# Integrability/regularity of Lyapunov exponents

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?

• So I don't know enough of the differential geometry to describe things well for the flow, so I want to focus my comments on the time one map: when you follow the flow for a unit time. In this case, the map looks like $(x,e)\mapsto (\phi_1(x),B_xe)$ where $B_x$ is the exponential(?) integral(?) (I'm not sure what the correct term is) of the flow map along the orbit. If your manifold were compact, you would have, I believe $\|B_x\|$ and $\|B_x^{-1}\|$ uniformly bounded over the manifold. In this case, what you can say from Oseledets is that – Anthony Quas Aug 30 at 18:12
• For question 1, the limit is an $L^\infty$ function; for question 2, the limit is in the sense of an $L^1$ limit. At the ergodic theory level, the limits are not uniform (see mathoverflow.net/questions/303697/…). It is conceivable that in the differential geometry setting that you describe, you may have better limiting properties, but this would probably depend on detailed properties of the manifold. – Anthony Quas Aug 30 at 18:15
• Hi Anthony, thanks for the comments! Is there any chance that you have time to post an answer with citations? A source (even for dynamics on $\mathbb{R}^n$) would be awesome :) – Julian Chaidez Aug 30 at 18:36
• HI Julian, I don't think my answers really go beyond a direct application of Oseledets theorem once you have converted the problem to a problem of maps rather than flows, by taking the time one map. So the work amounts to showing that what I called $B_x$ has the properties I describe. I think you will probably find this done in Barreira and Pesin's AMS Graduate Study in Mathematics (section 2.3). – Anthony Quas Aug 30 at 19:03
• Ok thanks, I'll take a look there. – Julian Chaidez Aug 30 at 19:11

Your question is actually more about the ergodic properties of the smooth invariant measure rather than about the Lyapunov exponents per se. Indeed, since the exponents are flow invariant, in the ergodic case they are just almost everywhere constant, whereas in the general situation they are measurable with respect to the partition of the state space into ergodic components.

By the way, Theorem 1 is due to Furstenberg - Kesten several years before Oseledets. The point is that the Lyapunov regularity (established by Oseledets) is a priori a much stronger property than just existence of the limits (*). However, as mentioned by Anthony in his first comment, in your situation these two properties are equivalent because of the uniform boundedness of the cocycle. This is also the reason why there is no difference between the time 1 map and the flow.

PS This answer was intended to be a comment but turned out to be too long for that.

EDIT I have to add this edit in view of the misleading comments of John B. OP's Theorem 1, i.e., the existence of the numbers $$\lambda_i$$ as introduced by the OP in equation ($$\ast$$), is an immediate consequence of the theorem of Furstenberg - Kesten (Theorem 1 from their 1960 paper) applied to the exterior powers of the matrices $$A_t$$. Furstenberg - Kesten theorem does not require any additional conditions on the cocycle other than the natural integrability of the logarithm of the norm of the cocycle. Lyapunov regularity of a sequence of matrices is a priori a much stronger condition than just the existence of the limits ($$\ast$$). Neither Furstenberg - Kesten nor Oseledets were aware of the fact that in the ergodic setup the Lyapunov regularity is actually an automatic consequence of the existence of the limits ($$\ast$$), which was first realized by Raghunathan in 1979, and later led to the geometric interpretation of the Lyapunov regularity as sublinear geodesic tracking property.

• The result is due to Oseledets. As you may recall Furstenberg and Kesten make certain assumptions on the positivity of the entries, rather unnatural when all should come from ergodic theory. – John B Sep 15 at 9:08
• Could you be more specific? – R W Sep 15 at 11:01
• You wrote "Theorem 1 is due to Furstenberg - Kesten several years before Oseledets". I am saying that they only proved it under additional positivity (and in fact also bounded distortion) assumptions. – John B Sep 15 at 21:29
• I have already asked you to be more specific about these alleged assumptions. Have a look at Theorem 2 (p. 460) of the orginal paper which does not use the assumptions made later (in Section 3) for other purposes. – R W Sep 15 at 23:18
• I think that you understand my former comment. It is crystal clear. Please correct your wrong answer accordingly. You should instead look at the corollary in page 462, we are not talking about norms here. – John B Sep 15 at 23:31