Today’s post brings the lectures notes I used in my second (and last) exposition on Marcelo Viana’s article “Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents” (as a part of a “Groupe de Travail” organized by S. Crovisier and J. Buzzi).
In this previous post, we started with a cocycle (, ) over a non-uniformly hyperbolic basis . Then, we saw that the vanishing of the top exponent at -a.e. (that is the same as all Lyapunov exponents vanish because takes values in ) implies some sort of domination (i.e., the dynamics of the cocycle along the fibers is dominated by the dynamics of the basis) and this last property (actually, 3-domination) allows for the construction of stable and unstable holonomies, e.g., , , depending in a Lispchitz way on and in a way on . Before putting these holonomies to work, we will state (without proof) the following result justifying the terminology “holonomy” (but which will be not used in the sequel):
Lemma 1 Let be a -dominated point, say with . For any and , one has
- if and only if .
Here, as usual, is the projective cocycle associated to , .
For later use, we denote by the projective stable/unstable holonomy associated to the stable/unstable holonomy .
Now, we proceed to discuss the 2nd step of the strategy of proof of Theorem 5 of the first post on Marcelo’s article whose statement we recall below:
Theorem 5. For all and with , and ergodic measure with local product structure, the set of cocycles with for -a.e. is open, dense and it has -codimension.
–2nd step of the proof of Theorem 5: holonomies and projective cocycles–
We start our considerations with the following notion:
Definition 2 We say that a compact set , , is a holonomy block.
During this entire section, we’ll make the following assumption:
Now, we take a -invariant probability measure on projecting to (such a always exists by compactness of ). By Rokhlin’s disintegration theorem (see also this link here for a modern exposition of this theorem), can be disintegrated into a family of probabilities on such that
Moreover, Rokhlin’s theorem also ensures that the disintegration is essentially uniquein the sense that if also verifies
then for -a.e. .
Next, let us fix a holonomy block of positive -measure, say , and let be a small positive real number such that for every with . Here, we denote by the unique point whenever , . For each , we define
Similarly, we define , and by replacing the condition by in the definitions above.
The main result of this section is the following proposition:
for all in a fixed stable leaf .
As we hinted in the description of the strategy of proof of Theorem 5 (in this previous post here), the key tool in the proof of this proposition is the following result of F. Ledrappier (whose proof will be omitted):
Theorem 4 (F. Ledrappier) Let be an injective measurable transformation of a Lebesgue space, and a -integrable cocycle (i.e., is -integrable). Let is a -algebra such that
- (mod ) and generates (mod );
- the -algebra generated by is contained in (mod ).
Then, if for -a.e. , we have that the map is -measurable (mod ), where is the disintegration of a -invariant probability on (and is the projective cocycle corresponding to and ).
Once one disposes of Ledrappier’s theorem, the idea of the proof of Proposition 3 is quite simple: one “translates” the invariance by holonomies condition into a measurability statement with respect to an adequate -algebra. To formalize this idea, we need the following “Markov partition like statement”:
- The subsets are not very small: for all ;
- Markov-like property: for any , , if , then .
At this point, I asked the audience of my second talk whether they wanted to hear about the details of this proposition. After a little discussion, Sylvain and Francois voted for skipping it since the arguments are very close to Bowen’s classical construction of Markov partitions (for uniformly hyperbolic systems), and I ended up by accepting their (sensate) decision. However, for this blog post, I believe that we could take the opportunity to discuss this topic, and we’ll do so here. But, in order to avoid interrupting the proof of Theorem 5, we’ll postpone it to the “Appendix” (last section of this post).
Anyway, by assuming this proposition, we consider as above, and, for each , denote by the biggest such that . Observe that, without loss of generality, we can also assume that in the proposition above.
Let be the -algebra generated by , i.e.,
Finally, let us introduce the cocycle ,
The lemma allowing us to convert the invariance by holonomies statement for into a measurability statement for is the following one:
- (i) and generates the Borel -algebra of ;
- (ii) the -algebra generated by is contained in ;
- (iii) is -integrable;
- (iv) the cocycles and have the same Lyapunov exponents.
Proof: Note that is the -algebra generated by . By the Markov-like property (see the 2nd item of Proposition 5), , i.e., . Also, since is the -algebra generated by , and as (as , ). It follows that generates . This proves (i) above.
By definition of , one can check that whenever is measurable (this is essentially the fact that is constant equal to on each with ). In other words, the -algebra generated by is contained in . This proves (ii) above.
The -integrability of is “almost obvious” from the -integrability of , except maybe by the case and . In the latter case, one notices that and are uniformly close to the identity (by Proposition 12 and Corollary 14 of the previous post). This proves (iii) above.
Finally, it suffices to show that the cocycles and are conjugated by a conjugacy at a bounded distance of the identity to get that they have the same Lyapunov exponents. In our case, we have where
Again by Proposition 12 and Corollary 14 of the previous post, we have that the conjugation is uniformly close to the identity. This proves (iv).
From this lemma, we can translate the invariance by holonomy condition for into a measurability condition for to get a proof of Proposition 3 as follows:
Proof: In the case of a -valued cocycle (), one has that for -a.e. if and only if for -a.e. . By the previous lemma, this implies that for -a.e. . Also, we observe that, by Rokhlin’s theorem, is -invariant if and only for -a.e. .
Now, we construct the following family of probabilities :
We claim that the probability measure on with disitegration is -invariant. Indeed,
- if , , we have
- if , ,
- the remaining case is evident (as and ).
In any case, we see that is -invariant. This discussion combined with Lemma 6 say that the cocycle and the -algebra fulfill the hypothesis of Ledrappier’s theorem. Hence, we conclude that the map
is -measurable. By definition of , this -measurability statement implies the following property: there is a subset of full -measure such that
whenever . By definition of and the nice “composition” properties of the stable holonomies, we have that the preivous equality means
Next, we combine Proposition 3 of invariance by holonomy with the local product structure of to deduce the following continuity result:
- (a) the map is continuous;
- (b) is invariant under stable and unstable holonomies on the whole support of .
Proof: Let , where and are the subsets provided by Proposition 3. We have that . Since has local product structure (), we have that for -a.e. . Fix a -generic point and define
when and , and otherwise. (This task would be easier if we could take , but we can’t do that in general). By definition of and local product structure, for -a.e. , i.e., is a disintegration of . Moreover, varies continuously along unstable leaves because, for each , the unstable holonomy is a Lipschitz function of (see Proposition 12 of the previous post). Now, we fix a -generic point and let the disintegration of obtained from by forcing stable-holonomy invariance from , i.e.,
when and , and otherwise. Again by definition of and local product structure, it follows that is a disintegration of such that is invariant by stable holonomies and varies continuously on the whole!
By a dual procedure, we can also construct a disintegration of such that is invariant by unstable holonomies and varies constinuously on the whole . By the uniqueness of the disintegration, we have that for -a.e. . By continuity, actually we must have for every . This completes the proof.
–3rd step of the proof of Theorem 5: contamination by periodic points (part I)–
Recall (from the previous post) that a periodic point of is dominated for when with .
is . Moreover, given linearly independent, the map
is a submersion.
For the proof of this proposition, we will need the following fact about the Lie group , :
is a submersion.
Using this remark, the proof of Proposition 8 goes as follows:
Proof: Fix be a neighborhood of such that . By the usual partition of unity argument, for any (i.e., a traceless matrix), there exists such that and for all . By the formula of Lemma 15 of the previous post (see also the remark below), the choice of and the fact that when imply
Since the directions are linearly independent (because are linearly independent), the proposition follows from Remark 1.
Remark 2 In the previous argument, the computation of derivative of with respect to the variable (provided by the formula of Proposition bla of the previous post) was simplified by the choice of with for all . However, this choice of depends on a partition of unity argument, and hence it is not well suited to deal with real/complex-analyticcocycles (even though it is well-adapted to cocycles). In principle, one could work with “more global” choices of , but this amounts to deal with the complete expression of , namely,
and it is not clear how to use this to get that the desired map is a submersion. Actually, to the best of my knowledge, the question of extending Marcelo’s theorem to real/complex-analytic cocycles is open, and, as far as I can see, the choice of “good” ‘s (with respect to the formula for ) is the sole issue here.
A corollary of the proof of Proposition 8 is the following result:
is a submersion.
Proof: For each , let . Given in , we fix a neighborhood of such that they are mutually disjoint and is the unique point in the intersection of and the several periodic orbits. By the usual partition of unity argument, we can take a curve such that
- and for every , ;
- for each .
It follows that . Since are arbitrary, the proof of the corollary is complete.
The discussion above prepares the “contamination by periodic points” argument: roughly speaking, we just saw that one has sufficient freedom to deform cocycles nearby any finite family of periodic points. Of course, in order to profit of this “freedom”, we need to produce “lots” of periodic points. This will be the main concern of the remainder of this section (where we will construct holonomy blocks with “several” periodic points).
In the sequel, we denote by and we assume that .
- (1) the periodic points are heteroclinically related:
- (2) the periodic points are “generic”: , where is the period of .
The proof of this proposition is an adaptation of the so-called Katok’s shadowing lemma allowing to construct “lots” of periodic points inside hyperbolic blocks for non-uniformly hyperbolic systems:
Theorem 11 (Katok’s shadowing lemma) For every and , there are , , and compact subsets with as such that for any and with and , there is a periodic point of period verifying the following properties:
- is hyperbolic, for each eigenvalue of , and for all , ;
- has size and it is uniformly transverse (i.e., it makes an angle ) with all unstable disks centered at ; in particular, for all ;
- for each .
Morally speaking, the difference between Katok’s shadowing lemma and Proposition 10 is the fact that the compact subsets in Katok’s result are closely related to hyperbolic blocks, while the compact subsets in Proposition 10 are holonomy blocks (i.e., subsets of hyperbolic blocks given by certain domination constraints), and hence are “usually” smaller than . However, from the technical point of view, this difference is not a big deal, but the proof of Proposition 10 from Katok’s shadowing lemma is not exciting (in the sense that we don’t need to introduce new ideas during the process). So, I will proceed as in my lecture and I will skip it (recommending to the curious reader to consult pages 667–671 of Marcelo’s article).
Corollary 12 Let periodic points as in Proposition 10, and let , . Let , , such that, for each , are linearly independent. Then,
is a submersion (where is a sufficiently small neighborhood of ).
–4th step of the proof of Theorem 5: contamination by periodic points (part II)–
We start the proof of Theorem 5 with the case . Given , we take a holonomy block with dominated periodic points (with periods ). By Corollary 9, the map
is a submersion. On the other hand, denoting by the subset of matrices with a pair of eigenvalues with the same norm. We have that is contained in a finite union of submanifolds of codimension .
Therefore, the set
has codimension . Without loss of generality, we can assume that for all and . For each , let , , be the eigenspaces of , and .
By Corollary 12, the map
is a submersion. Hence, the subset
has codimension .
Fix . We claim that for in subset of positive -measure set. Indeed, suppose that this is not the case. Then, we will reach a contradiction by the following contamination by periodic points argument.
for all and . On the other hand, by Proposition 7, we know that if for -a.e. , then any -invariant probability on (projecting to ) admits a disintegration such that
- the map is continuous,
- is invariant by holonomies.
Now, we observe that if is -invariant, then for -a.e. . Hence, by continuity, it follows that
for all . Here, we also used that (by Proposition 10). In particular, if , we get that, for each , is a convex combination of Dirac measures supported on ‘s. Combining this with the invariance by holonomies, we obtain that, for any such that and for every , one can find with
Of course, this is a contradiction with (1)showing that if satisfies for -a.e. , then . Since and have codimension and is arbitrary, the proof of Theorem 5 in the case is complete.
Next, we consider the case . Here, the subset of matrices with a pair of eigenvalues with the same norm hasn’t codimension : actually, the subset of matrices of with a pair of complex (non-real) conjugated eigenvalues is open! Hence, the argument employed above to deal with the case can’t be mimicked here.
To overcome this technical difficulty, we recall the following standard trick to “convert” a pair of complex conjugated eigenvalues of a matrix into two real eigenvalues of distinct norms of a matrix close to some power of : if is a (complex) eigenvalue of , then is an eigenvalue of ; by appropriately choosing , we have that can be made arbitrarily close to (i.e., close mod ); then, by a preliminary small perturbation of , we can make and its complex conjugate both equal to , and, by another small perturbation, we can “separate” them into two real eigenvalues of distinct norms (say one with norm and the other with norm ).
The idea in the general setting of cocycles over a non-uniformly hyperbolic basis is simply to adapt the previous trick, namely, if some of the periodic points considered above (in a holonomy block ) is “problematic”, i.e., has some pair of complex conjugated eigenvalues, one uses a result of C. Bonatti and M. Viana allowing to replace by a nearby periodic point whose period is a (well-chosen) multiple of so that has a pair of real eigenvalues of distinct norms, up to excluding a subset of cocycles of positive codimension. Then, up to performing finitely many of such “exclusions” of positive codimension subsets of cocycles, we can assume that the periodic points in the holonomy block were chosen so that only has real eigenvalues of distinct norms, and thus the same argument of the case can be applied from this point on. For more details, we recommend reading pages 674-675 of Marcelo’s article.
At this point, our considerations on Theorem 5 of the previous post are finished. Closing this section, let us say a few words on the proofs of Corollaries 6 and 7 of the previous post. In these corollaries, the measure was not assumed to be ergodic, but a classical argument based on the local product structure assumption on and a Hopf-like argument (see Lemma 5.1 of Marcelo’s article) says that has countably many ergodic components , in the case of non-uniformly hyperbolic systems and has finitely many ergodic components in the case of uniformly hyperbolic homeomorphisms , i.e., with countable or finite. Moroever, each , , has local product structure, and hence Theorem 5 can be applied to each , . As a consequence, we get an open and dense subset of with -codimension such that, for every , it holds for -a.e. .
Therefore, the subset of is residual when is countable (the case of non-uniformly hyperbolic) and open, dense with -codimension when is finite (the case of uniformly hyperbolic). Moreover, any satisfies for -a.e. . This proves the desired corollaries.
–Some comments and open questions–
In this section, we collect some natural open questions motivated by Marcelo’s theorem.
During our discussion, we dealt exclusively with the group . In fact, the main properties we used about this group (see Proposition 8) were:
- Every can be approximated by matrices in with eigenvalues of distinct norms;
- the map is a submersion whenever are linearly independent.
This last property requires . In particular, we “miss” important classical groups in Dynamics, such as the symplectic group (whose dimension is ). So, one natural question is: classify the Lie groups such that “typical” -valued cocycles satisfy the conclusion of Marcelo’s theorem.
Another natural question is about the “topological nature” of the subset of cocycles with vanishing top exponent. For instance, is closed? if not, does the closure of have -codimension? Of course, the first question is stronger than the second one, and both of them would follow easily if we knew that the Lyapunov exponents vary continuously with the cocycle in the -topology (). In particular, this motivates the question: when do the Lyapunov exponents vary continuously? To the best of my knowledge, the current scenario is the following: for -topology, we know (after the works of Bochi and Viana, see here and here) that is a point of -continuity whenever the Oseledets splitting is dominated or trivial. Moreover, Bochi and Viana have examples of discontinuity for the -topology with small. Also, it was recently shown by Bocker and Viana that Lyapunov exponents of locally constant cocycles over Bernoulli shifts vary continuously with the cocycle (and the probability weights).
Closing this section, we recall that Bonatti, Gomez-Mont, Viana showed that the subset of cocycles with non-vanishing Lyapunov exponents can be chosen independently of the measure if we consider exclusively dominated cocycles (in an appropriated sense). Of course, a natural question is whether such a choice of (independently of ) can be made in general, i.e., in the statement of Theorem 5.
–Appendix: proof of Proposition 5–
We end today’s post with the proof of Proposition 5 (on the construction of Markov-like partitions). Take with and set . For each , let and define inductively
where . By definition, and for all .
Definition 13 Let and , i.e.,
Definition 14 Let where .
We claim that satisfies the conclusions of Proposition 5. Indeed, we begin by showing that is not very small:
Lemma 15 for all .
Proof: By definition of and , one has for all . Hence, . Now, given and , let
We know that as (since ). So, if is sufficiently small, we have for all . On the other hand, for all and . Therefore,
so that for all . In particular, it follows that . This completes the proof of the lemma.
Next, we will show that satisfies the Markov-like property. To do so, we need the following lemma:
Lemma 16 Suppose that . Then,
- (a) and
- (b) for all , if , then .
Proof: Let us prove item (a). If intersects , then , and hence . Thus, our task is reduced to consider the case for some .
If , one gets . Thus, (since, otherwise, , and hence ). Therefore, . In particular, (because implies ) and, a fortiori, .
The remaining case is similar: one has , and so we fit into the hypothesis of the lemma with replaced by and replaced by . Hence, by induction, we complete the proof of item (a) (as the case is immediate).
Now, let us prove item (b). If , we have , and hence, by definition of , one gets . By item (a) (we just proved), it follows that , a contradiction with the fact that . If , we have , so that, by item (a),
Thus, , and the proof of the lemma is complete.
Finally, we end the proof of Proposition 5 by verifying the Markov-like property. Assume that . By item (a) of the previous lemma, it follows that . Therefore, by definition of , it suffices to check that for all . By item (b) of the previous lemma, if for some , then , and hence, by definition of and , , a contradiction. This ends the argument.