in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book.
Theorem discrete Oseledets (A) :
Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be a diffeomorphism of class $ C^1$ and $\mu$ invariant probability measure for $f$. For $\mu$ a.e $ x \in M $ there are real numbers $\lambda_1(x)>\ldots >\lambda_r(x)$ and decomposition $T_xM=E_1(x)\oplus \ldots \oplus E_r(x)$ such that $$\displaystyle{\lim_{ k \rightarrow \pm \infty }\frac{1}{k}\log\Vert Df^k(x)u\Vert=\lambda_j(x)}$$ for all $u\in E_j(x)\backslash \lbrace 0\rbrace$ and $1\leq j\leq r$.
I am curious if this result can be adapted to flows. Let me know if my reasoning is correct:
Theorem continuous Oseledets (B) [I think this should be]:
Let $ \phi: \mathbb{R}\times M \rightarrow M $ a flows and $d\phi_x^t: T_xM \rightarrow T_{\phi^t(x)}M$ where $ \phi^t=\phi(t,.): M \rightarrow M $ then $\mu$ a.e $ x \in M $ there are real numbers $\lambda_1(x)>\ldots >\lambda_r(x)$ and decomposition $T_xM=E_1(x)\oplus \ldots \oplus E_r(x)$ such that $$\displaystyle{\lim_{ t \rightarrow \pm \infty }\frac{1}{t}\log\Vert d\phi_x^tu\Vert=\lambda_j(x)} \ \ \ (*)$$ for all $u\in E_j(x)\backslash \lbrace 0\rbrace$ and $1\leq j\leq r$.
Also appreciate any suggestions to prove this assertion (B), in particular the existence of limit (*) using the result (A).