properties and several applications of regularity coefficients, and specifically of the Grobman coefficient. More precisely, we consider a non-autonomous dynamics with discrete time defined by a tempered sequence of upper-triangular matrices and we obtain lower and upper bounds for their Grobman coefficients. The applications of these bounds include a sharp upper bound for the Grobman coefficients of the exterior powers of a tempered sequence of upper-triangular matrices as well as a simple proof of Oseledets’ multiplicative ergodic theorem in [8] for cocycles over a measure-preserving transformation. 1.1. Lyapunov regularity. We start with a brief discussion about the relevance and in fact the importance of Lyapunov regularity. The notion was introduced by Lyapunov in [6] (see [7] for an English translation) and plays an important role in the Lyapunov stability theory. It turns out that it allows one to study the persistence of the stability and of the conditional stability of a linear dynamics under sufficiently small nonlinear perturbations (see for example [2]). This is particularly effective in the context of ergodic theory since for a smooth dynamics preserving a finite measure satisfying a mild integrability assumption, the linearization along almost all trajectories is Lyapunov regular. This is a consequence of Oseledets’ multiplicative ergodic theorem (further details are given below). We illustrate the role of Lyapunov regularity with a simple example in the case of continuous time. Consider a linear equation v′ = A(t)v (1.1) on Rq for some q × q matrices A(t) whose entries vary continuously with t ∈ R. We assume that supt∈R ‖A(t)‖ < +∞. Its Lyapunov exponent is the function 2020 Mathematics Subject Classification. 37C75. Key words and phrases. Lyapunov regularity; exterior powers; cocycles. ©2022. This work is licensed under a CC BY 4.0 license. Submitted February 11, 2021. Published March 18, 2022. 1 2 L. BARREIRA, C. VALLS EJDE-2022/19 λ : Rq → R ∪ {−∞} defined by λ(v0) = lim sup t→∞ 1 t log ‖v(t)‖, (1.2) where v(t) denotes the solution of (1.1) with v(0) = v0 (note that all solutions are global), with the convention that log 0 = −∞. The Lyapunov coefficient of equation (1.1) is defined by σ(A) = min q∑ i=1 λ(vi)− lim inf t→∞ 1 t ∫ t 0 trA(τ) dτ, where the minimum is taken over all bases v1, . . . , vq for Rq (see [7]). One can easily verify that σ(A) ≥ 0. The equation is said to be Lyapunov regular or simply regular if σ(A) = 0 or, equivalently, if lim t→∞ 1 t ∫ t 0 trA(τ) dτ = q∑ i=1 λ(vi) for some basis v1, . . . , vq for Rq. If the Lyapunov exponent λ takes only negative val- ues, then equation (1.1) is asymptotically stable: any solution v(t) of the equation tends to zero when t→ +∞. However, the equation need not be uniformly asymp- totically stable: in order that a given solution is contained in an ε-neighborhood of the origin, one may need to take the initial condition sufficiently small depending on the initial time. A consequence is that the type of stability may change under arbitrarily small nonlinear perturbations. Indeed, it was shown by Perron in [10] that there are examples of asymptotically stable equations v′ = A(t)v for which some perturbation u′ = A(t)u+ f(u) has a solution u(t) with lim sup t→∞ 1 t log ‖u(t)‖ > 0. On the other hand, Lyapunov [7] showed that for a regular equation the asymp- totic stability persists under sufficiently small nonlinear perturbations, such as for example when ‖f(u)− f(v)‖ ≤ c‖u− v‖(‖u‖r + ‖v‖r) for all t ∈ R and u, v ∈ Rq, for some constants c, r > 0 (see [2] for details). 1.2. Context of ergodic theory. As noted above, the notion of Lyapunov regu- larity is particularly effective in the study of the persistence of stability and con- ditional stability under sufficiently small nonlinear perturbations in the context of ergodic theory. More precisely, Lyapunov regularity turns out to be typical under fairly general assumptions in ergodic theory. To formulate a precise statement, recall that a measure µ on Rq is invariant under a flow (φt)t∈R if µ(φt(A)) = µ(A) for every measurable set A ⊂ Rq and t ∈ R. In particular, any flow defined by a differentiable vector field with zero divergence, such as any Hamiltonian vector field, preserves the Lebesgue measure. Now consider a differential equation x′ = F (x) on Rq for some C1 vector field F . We assume that EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 3 the equation generates a flow (φt)t∈R preserving a finite measure µ with compact support on Rn. Then for µ-almost every x ∈ Rq the linear variational equation v′ = Ax(t)v with Ax(t) = dφt(x)F (1.3) is Lyapunov regular. This statement is a special case of Oseledets’ multiplicative ergodic theorem in [8]. On the other hand, since the general solution of equation (1.3) is v(t) = (dxφt)v0, with t ∈ R and v0 ∈ Rq, the Lyapunov exponent λ in (1.2) becomes λ(v0) = lim sup t→+∞ 1 t log ‖(dxφt)v0‖ for a given x ∈ Rq. Therefore, the values of the Lyapunov exponent are naturally related to the exponential growth rates of contraction and expansion of the matrices dxφt and so to the hyperbolicity of the linear variational equation in (1.3). Indeed, as noted above, if the Lyapunov exponent λ takes only negative values, then the equation is asymptotically stable (although not necessarily uniformly asymptoti- cally stable). This leads naturally to the notion of nonuniform hyperbolicity. The classical notion of uniform hyperbolicity, essentially introduced by Perron in [11], plays an important role in a large part of the theory of differential equations and dynamical systems. Some of its consequences are the existence of topological conjugacies and of stable and unstable invariant manifolds under sufficiently small nonlinear perturbations. On the other hand, the existence of uniform hyperbolicity is a rather stringent condition. The notion of nonuniform hyperbolicity introduced by Pesin in [12] is a much weaker requirement and so it is also much more common. Among the most important properties due to nonuniform hyperbolicity are the existence of stable and unstable manifolds and their absolute continuity, the ergodic properties of dynamical systems preserving a finite measure that is absolutely continuous with respect to the volume, a formula for the Kolmogorov-Sinai entropy in terms of the Lyapunov exponents, a rich orbit structure, an exponential growth rate for the number of periodic points, and an approximation by uniformly hyperbolic horseshoes. We refer the reader to the book [3] for references and for a detailed presentation of a large part of the theory. 1.3. Regularity coefficients. There are several regularity coefficients that can be used to define Lyapunov regularity, such as those introduced by Perron (see [9, 11]) and Grobman (see [5]). Some of them are more amenable than others in specific situations, although all of them are related. More precisely, not only all of them vanish if and only if the dynamics is regular, but also the ratio of any two of them (over all nonregular dynamics) is bounded and bounded away from zero. From this point of view, it is irrelevant to obtain lower and upper bounds for any specific regularity coefficient since then these can be transferred to any other regularity coefficient. Here we use the regularity coefficient due to Grobman. We refer the reader to the books [1, 5] for detailed accounts of the theory. Also as noted above, the type of stability persists when the Lyapunov coefficient is sufficiently small when compared to the Lyapunov exponents, but not necessarily zero. Since the Lyapunov coefficient may be hard to compute, it is convenient to have sharp bounds (the best would be to have in terms of the Lyapunov exponents but these are also hard to compute). This justifies the interest of the present work. 4 L. BARREIRA, C. VALLS EJDE-2022/19 Now we formulate briefly our results that give lower and upper bounds for the Grobman coefficients. Consider a sequence of invertible upper-triangular q × q matrices (Am)m∈N and let e1, . . . , eq be the canonical basis for Rq. Since the ma- trices Am are upper-triangular, the space Ek ⊂ Rq generated by the first k vectors e1, . . . , ek, for k = 1, . . . , q, is invariant under all the matrices. Moreover, Ek is isomorphic to Rk and the restriction of each matrix Am to Ek can be identified with a k × k upper-triangular matrix Am,k. We denote by γk the Grobman coeffi- cient of the pair of Lyapunov exponents associated with the sequences of matrices (Am,k)m∈N and (Bm,k)m∈N, where Bm,k = (A∗m,k)−1 (see Section 2 for the defini- tion of the Grobman coefficient). The sequence (Am)m∈N is said to be tempered if lim sup m→+∞ 1 m log+ ‖Am‖ = 0, (1.4) where log+ x = max{0, log x}. Moreover, denoting by amij the entries of Am, for each i = 1, . . . , q let αi = lim inf m→+∞ 1 m log m∏ l=1 |alii| and αi = lim sup m→+∞ 1 m log m∏ l=1 |alii|. The following result is a consequence of Theorems 3.1 and 4.7 below. Theorem 1.1. Let (Am)m∈N be a tempered sequence of invertible q × q matrices whose Lyapunov exponent λ takes only finite values on Rq \ {0}. If Am is upper- triangular for every m ∈ N, then max k=1,...,q γk ≥ 1 q2 q∑ i=1 (αi − αi) and γq ≤ q∑ i=1 (αi − αi). (1.5) We emphasize that there is no loss of generality in considering only upper- triangular (or lower-triangular) matrices. Indeed, given a sequence of invertible q×q matrices (Am)m∈N there exists a sequence of orthogonal q × q matrices (Um)m∈N such that Cm = U∗m+1AmUm is upper-triangular for each m ∈ N (see [1, Theo- rem 3.2.1]). Note that since the matrices Um are orthogonal, the values of the Lyapunov exponent of the sequence (Cm)m∈N are the same as those of the Lya- punov exponent of the sequence (Am)m∈N, together with their multiplicities. To the best of our knowledge, the lower bound in (1.5) is new —a different lower bound, although not expressed in terms of the numbers αi and αi, was given in [4]. The upper bound in (1.5) appeared before, although here we correct a mistake in the original proof (namely, it is tacitly assumed in [4, Lemma A.1] that the limit of the left-hand side of equation (A.6) is zero). The methods of proof to obtain the lower and upper bounds in (1.5) are different. The lower bound is obtained through estimates for the numbers αi and αi in terms of the exponential growth rate of volumes for the restrictions of the dynamics to the subspaces Ek. On the other hand, the upper bound is obtained by first constructing dual bases with specific estimates for their Lyapunov exponents that then are used to obtain the upper bound for the Grobman coefficient. More precisely, we show that there exist dual bases v1 . . . , vq and w1 . . . , wq for Rq such that λ(vj) ≤ αj + j−1∑ k=1 (αk − αk), EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 5 µ(wj) ≤ −αj + q∑ k=j+1 (αk − αk) for j = 1, . . . , q (see Theorems 4.1 and 4.4). These estimates are then used to obtain the sharp upper bound for the Grobman coefficient in (1.5). The lower and upper bounds in (1.5) allow one to deduce the following charac- terization of Lyapunov regularity. Theorem 1.2. Let (Am)m∈N be a tempered sequence of invertible q × q upper- triangular matrices whose Lyapunov exponent λ takes only finite values on Rq \{0}. Then the sequence is Lyapunov regular if and only if αi = αi for i = 1, . . . , q. Theorem 1.2 is proved in Section 5. Since all norms on the space Rq are equiva- lent, all the results in the paper are independent of the particular choice of norm. So we can fix any norm on Rq from the beginning. 1.4. Applications of the results. We give two applications of the former results: we obtain an upper bound for the Grobman coefficient of the exterior powers of a tempered sequence of upper-triangular matrices and we give a simple proof of Oseledets’ multiplicative ergodic theorem for cocycles over a measure-preserving transformation. The exterior powers of a sequence of matrices (see Section 6 for the definition) relate well to the k-volumes in Rq with k ≤ q. Indeed, the k-volume determined by the vectors v1, . . . , vk is given by vol(v1, . . . , vk) = ‖v1 ∧ · · · ∧ vk‖. In view of the importance of the exponential growth rate of volumes in ergodic theory (see also [13, 14] for the relation with the entropy conjecture), it is quite relevant a corresponding study to that in the former section for exterior powers. In Section 6 we show that the Grobman coefficient γ(λk, µk) of the sequence of matrices (Λk(Am))m∈N satisfies γ(λk, µk) ≤ ( q − 1 k − 1 ) q∑ i=1 (αi − αi) for each integer k ∈ [1, q] (see Theorem 6.1). Together with Theorem 1.2, this implies that if the sequence of matrices (Am)m∈N is regular, then so is the sequence (Λk(Am))m∈N for k = 1, . . . , q. Finally, in Section 7 we give a proof of Oseledets’ multiplicative ergodic theo- rem for cocycles over a measure-preserving transformation, which says that under a certain integrability assumption the set of all regular points has full measure. The proof is based on the upper bound for the Grobman coefficient after making an ap- propriate upper-triangular reduction of the cocycle. Although the upper-triangular reduction is reminiscent of the original proof by Oseledets, our argument does not require Kingman’s subadditive ergodic theorem and is streamlined after having the upper bound. We also provide full technical details for this reduction, which is obtained through a lift of the cocycle to a linear extension on the product of the original space and the orthogonal group. 6 L. BARREIRA, C. VALLS EJDE-2022/19 2. Lyapunov regularity In this section we introduce the notion of Lyapunov regularity for the nonau- tonomous dynamics defined by a sequence of matrices and we recall one of its main characterizations. This will be used later on in the paper. Consider a sequence of invertible q × q matrices (Am)m∈N. We shall always assume that lim sup m→+∞ 1 m log ‖A±1m ‖ < +∞, (2.1) where Am = { Am−1 · · ·A1 if m > 1, Id if m = 1. This happens for example if the sequences (Am)m∈N and (A−1m )m∈N are bounded. The Lyapunov exponent λ : Rq → R∪{−∞} associated with (Am)m∈N is defined by λ(v) = lim sup m→+∞ 1 m log ‖Amv‖. Condition (2.1) implies that λ does not take infinite values on Rq\{0}. The sequence of matrices (Am)m∈N is said to be (Lyapunov) regular if lim m→+∞ 1 m log |detAm| = min q∑ i=1 λ(vi) (2.2) with the minimum taken over all bases v1, . . . , vq of Rq. We emphasize that the notion of regularity includes the requirement that the limit in the left-hand side of (2.2) exists. We also consider the sequence of matrices Bm = (A∗m)−1 for m ∈ N, where A∗ denotes the adjoint of A, and we let Bm = (A∗m)−1. The Lyapunov exponent µ : Rq → R ∪ {−∞} associated with (Bm)m∈N is defined by µ(w) = lim sup m→+∞ 1 m log ‖Bmw‖ Condition (2.1) implies that µ does not take infinite values on Rq \ {0}. We define the Grobman coefficient of λ and µ by γ(λ, µ) = min max { λ(vi) + µ(wi) : 1 ≤ i ≤ q } with the minimum taken over all dual bases v1, . . . , vq and w1, . . . , wq of Rq, that is, such that 〈vi, wj〉 = δij for all i and j, where δij is the Kronecker symbol. Of course, condition (2.1) ensures that the Grobman coefficient is well defined. We recall that a sequence of matrices (Am)m∈N is said to be tempered if (1.4) holds. For example, any bounded sequence is tempered. It can be shown that a tempered sequence of matrices (Am)m∈N is regular if and only if γ(λ, µ) = 0 (see for example [4]). We shall use this characterization of regularity together with an upper bound for the Grobman coefficient (see Theorem 4.7) to give a new streamlined proof of Oseledets’ multiplicative ergodic theorem in [8] (see Theorem 7.1). EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 7 3. Lower bound for the Grobman coefficients In this section we establish a lower bound for the Grobman coefficients of a sequence of upper-triangular matrices (Am)m∈N expressed in terms of the entries in the main diagonal. For each k = 1, . . . , q, let Am,k be the k × k upper-triangular matrix obtained from Am considering only the first k columns of its first k rows. We denote by γk = γk(λ, µ) the Grobman coefficient of the pair of Lyapunov exponents associated with the sequences of matrices (Am,k)m∈N and (Bm,k)m∈N, where Bm,k = (A∗m,k)−1. Moreover, denoting by amij the entries of Am, for each i = 1, . . . , q let αi = lim inf m→+∞ 1 m log m∏ l=1 |alii| and αi = lim sup m→+∞ 1 m log m∏ l=1 |alii|. (3.1) Theorem 3.1. Let (Am)m∈N be a sequence of invertible q × q matrices satisfy- ing (2.1). If Am is upper-triangular for every m ∈ N, then max i=k,...,q γk(λ, µ) ≥ 1 q2 q∑ i=1 (αi − αi). Proof. Given a basis v1, . . . , vq for Rq such that v1, . . . , vk is a basis for Ek (the space generated by the vectors e1, . . . , ek) for each k = 1, . . . , q, one can always assume that q∑ i=1 λ(v′i) ≥ q∑ i=1 λ(vi) (3.2) for all bases v′1, . . . , v ′ q for Rq. Moreover, let w1, . . . , wq be a basis for Rq such that q∑ i=1 µ(w′i) ≥ q∑ i=1 µ(wi) for all bases w′1, . . . , w ′ q for Rq (note that v1, . . . , vq and w1, . . . , wq need not be dual bases). By Hadamard’s inequality, we have |detBm| ≤ |detW |−1 q∏ i=1 ‖Bmwi‖, where W is the matrix with columns w1, . . . , wq, and so lim inf m→+∞ 1 m log |detAm| = − lim sup m→+∞ 1 m log |detBm| ≥ − q∑ i=1 µ(wi) = q∑ i=1 λ(vi)− q∑ i=1 (λ(vi) + µ(wi)). Now let λ′1 ≤ · · · ≤ λ′q and µ′1 ≥ · · · ≥ µ′q be the ordered values of the numbers λ(v1), . . . , λ(vq) and µ(w1), . . . , µ(wq), respec- tively, counted with their multiplicities. The number π(λ, µ) = max { λ′i + µ′i : 1 ≤ i ≤ q } 8 L. BARREIRA, C. VALLS EJDE-2022/19 satisfies π(λ, µ) ≤ γ(λ, µ) (see [1, Theorem 2.4.2]). Therefore, q∑ i=1 (λ(vi) + µ(wi)) = q∑ i=1 (λ′i + µ′i) ≤ qπ(λ, µ) ≤ qγ(λ, µ) and so lim inf m→+∞ 1 m log |detAm| ≥ q∑ i=1 λ(vi)− qγ(λ, µ). Letting Am,k = { Am−1,k · · ·A1,k if m > 1, Id if m = 1, one can show in a similar manner that lim inf m→+∞ 1 m log |detAm,k| ≥ k∑ i=1 λ(vi)− kγk for k = 1, . . . , q. On the other hand, since detAm,k = (detAm,k−1) m−1∏ l=1 alkk, (3.3) we obtain αk ≤ lim sup m→+∞ 1 m log |detAm,k| − lim inf m→+∞ 1 m log |detAm,k−1| ≤ k∑ i=1 λ(vi)− k−1∑ i=1 λ(vi) + (k − 1)γk−1 = λ(vk) + (k − 1)γk−1. Similarly, αk ≥ lim inf m→+∞ 1 m log |detAm,k| − lim sup m→+∞ 1 m log |detAm,k−1| ≥ k∑ i=1 λ(vi)− kγk − k−1∑ i=1 λ(vi) = λ(vk)− kγk. Therefore, αk − αk ≤ λ(vk) + (k − 1)γk−1 − λ(vk) + kγk = (k − 1)γk−1 + kγk ≤ (2k − 1) max k γk, which yields the inequality q∑ k=1 (αk − αk) ≤ q∑ k=1 (2k − 1) max k γk = q2 max k γk. This completes the proof of the theorem. � EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 9 4. Upper bound for the Grobman coefficients In this section we establish a corresponding upper bound for the Grobman coef- ficient of a tempered sequence of upper-triangular matrices in terms of the entries in the main diagonal. We start by constructing certain special bases that are then used to establish an upper bound for γ(λ, µ) in terms of the numbers in (3.1). Theorem 4.1. Let (Am)m∈N be a tempered sequence of invertible q × q matrices satisfying (2.1). If Am is upper-triangular for every m ∈ N, then there exists a basis v1 . . . , vq for Rq such that λ(vj) ≤ αj + j−1∑ k=1 (αk − αk) for j = 1, . . . , q. Proof. For each i, j = 1, . . . , q with i ≤ j, let βij = αj − αi + j−1∑ k=i+1 (αk − αk), (4.1) κij = βij + max i≤l≤j (αl − αl). (4.2) In what follows we make the convention that ∏q l=p cl = 1 when q < p. For each m ∈ N we consider the matrix Z(m) with entries zmij defined recursively as follows: (1) for i > j let zmij = 0; (2) for i = j let zmii = ∏m−1 l=1 alii; (3) for i < j with κij ≥ 0 let z1ij = 0 and for m ≥ 2 let zmij = m−2∑ p=0 j∑ l=i+1 am−p−1il zm−p−1lj m−1∏ r=m−p arii; (4.3) (4) for i < j with κij < 0 let zmij = − +∞∑ p=1 j∑ l=i+1 am+p−1 il zm+p−1 lj m+p−1∏ r=m (arii) −1. (4.4) Lemma 4.2. Assuming that the series in (4.4) converge whenever κij < 0, we have Z(m+ 1) = AmZ(m) for all m ∈ N. Proof. We first assume that κij ≥ 0. Then zm+1 ij = m−1∑ p=0 j∑ l=i+1 am−pil zm−plj m∏ r=m−p+1 arii = j∑ l=i+1 amil z m lj + m−1∑ p=1 j∑ l=i+1 am−pil zm−plj m∏ r=m−p+1 arii. We have m−1∑ p=1 j∑ l=i+1 am−pil zm−plj m∏ r=m−p+1 arii = amii m−1∑ p=1 j∑ l=i+1 am−pil zm−plj m−1∏ r=m−p+1 arii = amii z m ij 10 L. BARREIRA, C. VALLS EJDE-2022/19 since κij ≥ 0 and so zm+1 ij = j∑ l=i+1 amil z m lj + amii z m ij = j∑ l=i amil z m lj . Now we assume that κij < 0. Then zm+1 ij (amii ) −1 = − +∞∑ p=1 j∑ l=i+1 am+p il zm+p lj m+p∏ r=m (arii) −1 = − +∞∑ p=0 j∑ l=i+1 am+p il zm+p lj m+p∏ r=m (arii) −1 + j∑ l=i+1 amil z m lj (amii ) −1. Since κij < 0, we have − +∞∑ p=0 j∑ l=i+1 am+p il zm+p lj m+p∏ r=m (arii) −1 = zmij and so zm+1 ij (amii ) −1 = zmij + j∑ l=i+1 amil z m lj (amii ) −1, which gives zm+1 ij = amii z m ij + j∑ l=i+1 amil z m lj = j∑ l=i amil z m lj . This completes the proof. � Now let λij = lim sup m→+∞ 1 m log |zmij |. Lemma 4.3. For i, j = 1, . . . , q, we have λii = αi and for each i < j the entry zmij is well defined for all m ∈ N and satisfies λij ≤ αj + j−1∑ k=i (αk − αk). Proof. Clearly, λii = αi for i = 1, . . . , q. For the entries with i < j we proceed by backwards induction on i. Given i < q, assume that for i+ 1 ≤ l ≤ j the entry zmlj is well defined for all m ∈ N and satisfies λlj ≤ αj + j−1∑ k=l (αk − αk). We show that zmij is well defined for all m ∈ N (which is immediate when κij ≥ 0 because it is given by a finite sum) and λij ≤ αj + j−1∑ k=i (αk − αk). (4.5) EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 11 Assume that κij < 0. By (1.4) and the induction hypothesis, for each ε > 0 there exists D > 0 such that |anil| ≤ Deεn, D−1e(αi−ε)n ≤ n∏ l=1 |alii| ≤ De(αi+ε)n, |znlj | ≤ De[αj+ ∑j−1 k=l (αk−αk)+ε]n for all n ∈ N and i+ 1 ≤ l ≤ j. Therefore, A := +∞∑ p=1 j∑ l=i+1 |am+p−1 il zm+p−1 lj | m+p−1∏ r=m |arii|−1 ≤ +∞∑ p=1 j∑ l=i+1 Deε(m+p−1)De[αj+ ∑j−1 k=l (αk−αk)+ε](m+p−1)|arii|−1 m−1∏ r=1 |arii| ≤ D′ +∞∑ p=1 j∑ l=i+1 e[αj−αi+ ∑j−1 k=l (αk−αk)+3ε](m+p) m−1∏ r=1 |arii| for some constant D′ > 0. Since κij < 0, we have βij < 0. Now take ε > 0 such that βij + 3ε < 0. Then A ≤ D′q +∞∑ p=1 e(βij+3ε)(m+p) m−1∏ r=1 |arii| < D′q 1− eβij+3ε m−1∏ r=1 |arii| < +∞. (4.6) This shows that zmij is also well defined when κij < 0. Finally, we establish the estimate in (4.5). First assume that κij ≥ 0. It follows from (4.3) that λij ≤ lim sup m→+∞ 1 m log m−2∑ p=0 j∑ l=i+1 |am−p−1il zm−p−1lj | m−1∏ r=m−p |arii| ≤ lim sup m→+∞ 1 m log m−2∑ p=0 j∑ l=i+1 D2eε(m−p−1)+[αj+ ∑j−1 k=l (αk−αk)+ε](m−p−1) × m−1∏ r=1 |arii|De(−αi+ε)(m−p−1) ≤ αi + lim sup m→+∞ 1 m log ( D′e3εm m−2∑ p=0 eβij(m−p) ) ≤ αi + lim sup m→+∞ 1 m log(meβijm) + 3ε = αj + j−1∑ k=i (αk − αk) + 3ε for some constant D′ > 0. Since ε is arbitrary, this yields inequality (4.5). Now assume that κij < 0, which implies that βij < 0. Take ε > 0 such that βij +3ε < 0. By (4.6) we have λij ≤ lim sup m→+∞ 1 m log ( D′q e(βij+3ε)(m+1) 1− eβij+3ε m−1∏ r=1 |arii| ) 12 L. BARREIRA, C. VALLS EJDE-2022/19 = βij + 3ε+ lim sup m→+∞ 1 m log(De(αi+ε)m) = αj + j−1∑ k=i (αk − αk) + 3ε. Again, since ε is arbitrary, this yields inequality (4.5). � Now consider the basis formed by the columns of Z(1), that is, the vectors vj = Z(1)ej for j = 1, . . . , q. By Lemma 4.3 we obtain λ(vj) = max { λij : 1 ≤ i ≤ q } ≤ αj + j−1∑ k=1 (αk − αk) for j = 1, . . . , q, which completes the proof of the theorem. � Now we consider the matrices Bm. Theorem 4.4. Let (Am)m∈N be a tempered sequence of invertible q × q matrices satisfying (2.1). If Am is upper-triangular for every m ∈ N, then there exists a basis w1 . . . , wq for Rq such that µ(wj) ≤ −αj + q∑ k=j+1 (αk − αk) for j = 1, . . . , q. Proof. We continue to define βij and κij by (4.1) and (4.2). For each m ∈ N we consider the matrix W (m) with entries wmij defined recursively by: (1) for i < j let wmij = 0; (2) for i = j let wmii = ∏m−1 l=1 (alii) −1; (3) for i > j with κji ≥ 0 let w1 ij = 0 and for m ≥ 2 let wmij = − m−2∑ p=0 i−1∑ l=j am−p−1li wm−plj m−1∏ r=m−p−1 (arii) −1; (4.7) (4) for i > j with κji < 0 let wmij = +∞∑ p=1 i−1∑ l=j am+p−1 li wm+p lj m+p−2∏ r=m arii. (4.8) Lemma 4.5. Assuming that the series in (4.8) converge whenever κji < 0, we have W (m+ 1) = BmW (m) for all m ∈ N. Proof. In view of the definition of Bm, the statement is equivalent to show that W (m) = B−1m W (m+ 1) = A∗mW (m+ 1). We first assume that κji ≥ 0. Then wm+1 ij = − m−1∑ p=0 i−1∑ l=j am−pli wm+1−p lj m∏ r=m−p (arii) −1 = − i−1∑ l=j amli w m+1 lj (amii ) −1 − m−1∑ p=1 i−1∑ l=j am−pli wm+1−p lj m∏ r=m−p (arii) −1. EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 13 We have − m−1∑ p=1 i−1∑ l=j am−pli wm+1−p lj m∏ r=m−p (arii) −1 = −(amii ) −1 m−1∑ p=1 i−1∑ l=j am−pli wm+1−p lj m−1∏ r=m−p (arii) −1 = (amii ) −1wmij since κji ≥ 0 and so wm+1 ij = − i−1∑ l=j amli w m+1 lj (amii ) −1 + (amii ) −1wmij , which yields wmij = amiiw m+1 ij + i−1∑ l=j amli w m+1 lj = i∑ l=j amli w m+1 lj . Now we assume that κji < 0. Then amiiw m+1 ij = +∞∑ p=1 i−1∑ l=j am+p li wm+p+1 lj m+p−1∏ r=m arii = +∞∑ p=0 i−1∑ l=j am+p li wm+p+1 lj m+p−1∏ r=m arii − i−1∑ l=j amli w m+1 lj . Since κji < 0, we have +∞∑ p=0 i−1∑ l=j am+p li wm+p+1 lj m+p−1∏ r=m arii = wmij and so wm+1 ij amii = wmij − i−1∑ l=j amli w m+1 lj , which gives wmij = amiiw m+1 ij + i−1∑ l=j amli w m+1 lj = i∑ l=j amli w m+1 lj . This completes the proof. � Now let µij = lim sup m→+∞ 1 m log |wmij |. Lemma 4.6. For i, j = 1, . . . , q, we have µii = −αi and for each i > j the entry wmij is well defined for all m ∈ N and µij ≤ −αj + i∑ k=j+1 (αk − αk). 14 L. BARREIRA, C. VALLS EJDE-2022/19 Proof. Clearly, µii = −αi for i = 1, . . . , q. For the entries with i > j we proceed by induction on i. Given i < q, assume that for j ≤ l ≤ i− 1 the entry wmlj is well defined for all m ∈ N and satisfies µlj ≤ −αj + l∑ k=j+1 (αk − αk). We show that wmij is well defined for all m ∈ N (which is immediate when κji ≥ 0 because it is given by a finite sum) and µij ≤ −αj + i∑ k=j+1 (αk − αk). (4.9) Assume that κji < 0. By (1.4) and the induction hypothesis, for each ε > 0 there exists D > 0 such that |anli| ≤ Deεn, D−1e(αi−ε)n ≤ n∏ l=1 |alii| ≤ De(αi+ε)n, |wnlj | ≤ De[−αj+ ∑l k=j+1(αk−αk)]n for all n ∈ N and j ≤ l ≤ i− 1. Therefore, B : = +∞∑ p=1 i−1∑ l=j |am+p−1 li wm+p lj | m+p−2∏ r=m |arii| ≤ +∞∑ p=1 i−1∑ l=j Deε(m+p−1)De[−αj+ ∑l k=j+1(αk−αk)](m+p) m+p−2∏ r=1 |arii| m−1∏ r=1 |arii|−1 ≤ D′ +∞∑ p=1 i−1∑ l=j e[αi−αj+ ∑l k=j+1(αk−αk)+3ε](m+p)| m−1∏ r=1 |arii|−1 for some constant D′ > 0. Since κji < 0, we have βji < 0. Now take ε > 0 such that βij + 3ε < 0. Then B ≤ D′q +∞∑ p=1 e(βji+3ε)(m+p) m−1∏ r=1 |arii|−1 < D′q 1− eβji+3ε m−1∏ r=1 |arii|−1 < +∞. (4.10) This shows that wmij is also well defined when κji < 0. Finally, we establish the estimate in (4.9). First assume that κji ≥ 0. It follows from (4.7) that µij ≤ lim sup m→+∞ 1 m log m−2∑ p=0 i−1∑ l=j |am−p−1li wm−plj | m−1∏ r=m−p−1 |arii|−1 ≤ lim sup m→+∞ 1 m log m−2∑ p=0 i−1∑ l=j D2eε(m−p−1)+[−αj+ ∑l k=j+1(αk−αk)+ε](m−p) × m−1∏ r=1 |arii|−1De(αi+ε)(m−p−2) EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 15 ≤ −αi + lim sup m→+∞ 1 m log ( D′e3εm m−2∑ p=0 eβji(m−p) ) ≤ −αi + lim sup m→+∞ 1 m log(meβjim) + 3ε ≤ −αj + i∑ k=j+1 (αk − αk) + 3ε for some constant D′ > 0. Since ε is arbitrary, this yields inequality (4.9). Now assume that κji < 0, which implies tat βij < 0. Take ε > 0 such that βji + 3ε < 0. By (4.10) we have µij ≤ lim sup m→+∞ 1 m log ( D′q e(βji+3ε)(m+1) 1− eβji+3ε m−1∏ r=1 |arii|−1 ) = βji + 3ε+ lim sup m→+∞ 1 m log(De(−αi+ε)m) = −αj + i∑ k=j+1 (αk − αk) + 3ε. Again, since ε is arbitrary, this yields inequality (4.9). � Now consider the basis formed by the columns of W (1), that is, the vectors wj = W (1)ej for j = 1, . . . , q. By Lemma 4.6 we obtain µ(wj) = max { µij : 1 ≤ i ≤ q } ≤ −αj + q∑ k=j+1 (αk − αk) for j = 1, . . . , q, which completes the proof of the theorem. � Finally, we establish the upper bound for γ(λ, µ) using the bases v1, . . . , vq and w1, . . . , wq constructed in Theorems 4.1 and 4.4. Theorem 4.7. Let (Am)m∈N be a tempered sequence of invertible q × q matrices satisfying (2.1). If Am is upper-triangular for every m ∈ N, then γ(λ, µ) ≤ q∑ i=1 (αi − αi). (4.11) Proof. Let Z(m) and W (m) be the matrices constructed in the proofs of Theo- rems 4.1 and 4.4 (whose columns are the vectors, respectively, Amvi and Bmwi for i = 1, . . . , q). By Lemmas 4.2 and 4.5, for each m ∈ N we have W (m+ 1)∗Z(m+ 1) = (BmW (m))∗AmZ(m) = W (m)∗A−1m AmZ(m) = W (m)∗Z(m) and it follows by induction that W (m)∗Z(m) = W (1)∗Z(1). (4.12) Since z1ij = 0 for i > j and w1 ij = 0 for i < j, we have (W (m)∗Z(m))ij = 0 for i > j, 16 L. BARREIRA, C. VALLS EJDE-2022/19 (W (m)∗Z(m))ij = q∑ l=1 wmli z m lj = j∑ l=i wmli z m lj for i ≤ j. In particular, since z1ii = w1 ii = 1 for i = 1, . . . , q, we have (W (m)∗Z(m))ii = (W (1)∗Z(1))ii = w1 iiz 1 ii = 1 for i = 1, . . . , q. Now take i < j. When κij ≥ 0, since βij = βil + βlj , we have 0 ≤ κij = βij + max i≤k≤j (αk − αk) = βil + βlj + max i≤k≤j (αk − αk) ≤ βil + max i≤k≤l (αk − αk) + βlj + max l≤k≤j (αk − αk) = κil + κlj . Hence, for each l either κil ≥ 0 or κlj ≥ 0, and so either w1 li = 0 or z1lj = 0. Therefore, (W (1)∗Z(1))ij = j∑ l=i w1 liz 1 lj = 0 because each term in the sum vanishes. Finally, when κij < 0, by Lemmas 4.3 and 4.6 we have lim sup m→∞ 1 m log |(W (m)∗Z(m))ij | = lim sup m→∞ 1 m log j∑ l=i |wmli zmlj | ≤ max i≤l≤j lim sup m→∞ 1 m log |wmli zmlj | ≤ max j≤l≤i (µli + λlj) ≤ max i≤l≤j ( −αi + l∑ k=i+1 (αk − αk) + αj + j−1∑ k=l (αk − αk) ) = αj − αi + j−1∑ k=i+1 (αk − αk) + max i≤l≤j (αl − αl) = βij + max i≤l≤j (αl − αl) = κij < 0 and so it follows from (4.12) that (W (1)∗Z(1))ij = lim m→+∞ (W (m)∗Z(m))ij = 0. Summing up, W (1)∗Z(1) is the identity matrix and so 〈vi, wj〉 = 〈Z(1)ei,W (1)ej〉 = e∗jW (1)∗Z(1)ei = δij , where δij is the Kronecker symbol. In other words, the bases v1, . . . , vq and w1, . . . , wq are dual. On the other hand, by Lemmas 4.3 and 4.6 we have λ(vj) ≤ αj + j−1∑ k=1 (αk − αk), EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 17 µ(wj) ≤ −αj + q∑ k=j+1 (αk − αk). Hence, for j = 1, . . . , q we obtain λ(vj) + µ(wj) ≤ q∑ k=1 (αk − αk) and since the bases are dual, inequality (4.11) follows readily from the definition of the Grobman coefficient. � 5. Proof of Theorem 1.2 Let (Am)m∈N be a tempered sequence of invertible q×q upper-triangular matrices whose Lyapunov exponent λ takes only finite values on Rq \ {0}. Proof of Theorem 1.2. First assume that αi = αi for i = 1, . . . , q. Then it follows readily from Theorem 4.7 that γ(λ, µ) = 0 and so the sequence of matrices (Am)m∈N is regular. Now assume that the sequence (Am)m∈N is regular. As in the proof of Theo- rem 3.1, let v1, . . . , vq be a basis for Rq such that v1, . . . , vk is a basis for Ek (the space generated by the vectors e1, . . . , ek) for each k = 1, . . . , q that satisfies (3.2) for all bases v′1, . . . , v ′ q for Rq. Then lim m→+∞ 1 m log |detAm| = q∑ i=1 λ(vi). (5.1) Let Am,k be the k× k upper-triangular matrix obtained from Am considering only the first k columns of its first k rows. Taking k = q − 1 in (3.3) we obtain detAm,q−1 = (detAm) m−1∏ l=1 (alqq) −1 and so lim inf m→+∞ 1 m log |detAm,q−1| = lim m→+∞ 1 m log |detAm|+ lim inf m→+∞ 1 m log m−1∏ l=1 |alqq|−1 = lim m→+∞ 1 m log |detAm| − αq. (5.2) Since v1, . . . , vq−1 is a basis for Eq−1, the vector vq has a nonzero component along eq and so the component of Amvq along eq is equal to c ∏m−1 l=1 alqq for some constant c 6= 0 independent of m. This readily implies that λ(vq) ≥ αq. Combining (5.1) and (5.2), we conclude that lim inf m→+∞ 1 m log |detAm,q−1| = q∑ i=1 λ(vi)− αq ≥ q−1∑ i=1 λ(vi). On other hand, we also have lim sup m→+∞ 1 m log |detAm,q−1| ≤ q−1∑ i=1 λ(vi) 18 L. BARREIRA, C. VALLS EJDE-2022/19 and so lim m→+∞ 1 m log |detAm,q−1| = q−1∑ i=1 λ(vi). This shows that the sequence (Am,q−1)m∈N is regular. One can proceed inductively to show that the same happens to each sequence of matrices (Am,k)m∈N and so γk(λ, µ) = 0 for k = 1, . . . , q. Hence, it follows readily from Theorem 3.1 that αi = αi for i = 1, . . . , q. � 6. Regularity of exterior powers In this section we consider the non-autonomous dynamics defined by the exterior powers of a tempered sequence of upper-triangular matrices and we obtain a sharp upper bound for its Grobman coefficient. We first recall some basic notions. For each integer k ∈ [1, q], let Λk(Rq) be the kth exterior power of Rq, that is, the vector space of dimension ( q k ) spanned by the k-vectors v1 ∧ · · · ∧ vk with v1, . . . , vk ∈ Rq. We define a scalar product on Λk(Rq) by requiring that 〈v1 ∧ · · · ∧ vk, w1 ∧ · · · ∧ wk〉 = det J, where J is the k× k matrix with entries bij = 〈vi, wj〉 for i, j = 1, . . . , k. Any q× q matrix A induces a linear map Λk(A) on Λk(Rq) by requiring that Λk(A)(v1 ∧ · · · ∧ vk) = Av1 ∧ · · · ∧Avk (6.1) for all v1, . . . , vk ∈ Rq. Now consider a sequence of invertible q × q matrices (Am)m∈N satisfying prop- erty (2.1). The Lyapunov exponent λk : Λk(Rq) → R ∪ {−∞} associated with the sequence (Λk(Am))m∈N is given by λk(v) = lim sup m→+∞ 1 m log ‖Λk(Am)v‖. Indeed, it follows readily from (6.1) that Λk(Am) = { Λk(Am−1) · · ·Λk(A1) if m > 1, Id if m = 1. Moreover, since ‖Λk(Am)‖ ≤ ‖Am‖k, it follows from condition (2.1) that the Lya- punov exponent λk does not take infinite values on Rq \ {0}. We also consider the sequence of matrices Bm = (A∗m)−1 for m ∈ N. The Lya- punov exponent µk : Λk(Rq)→ R∪{−∞} associated with the sequence (Λk(Bm))m∈N is given by µk(w) = lim sup m→+∞ 1 m log ‖Λk(Bm)w‖, where Bm = (A∗m)−1 (and condition (2.1) ensures that µk does not take infinite values on Rq \ {0}). The Grobman coefficient of λk and µk is given by γ(λk, µk) = min max { λk(vi) + µk(wi) : 1 ≤ i ≤ q } with the minimum taken over all dual bases v1, . . . , vq and w1, . . . , wq of Λk(Rq). In the following result we establish an upper bound for γ(λk, µk) when the matri- ces are upper-triangular. We continue to denote by αi and αi the numbers defined by (3.1) for i = 1, . . . , q. EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 19 Theorem 6.1. Let (Am)m∈N be a tempered sequence of invertible q × q matrices satisfying (2.1). If Am is upper-triangular for every m ∈ N, then for every integer k ∈ [1, q] we have γ(λk, µk) ≤ ( q − 1 k − 1 ) q∑ i=1 (αi − αi). (6.2) Proof. Consider the canonical basis e1, . . . , eq for Rq. Then the k-vectors ei1···ik = ei1 ∧ · · · ∧ eik with i1 < · · · < ik form a basis for Λk(Rq). Note that a matrix A is upper-triangular if and only if Aei is a linear combination of the vectors e1, . . . , ei for i = 1, . . . , q. Hence, since Am is upper-triangular, the k-vector Amei1 ∧ · · · ∧Ameik is a linear combination of the k-vectors ej1···jk such that j1 < · · · < jk and jl ≤ il for l = 1, . . . , k. (6.3) In other words, the linear map Λk(Am) is upper-triangular with respect to the basis ei1···ik for Λk(Rq) ordered so that a k-vector ej1···jk comes before another k-vector ei1···ik if and only if condition (6.3) holds. In particular, this implies that the elements on the main diagonal of the linear representation of Λk(Am) with respect to the basis ei1···ik are given by m−1∏ l=1 ai1i1(l)ei1 ∧ · · · ∧ m−1∏ l=1 aikik(l)eik = k∏ p=1 m−1∏ l=1 aipip(l)ei1 ∧ · · · ∧ eik . This allows one to apply Theorem 4.7 to deduce that γ(λk, µk) ≤ ∑ i1<··· 0 there exists a compact set K ⊂ X with µ(K) ≥ 1− ε. By considering the transformation induced by f on K (which in view of Poincaré’s recurrence theorem is well defined), we may assume from the beginning without loss of generality that X is compact. We denote by GLq the set of all invertible q × q matrices with real entries. A measurable function A : X × N → GLq is called a cocycle over f if for every x ∈ X and n,m ∈ N we have A(x, 0) = Id and A(x, n+m) = A(fn(x),m)A(x, n). Given a measurable function A : X → GLq, we define a cocycle over f by A(x,m) = { A(fm−1(x)) · · ·A(x) if m > 0, Id if m = 0. (7.1) Conversely, any cocycle A can be obtained as in (7.1) taking A(x) = A(x, 1). The function A is called the generator of the cocycle A. Now let A : X×N→ GLq be a cocycle over f . The Lyapunov exponent of a pair (x, v) ∈ X × Rq is defined by λ(x, v) = lim sup m→+∞ 1 m log ‖A(x,m)v‖. Consider a point x ∈ X such that λ(x, v) is finite for all v 6= 0. We say that x is (Lyapunov) regular for A if the sequence of matrices (Am(x))m∈N given by Am(x) = A(fm(x)) is regular, that is, if and only if (see (2.2)) lim m→+∞ 1 m log |detA(x,m)| = min q∑ i=1 λ(x, vi) with the minimum taken over all bases v1, . . . , vq of Rq. Finally, we give a simple proof of Oseledets’ multiplicative ergodic theorem for cocycles over a measure-preserving transformation. EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 21 Theorem 7.1 (see [8]). Let A be a cocycle over a measure-preserving transforma- tion f . If its generator satisfies log+ ‖A±1‖ ∈ L1(X,µ), then the set of all regular points for A has full µ-measure. Proof. Consider the µ-integrable functions φ±(x) = log+ ‖A(x)±1‖. It follows from Birkhoff’s ergodic theorem that the limit ψ±(x) = lim m→+∞ 1 m m−1∑ k=0 φ±(fk(x)) exists for µ-almost every x ∈ X. Moreover, each ψ± is µ-integrable and∫ X ψ± dµ = ∫ X φ± dµ. In particular, ψ± is finite µ-almost everywhere. Note also that ‖v‖ ≤ ‖A(x,m)−1‖ · ‖A(x,m)v‖. Therefore, λ(x, v) ≤ lim sup m→+∞ 1 m log+ ‖A(x,m)‖ ≤ ψ+(x) < +∞ and λ(x, v) ≥ lim sup m→+∞ 1 m log(‖A(x,m)−1‖−1) ≥ − lim sup m→+∞ 1 m log ‖A(x,m)−1‖ ≥ −ψ−(x) > −∞ for µ-almost every x ∈ X and any v 6= 0. This allows one to compute the corre- sponding Grobman coefficient. In addition, we have 1 m m−1∑ k=0 φ±(fk(x)) = m+ 1 m · 1 m+ 1 m∑ k=0 φ±(fk(x))− 1 m φ±(fm(x)) and again it follows from Birkhoff’s ergodic theorem that lim m→+∞ 1 m φ±(fm(x)) = 0 for µ-almost every x ∈ X. In particular, the sequence (Am(x))m∈N is tempered for µ-almost every x ∈ X. Therefore, one would be able to apply Theorem 4.7 provided that the matrices Am(x) were upper-triangular. Now we make an upper-triangular reduction of the matrices. Let SOq be the set of orthogonal q× q matrices and let Y = X×SOq. Given (x, U) ∈ Y , we apply the Gram–Schmidt process to the columns of A(x)U to obtain A(x)U = V (x, U)B(x, U), (7.2) where V (x, U) is orthogonal and B(x, U) is upper-triangular with positive entries on the main diagonal. Note that these matrices are uniquely defined. Moreover, 22 L. BARREIRA, C. VALLS EJDE-2022/19 it follows readily from the steps in the Gram–Schmidt process that both are mea- surable in (x, U). We consider the measurable linear extension F : Y → Y of the map f defined by F (x, U) = (f(x), V (x, U)). For the projection π : (x, U) 7→ U , it follows from (7.2) that B(x, U) = π(F (x, U))−1A(x)π(x, U). (7.3) Let Ā and B be the cocycles over F with generators, respectively, Ā(x, U) = A(x) and B(x, U). We have Ā((x, U),m) = A(x,m) and it follows from (7.3) that B((x, U),m) = π(Fm−1(x, U))−1A(x,m)π(x, U). (7.4) In particular, detB((x, U),m) = detA(x,m). Moreover, since ‖U‖ = 1 for U ∈ SOq, it follows from (7.4) that lim sup m→+∞ 1 m log ‖B((x, U),m)v‖ = λ(x, v) for all (x, U) ∈ Y and v ∈ Rq. This readily implies that given x ∈ X, the sequence (Am(x))m∈N is regular if and only if the sequence of upper-triangular matrices Bm(x, U) = B(Fm−1(x, U)) for m ∈ N (7.5) is regular for some U ∈ SOq (and so for all U ∈ SOq). By (7.3) we have Bm(x, U) = π(Fm(x, U))−1Am(x)π(x, U) and since (Am(x))m∈N is tempered for µ-almost every point x ∈ X, the sequence (Bm(x, U))m∈N is tempered for µ-almost every x ∈ X and every U ∈ SOq. Now let ν be a probability measure on Y such that ν(B × SOq) = µ(B) for any measurable set B ⊂ X and define measures νn = 1 n n−1∑ k=0 ν ◦ F−k for each n ∈ N. Then νn(B × SOq) = 1 n n−1∑ k=0 ν(F−k(B × SOq)) = 1 n n−1∑ k=0 ν(f−kB × SOq) = 1 n n−1∑ k=0 µ(f−kB) = µ(B) (7.6) since the measure µ is f -invariant. Let ν̄ be the weak* limit of some sequence of measures νkn . In order to show that ν̄ is F -invariant we proceed as follows. First observe that for any continuous function ψ : Y → R we have∫ Y ψ dνkn → ∫ Y ψ dν̄ (7.7) EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 23 when n → +∞ (recall that we are assuming that X is compact and so Y is also compact). Moreover,∫ Y (ψ ◦ F ) dνkn = ∫ Y ψ d(νkn ◦ F−1) = ∫ Y ψ d ( νkn + ν ◦ F−kn − ν kn ) and so ∫ Y (ψ ◦ F ) dνkn − ∫ Y ψ dνkn → 0 (7.8) when n→ +∞. We will show that∫ Y (ψ ◦ F ) dνkn → ∫ Y (ψ ◦ F ) dν̄ (7.9) when n→ +∞. Together with (7.7) and (7.8) this implies that∫ Y (ψ ◦ F ) dν̄ = ∫ Y ψ dν̄ for any continuous function ψ : Y → R and so ν̄ is F -invariant. Moreover, it follows from (7.6) that ν̄(B × SOq) = µ(B) (7.10) for any measurable set B ⊂ X. To establish (7.9), for each m ∈ N consider a compact set Km ⊂ X with measure µ(Km) ≥ 1− 1/m such that both f and the generator A of the cocycle are contin- uous on Km (which by Lusin’s theorem is always possible). Then F is continuous on K ′m := Km × SOq and νn(K ′m) = µ(Km) ≥ 1− 1/m (7.11) for all n ∈ N. Let ψ : Y → R be a continuous function. Then κ = ψ◦F is continuous on K ′m and by Tietze’s extension theorem it has a continuous extension κ̄ to the whole Y . For each l ∈ N we consider the continuous function κl : Y → R defined by κl(x) = max { κ̄(x)− l dist(x,K ′m), 0 } . Given ε > 0, there exists l ∈ N such that∫ Y κl dν̄ < ∫ K′m κ dν̄ + ε. Then lim sup n→∞ ∫ K′m κ dνkn ≤ lim n→∞ ∫ Y κl dνkn = ∫ Y κl dν̄ < ∫ K′m κ dν̄ + ε and it follows from the arbitrariness of ε that lim sup n→∞ ∫ K′m κ dνkn ≤ ∫ K′m κ dν̄. Replacing ψ by −ψ, we also obtain lim inf n→∞ ∫ K′m κ dνkn ≥ ∫ K′m κ dν̄ and so lim n→∞ ∫ K′m κ dνkn = ∫ K′m κ dν̄. (7.12) 24 L. BARREIRA, C. VALLS EJDE-2022/19 One can now establish (7.9). It follows from (7.11) that∣∣∣ ∫ Y κ dνkn − ∫ Y κ dν̄ ∣∣∣ ≤ ∣∣∣ ∫ Y κ dνkn − ∫ K′m κ dνkn ∣∣∣+ ∣∣∣ ∫ K′m κ dνkn − ∫ K′m κ dν̄ ∣∣∣ + ∣∣∣ ∫ K′m κ dν̄ − ∫ Y κ dν̄ ∣∣∣ ≤ 1 m max |ψ|+ ∣∣∣ ∫ K′m κ dνkn − ∫ K′m κ dν̄ ∣∣∣+ 1 m max |ψ|. Hence, by (7.12) we obtain lim sup n→+∞ ∣∣∣ ∫ Y κ dνkn − ∫ Y κ dν̄ ∣∣∣ ≤ 2 m max |ψ| and letting m→ +∞ yields property (7.9) holds. Finally, we establish the regularity of the sequence of matrices Bm(x, U) in (7.5) for µ-almost every x ∈ X and all U ∈ SOq. Let bij(x, U) be the entries of B(x, U). Moreover, for each i = 1, . . . , q let β i (x, U) = lim inf m→+∞ 1 m log m−1∏ l=0 bii(F l(x, U)), βi(x, U) = lim sup m→+∞ 1 m log m−1∏ l=0 bii(F l(x, U)). Since log+ ‖A±1‖ ∈ L1(X,µ) and B(x, U) is upper-triangular with positive entries on the main diagonal, the functions log bii are ν̄-integrable and so it follows from Birkhoff’s ergodic theorem that β i (x, U) = βi(x, U) = lim m→+∞ 1 m log m−1∏ l=0 bii(F l(x, U)) for ν̄-almost every (x, U) ∈ Y . By Theorem 4.7, this implies that the tem- pered sequence of upper-triangular matrices Bm(x, U) is regular for ν̄-almost every (x, U) ∈ Y . On the other hand, we showed before that (Am(x))m∈N is regular if and only if the sequence Bm(x, U) is regular for some U ∈ SOq. The desired statement follows now readily from (7.10). � Acknowledgements. We were supported by FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020. References [1] L. Barreira; Lyapunov exponents, Birkhäuser/Springer, Cham, 2017. [2] L. Barreira, Ya. Pesin; Lyapunov exponents and smooth ergodic theory, University Lecture Series, vol. 23, American Mathematical Society, Providence, RI, 2002. [3] L. Barreira, Ya. Pesin; Nonuniform hyperbolicity. Dynamics of systems with nonzero Lya- punov exponents, Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge, 2007. [4] L. Barreira, C. Valls; Stability theory and Lyapunov regularity, J. Differential Equations, 232 (2007), 675–701. [5] D. Bylov, R. Vinograd, D. Grobman, V. Nemyckii; Theory of Lyapunov exponents and its application to problems of stability, Izdat. “Nauka”, Moscow, 1966, in Russian. [6] A. Liapounoff; Problème général de la stabilité du mouvement, Annals of Mathematics Stud- ies, vol. 17, Princeton University Press, Princeton, NJ, 1947. EJDE-2022/19 SOME APPLICATIONS OF LYAPUNOV REGULARITY 25 [7] A. Lyapunov; The general problem of the stability of motion, Taylor & Francis Group, Lon- don, 1992. [8] V. Oseledets; A multiplicative ergodic theorem. Liapunov characteristic numbers for dynam- ical systems, Trans. Moscow Math. Soc., 19 (1968), 197–221. [9] O. Perron; Über stabilität und asymptotishes Verhalten der Lösungen eines Systemes endlicher Differenzengleichungen, J. Reine Angew. Math., 161 (1929), 41–64. [10] O. Perron; Die Ordnungszahlen linearer Differentialgleichungssyteme, Math. Zs., 31 (1930), 748–766. [11] O. Perron; Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703–728. [12] Ya. Pesin; Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 10 (1976), 1261–1305. [13] Y. Yomdin; Volume growth and entropy, Israel J. Math., 57 (1987), 285–300. [14] Y. Yomdin; Ck-resolution of semialgebraic mappings. Addendum to: “Volume growth and entropy”, Israel J. Math.; 57 (1987), 301–317. Lúıs Barreira Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal Email address: barreira@math.tecnico.ulisboa.pt Claudia Valls Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal Email address: cvalls@math.tecnico.ulisboa.pt 1. Introduction 1.1. Lyapunov regularity 1.2. Context of ergodic theory 1.3. Regularity coefficients 1.4. Applications of the results 2. Lyapunov regularity 3. Lower bound for the Grobman coefficients 4. Upper bound for the Grobman coefficients 5. Proof of Theorem 1.2 6. Regularity of exterior powers 7. Regularity of cocycles Acknowledgements References