Posted by: matheuscmss | May 27, 2013

Cohomology of Schmidt-bounded cocycles

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 {a:X\rightarrow GL(d,\mathbb{R})} (over an invertible map {T:X\rightarrow X} preserving an ergodic probability measure {\mu}) is relatively simple in appropriate coordinates (at {\mu}-almost every point {x\in X}): there exists a decomposition {\mathbb{R}^d=\oplus_{i=1}^k V_i(x)} and a finite collection of numbers {\lambda_1>\dots>\lambda_k} such that {\|a(T^{n-1}(x))\dots a(x)v_i\|\sim \exp(\lambda_i+o(1))\|v_i\|} for every {v_i\in V_i-\{0\}} as {n\rightarrow\pm\infty}. 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 {\mathbb{R}^d} “adapted” to {\oplus_{i=1}^k V_i(x)}, the cocycle {a} as a “Jordan normal form”

\displaystyle a=\left(\begin{array}{ccc} c_1&\ast&\ast \\ 0 & \ddots & \ast \\ 0 & 0 & c_m\end{array}\right) \ \ \ \ \ (1)

where {c_n}, {n=1,\dots, m}, are conformal matrices (or rather cocycles), i.e., scalar multiples of orthogonal matrices.

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 {SL(2,\mathbb{R})}-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 {SL(2,\mathbb{R})}-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 {\ast} 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.

1. Preliminaries

Let {G} be a countable group acting ergodically on a {(X,\mathcal{B},\mu)} be a standard (in Rokhlin’s sense) non-atomic probability space.

Example 1 For {G=\mathbb{Z}}, an ergodic {G}-action on {(X,\mathcal{B},\mu)} is simply the action of a single ergodic invertible transformation {T:(X,\mathcal{B},\mu)\rightarrow (X,\mathcal{B},\mu)}.

Let {H} be a (second countable) locally compact group. A cocycle {a} is a map {a:G\times X\rightarrow H} such that {a(g\cdot h,x)=a(g,h(x))\cdot a(h,x)} for all {x\in X}, {g,h\in G}.

Example 2 For {G=\mathbb{Z}} and {H=GL(d,\mathbb{R})}, any cocycle {a:\mathbb{Z}\times X\rightarrow GL(d,\mathbb{R})} is determined by a map {A:X\rightarrow GL(d,\mathbb{R})}: indeed, from {A}, one can define (inductively) {a(0,x)=\textrm{Id}\in GL(d,\mathbb{R})}, {a(1,x)=A(x)} and {a(n,x)=a(1,T^{n-1}(x))\cdot a(n-1,x)}, i.e., {a(n,x)=A(T^{n-1}(x))\dots A(x)}.

The study of cocycles is an important subject in Dynamical Systems with an entire literature dedicated to it (see, e.g., these articles here and references therein).

Today, we will discuss a beautiful result of K. Schmidt characterizing the cocycles {a:G\times X\rightarrow H} taking values in compact subgroups of {H} up to a measurable change of coordinates.

2. Cohomology of “bounded” cocycles

Definition 1 We say that a cocycle {a:G\times X\rightarrow H} is bounded (in Schmidt’s sense) if, for every {\varepsilon>0}, there exists a compact set {K_{\varepsilon}\subset H} such that

\displaystyle \mu(\{x\in X: a(g,x)\notin K_{\varepsilon}\})<\varepsilon

for all {g\in G}.

In other words, a cocycle is bounded in Schmidt’s sense if it takes “most of its values” in compact subsets of {H}.

In this language, K. Schmidt (cf. Theorem 4.7 of this paper here) showed that:

Theorem 2 (K. Schmidt) A cocycle {a:G\times X\rightarrow H} is bounded (in Schmidt’s sense) if and only if it is cohomologous to a cocycle taking its values in a compact subgroup {K} of {H}, i.e., there exists a measurable change of coordinates {c: X\rightarrow H} such that {\widetilde{a}(g,x):=c(g(x))a(g,x)c(x)^{-1}\in K} for all {x\in X}, {g\in G}.

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 {a:G\times X\rightarrow H}, we can construct a natural unitary representation of {G} as follows. Denote by {\lambda} a left-invariant Haar measure on {H} and consider the space {\widehat{X}=X\times H} equipped with the measure {\widehat{\mu}=\mu\times\lambda}.

Given {a:G\times X\rightarrow H}, we can define a skew-product action of {G} on {\widehat{X}} via the formula

\displaystyle g(x,\alpha)=(g(x), a(g,x)\alpha)

and we can consider the natural unitary representation of {G} on {L^2(\widehat{X},\widehat{\mu})}

\displaystyle \rho_a(g)(f)(x,\alpha) = f(g^{-1}(x), a(g^{-1},x)\alpha)

given by the so-called Koopman-von Neumann operator {\rho_a} of the skew-product action of {G} on {\widehat{X}}.

Lemma 3 If {a:G\times X\rightarrow H} is a bounded cocycle (in Schmidt’s sense), then {\rho_a} contains the trivial representation.

Proof: Given a Schmidt-bounded cocycle {a:G\times X\rightarrow H}, let us select a compact set {K_{1/2}\subset H} such that

\displaystyle \mu(\{x\in X: a(g,x)\notin K_{1/2}\})<1/2

for all {g\in G}.

Fix {N(id)} a compact neighborhood of the identity element of {H} with {\lambda(N(id))=1}, and let us define

\displaystyle K^*:=K_{1/2}^{-1} N(id):=\{k^{-1}n: k\in K_{1/2}, n\in N(id)\},

\displaystyle f_1(x,\alpha):=\chi_{N(id)}(\alpha)=\left\{\begin{array}{ll}1, & \textrm{if } \alpha\in N(id), \\ 0, & \textrm{otherwise}\end{array}\right.


\displaystyle f_2(x,\alpha):=\chi_{K^*}(\alpha)=\left\{\begin{array}{ll}1, & \textrm{if } \alpha\in K^*, \\ 0, & \textrm{otherwise}\end{array}\right.

By our choice of {K_{1/2}}, we have that

\displaystyle \langle \rho_a(g)(f_1), f_2\rangle>1/2

for all {g\in G}.

Denote by {C} the closure (in the weak topology) of the convex hull of the subset {\{\rho_a(g)(f_1):g\in G\}} in {L^2(\widehat{X},\widehat{\mu})}. Note that {G} acts on {C} (via {\rho_a(g)}) by isometries. In this situation, the Ryll-Nardzewski fixed-point theorem implies that this action of {G} on {C} has a fixed point, i.e., there exists {f_0\in C} such that {\rho_a(g)(f_0)=f_0} for all {g\in G}.

It follows that the representation {\rho_a(g)} contains the trivial representation if we can show that the fixed-point {f_0\in C} constructed above is not zero. However, this is not a difficult task: indeed, every element {v\in C} satisfies {\langle v,f_2\rangle>1/2}, so that {0\notin C}. This completes the proof of the lemma. \Box

2.2. Barycenter method

Once we know that the representation {\rho_a(g)} associated to a Schmidt-bounded cocycle {a} contains the trivial representation, we will find a measurable change of coordinates making that {a} takes its values in a compact subgroup of {H} by a version of the barycenter method.

Lemma 4 Let {a} be a cocycle such that {\rho_a(g)} contains the trivial representation. Then, {a} is cohomologous to a cocycle taking its values in a compact subgroup {K} of {H}.

Proof: Let {f_0\in L^2(\widehat{X},\widehat{\mu})} be a non-trivial fixed point of {\rho_a(g)} and consider the level sets {\{(x,\alpha)\in\widehat{X}: |f_0(x,\alpha)|\geq 1/n\}} for {n\in\mathbb{N}} of {|f_0|}.

Since {f_0\in L^2(\widehat{X},\widehat{\mu})} is not trivial, we can choose {n\in \mathbb{R}} such that the set

\displaystyle B=\{(x,\alpha)\in\widehat{X}: |f_0(x,\alpha)|\geq 1/n\}

satisfies {0<\widehat{\mu}(B)<\infty}.

Note that the level set {B} is invariant with respect to the skew-product action of {G} on {\widehat{X}}: indeed, this follows from the fact that {f_0} is a fixed point of {\rho_a(g)} for all {g\in G}.

In particular, we can restrict the action of {G} to {B} and we can normalize {\widehat{\mu}} so that {\widehat{\mu}(B)=1}. In this way, we obtain that the action of {G} on {B} preserves the probability measure {\widehat{\mu}|_{B}}.

Let us now consider the ergodic decomposition {\widehat{\mu}_{\widehat{x}}}, {\widehat{x}\in\widehat{B}} of {\widehat{\mu}} with respect to the {G}-action on {B}.

Denote by {\pi:\widehat{X}\rightarrow X}, {\pi(x,\alpha)=x} the projection in the first coordinate. Since {\widehat{\mu}} projects to {\mu} under {\pi} and the {G}-invariant measure {\mu} on {X} is ergodic, it follows (from the uniqueness of the ergodic decomposition) that {\widehat{\mu}_{\widehat{x}}} projects to {\mu} under {\pi} for {\widehat{\mu}}-a.e. {\widehat{x}\in X}.

Let us fix once and for all some {\widehat{x}\in \widehat{X}} such that the ergodic component {\widehat{\mu}_{\widehat{x}}} of {\widehat{\mu}} projects to {\mu} under {\pi}, and, for sake of simplicity, let us denote {\sigma=\widehat{\mu}_{\widehat{x}}} such a probability measure. Since {\pi^{-1}(x)}, {x\in X}, is a measurable partition of {\widehat{X}} (in Rokhlin’s sense), by Rokhlin’s disintegration theorem, we deduce that {\sigma} has the form

\displaystyle \sigma = \int \sigma_x d\mu(x)

where each {\sigma_x} is supported on the fiber {\pi^{-1}(x)=\{(x,\alpha):\alpha\in H\}} of {\widehat{X}} over {x\in X}.

Note that the fibers {\pi^{-1}(x)} are naturally identified with {H}. In particular, we can think of each {\sigma_x} as a probability measure on {H}.

At this point, the plan is very simple: we will find a compact subgroup {K} of {H} (or rather its Haar measure) where the cocycle takes its values after a measurable change of coordinates as a barycenter of the probability measures {\sigma_x} on {H}.

More precisely, let us consider the (trivial) translation action of {H} on {\widehat{X}=X\times H} given by {R_{\alpha}(x,\beta)=(x,\beta\alpha^{-1})}. The reader can easily check that the action {R_{\alpha}} preserves {\widehat{\mu}} and it commutes with the skew-product {G}-action on {\widehat{X}}. It follows from the fact that {\sigma} comes from a ergodic decomposition that, if the push-forward {(R_{\alpha})_*\sigma} of {\sigma} with respect to any {R_{\alpha}}, {\alpha\in H}, is not singular with respect to {\sigma}, then {(R_{\alpha})_*\sigma = \sigma}.

Denote by {K=\{\alpha\in H: (R_{\alpha})_*\sigma = \sigma\}}. It is clear that {K} is a subgroup of {H}.

Let {D\subset H} be a countable dense subset of {H} such that {D\cap K} is dense in {K}. By the uniqueness of Rokhlin’s disintegration, from the (countably many) conditions {(R_{\alpha})_*\sigma=\sigma} or {(R_{\alpha})_*\sigma\perp\sigma} (i.e., either {(R_{\alpha})_*\sigma} coincides with {\sigma} or it is singular with respect to {\sigma}) , {\alpha\in D}, we deduce that, for {\mu}-a.e. {x\in X},

\displaystyle (R_{\alpha})_*\sigma_x = \sigma_x \textrm{ or } (R_{\alpha})_*\sigma\perp\sigma

for all {\alpha\in D}.

Actually, since the push-forward operators {(R_{\alpha})_*} depend continuously on {\alpha\in H}, we can improve our last conclusion to: for {\mu}-a.e. {x\in X},

\displaystyle (R_{\alpha})_*\sigma_x = \sigma_x

for all {\alpha\in K}, and

\displaystyle (R_{\alpha})_*\sigma_x \perp \sigma_x

for all {\alpha\notin K}.

It follows that, for {\mu}-a.e. {x\in X}, the probability measure {\sigma_x} is {R_{\alpha}}-invariant for all {\alpha\in K} and {\sigma_x} is supported in a translate {C(x)\cdot K}, {C(x)\in H}, of {K} (i.e., a coset in {H/K}). Moreover, since the family of probability measures {\sigma_x} coming from Rokhlin’s disintegration theorem depend measurably on {x}, we have a measurable map {X\rightarrow H/K} associating to each {x\in X} the coset {\textrm{supp}(\sigma_x)} in {H/K} corresponding to the support of {\sigma_x}. Because there is always a measurable map {H/K\rightarrow H} (more generally, such a measurable map from {N/M\rightarrow N} exists when {N} is a second countable locally compact and {M} is a closed subgroup of {N}, see this article here), it follows that we can select elements {C(x)\in H} in the support of {\sigma_x} depending measurably on {x\in X}.

Since (left) Haar measures of {K} are the unique (left) translation-invariant measures on {K} up to a multiplicative factor, we deduce that, for {\mu}-a.e. {x\in X}, the probability measure {\sigma_x} is a multiple of a translate of (left) Haar measures of {K}. Now, recall that {\sigma_x} are probability (hence, finite) measures, so that, a fortiori, all left Haar measures of {K} are finite and this implies that {K} is a compact subgroup of {H}. In summary, if we denote by {\rho} the Haar measure of the compact group {K} (normalized so that {\mu(K)=1}), then, for {\mu}-a.e. {x\in X}, one has {\sigma_x(B)=\rho((C(x)^{-1}\cdot B)\cap K)} for all {B\subset H}.

Finally, by inspecting the definitions of {\sigma} (ergodic component of the restriction of {\widehat{\mu}=\mu\times\lambda} to the {G}-invariant subset {B\subset \widehat{X}=X\times H}), {\sigma_x} (disintegration of {\sigma}), and by using the {G}-invariance of {\sigma}, we conclude that, for {\mu}-a.e. {x\in X},

\displaystyle C(g(x)) a(g,x) C(x)^{-1}\in K

for all {g\in G}, i.e., the cocycle {a(g,x)} is cohomologous to a cocycle {C(g(x)) a(g,x) C(x)^{-1}} taking its values in {K}. \Box

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 {K} of {H} is Schmidt-bounded is easy to derive (and it is left as an exercise to the reader).

So, it remains only to check the converse implication, but this is merely a concatenation of Lemmas 3 and 4 above.

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