My question is about some basic properties of Lyapunov exponents. Sorry if this stuff is basic, I just don't know where to find the statements that I'm looking for.

**Preliminaries.** Let $X$ be a closed smooth manifold. Let $V$ be a non-vanishing vector field on $X$ preserving a smooth measure $\mu$ on $X$ and let $E \to X$ denote the quotient bundle $TX/\text{span}(V)$. Let $\varphi:X \times \mathbb{R} \to X$ be the autonomous flow generated by $V$.

The flow can be lifted to a cocycle $\Phi:E \times \mathbb{R} \to E$ on the bundle $E$, given fiberwise by $$ (x,e,t) \mapsto (\varphi_t(x),D\varphi_{t,x}(e)) $$ Oseledet's theorem (as I understand it) says in this setting that

Theorem 1.There is a subset $S \subset X$ of full measure with respect to $\mu$ such that the Lyupanov exponents $\lambda_i(x)$ for $1 \le i \le n-1$ of the cocycle $(E,\Phi)$ are well-defined.

**Lyapunov Exponents.** Let's briefly review a definition of the Lyapunov exponents in a special case. Suppose that $E$ is isomorphic to a trivial bundle, and let $\tau:E \simeq X \times \mathbb{R}^{n-1}$ be a trivialization. Pick $x \in X$ and let $A$ denote the $\mathbb{R}$-family of $(n-1)$-dimensional matrices $A_t = \tau_{\varphi(x)} \circ D\varphi_{x,t} \circ \tau_x^{-1}$ acquired from the cocycle via this trivialization. Then
$$
(*) \qquad \lambda_i(x) := \lim_{t \to \infty} \lambda_i(t,x) \qquad\text{with} \qquad \lambda_i(t,x) := \frac{1}{2t} \ln(\text{Eig}_i(A_t^TA_t))
$$
Here $A_t^T$ is the transpose of $A_t$ and $\text{Eig}_i(\cdot)$ denotes taking the $i$th eigenvalue (ordered from largest to smallest) of a self-adjoint linear map.

**Questions.** I have a few questions about the analytical properties of $\lambda_i$ and about the limit above.

The first question is about the regularity of $\lambda_i$.

Question 1.Does $\lambda_i$ define a distribution? An $L^1$ function? An $L^\infty$ function? If not, what are some simple criteria for this being the case?

A stronger version of Question 1 is whether or not the limit $(*)$ converges in various topologies.

Question 2.Is the limit $(*)$ a distributional limit? An $L^1$ limit? An $L^\infty$ limit? Again, what are some criteria for this being the case?

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