One of the most important results for the measurable study of linear (dynamical) cocycles is Oseledets theorem. Very roughly speaking, this fundamental result says that the fiber dynamics of a linear cocycle (over an invertible map preserving an ergodic probability measure ) is relatively simple in appropriate coordinates (at -almost every point ): there exists a decomposition and a finite collection of numbers such that for every as . The content of Oseledets’ theorem become even clearer when combined with Zimmer’s amenable reduction theorem: by putting these results together, one essentially has that, by selecting a basis of “adapted” to , the cocycle as a “Jordan normal form”
For several applications, this Jordan normal form is satisfying, but sometimes it is desirable to “improve” this normal form in certain specific situations.
For example, the recent paper of A. Eskin and M. Mirzakhani shows a Ratner-like result for -actions on moduli spaces of Abelian differentials based on a certain exponential drift argument of Y. Benoist and J.-F. Quint (among several other arguments).
As it turns out, the exponential drift argument strongly relies on the fact that it is possible to perform time-changes for the Kontsevich-Zorich cocycle (over the -dynamics on moduli spaces) trying to make it look like a conformal by blocks linear cocycle. Here, the existence of such time-changes is not a consequence of Oseledets theorem and Zimmer’s amenable reduction theorem: indeed, it is possible to check that the existence of these time-changes are essentially equivalent to have that all blocks in (1) vanish.
In particular, we see that it is important (for some profound applications) to dispose of a criterion ensuring that a linear (dynamical) cocycle is conformal by blocks possibly after an adequate (measurable) change of coordinates.
In this post, we will follow this article of K. Schmidt (from 1981) to give a complete answer to the question in the previous paragraph.
Let be a countable group acting ergodically on a be a standard (in Rokhlin’s sense) non-atomic probability space.
Example 1 For , an ergodic -action on is simply the action of a single ergodic invertible transformation .
Example 2 For and , any cocycle is determined by a map : indeed, from , one can define (inductively) , and , i.e., .
Today, we will discuss a beautiful result of K. Schmidt characterizing the cocycles taking values in compact subgroups of up to a measurable change of coordinates.
2. Cohomology of “bounded” cocycles
Definition 1 We say that a cocycle is bounded (in Schmidt’s sense) if, for every , there exists a compact set such that
for all .
In other words, a cocycle is bounded in Schmidt’s sense if it takes “most of its values” in compact subsets of .
In this language, K. Schmidt (cf. Theorem 4.7 of this paper here) showed that:
Theorem 2 (K. Schmidt) A cocycle is bounded (in Schmidt’s sense) if and only if it is cohomologous to a cocycle taking its values in a compact subgroup of , i.e., there exists a measurable change of coordinates such that for all , .
The proof of Theorem 2 is not specially long, but, for the sake of exposition, we will divide it into two lemmas.
2.1. Bounded cocycles contain the trivial representation
From our cocycle , we can construct a natural unitary representation of as follows. Denote by a left-invariant Haar measure on and consider the space equipped with the measure .
Given , we can define a skew-product action of on via the formula
and we can consider the natural unitary representation of on
given by the so-called Koopman-von Neumann operator of the skew-product action of on .
Proof: Given a Schmidt-bounded cocycle , let us select a compact set such that
for all .
Fix a compact neighborhood of the identity element of with , and let us define
By our choice of , we have that
for all .
Denote by the closure (in the weak topology) of the convex hull of the subset in . Note that acts on (via ) by isometries. In this situation, the Ryll-Nardzewski fixed-point theorem implies that this action of on has a fixed point, i.e., there exists such that for all .
It follows that the representation contains the trivial representation if we can show that the fixed-point constructed above is not zero. However, this is not a difficult task: indeed, every element satisfies , so that . This completes the proof of the lemma.
2.2. Barycenter method
Once we know that the representation associated to a Schmidt-bounded cocycle contains the trivial representation, we will find a measurable change of coordinates making that takes its values in a compact subgroup of by a version of the barycenter method.
Proof: Let be a non-trivial fixed point of and consider the level sets for of .
Since is not trivial, we can choose such that the set
Note that the level set is invariant with respect to the skew-product action of on : indeed, this follows from the fact that is a fixed point of for all .
In particular, we can restrict the action of to and we can normalize so that . In this way, we obtain that the action of on preserves the probability measure .
Let us now consider the ergodic decomposition , of with respect to the -action on .
Denote by , the projection in the first coordinate. Since projects to under and the -invariant measure on is ergodic, it follows (from the uniqueness of the ergodic decomposition) that projects to under for -a.e. .
Let us fix once and for all some such that the ergodic component of projects to under , and, for sake of simplicity, let us denote such a probability measure. Since , , is a measurable partition of (in Rokhlin’s sense), by Rokhlin’s disintegration theorem, we deduce that has the form
where each is supported on the fiber of over .
Note that the fibers are naturally identified with . In particular, we can think of each as a probability measure on .
At this point, the plan is very simple: we will find a compact subgroup of (or rather its Haar measure) where the cocycle takes its values after a measurable change of coordinates as a barycenter of the probability measures on .
More precisely, let us consider the (trivial) translation action of on given by . The reader can easily check that the action preserves and it commutes with the skew-product -action on . It follows from the fact that comes from a ergodic decomposition that, if the push-forward of with respect to any , , is not singular with respect to , then .
Denote by . It is clear that is a subgroup of .
Let be a countable dense subset of such that is dense in . By the uniqueness of Rokhlin’s disintegration, from the (countably many) conditions or (i.e., either coincides with or it is singular with respect to ) , , we deduce that, for -a.e. ,
for all .
Actually, since the push-forward operators depend continuously on , we can improve our last conclusion to: for -a.e. ,
for all , and
for all .
It follows that, for -a.e. , the probability measure is -invariant for all and is supported in a translate , , of (i.e., a coset in ). Moreover, since the family of probability measures coming from Rokhlin’s disintegration theorem depend measurably on , we have a measurable map associating to each the coset in corresponding to the support of . Because there is always a measurable map (more generally, such a measurable map from exists when is a second countable locally compact and is a closed subgroup of , see this article here), it follows that we can select elements in the support of depending measurably on .
Since (left) Haar measures of are the unique (left) translation-invariant measures on up to a multiplicative factor, we deduce that, for -a.e. , the probability measure is a multiple of a translate of (left) Haar measures of . Now, recall that are probability (hence, finite) measures, so that, a fortiori, all left Haar measures of are finite and this implies that is a compact subgroup of . In summary, if we denote by the Haar measure of the compact group (normalized so that ), then, for -a.e. , one has for all .
Finally, by inspecting the definitions of (ergodic component of the restriction of to the -invariant subset ), (disintegration of ), and by using the -invariance of , we conclude that, for -a.e. ,
for all , i.e., the cocycle is cohomologous to a cocycle taking its values in .
2.3. End of proof of Schmidt’s theorem
The statement in Schmidt’s Theorem 2 that a cocycle that is cohomologous (measurably conjugated) to a cocycle taking its value in a compact subgroup of is Schmidt-bounded is easy to derive (and it is left as an exercise to the reader).