Last week, the conference “Random walks on groups” took place at IHP as part of the activities of a trimester on random walks and asymptotic geometry of groups (organized by Indira Chatterji, Anna Erschler, Vadim Kaimanovich, and Laurent Saloff-Coste) from January to March 2014.
Given the very interesting program of this conference, it was not surprising that Amphithéâtre Hermite (where the talks were delivered) was always full.
Today, we will discuss one of the talks of this conference, namely, the talk “On the continuity of Lyapunov spectrum for random products” of Alex Eskin about his joint work (in preparation) with Artur Avila and Marcelo Viana.
As usual, all mistakes/errors in this post are entirely my responsibility.
Remark 1 A video of a talk of Artur Avila on the same subject can be found here.
Update [February 11, 2014]: Last Friday, I was lucky enough to get some extra explanations concerning “costs of couplings” directly from Alex. At the end of this post (see the “Epilogue”), I will try to briefly summarize what I could understand from this conversation.
Let be a probability measure on , e.g., where is a (non-trivial) probability vector (i.e., and for all ) and are Dirac masses at .
Consider the random walk on induced by , i.e., let , , and, for each , , put
Remark 2 Of course, the intuition here is that the samples , , are describing a random walk on whenever we perform a random choice of with respect to (or, equivalently, random choices of ‘s with probability distribution ).
In this context, the Oseledets multiplicative ergodic theorem says that:
Theorem 1 (Oseledets) For -almost every , one has
where is a symmetric matrix with eigenvalues . (Here, is the transpose matrix of , and denotes the non-negative symmetric matrix such that .)
The numbers are called Lyapunov exponents.
Geometrically, Oseledets theorem says that the random walk almost surely tracks a geodesic of speed of the symmetric space (where is a maximal compact subgroup of ).
Remark 3 The top Lyapunov exponent can be recovered by the formula
for -a.e. , and the remaining Lyapunov exponents can be recovered by the following standard trick/observation: the top Lyapunov exponent of the action of on the -th exterior power is . For this reason, it is often (but not always!) the case that the results about the top Lyapunov exponent also provide information about all Lyapunov exponents.
Historically, the first results about the Lyapunov exponents of random products concerned their multiplicities for a fixed probability distribution . A prototypical theorem in this direction is the following result of Guivarch-Raugi and Goldsheid-Margulis providing sufficient conditions for the simplicity (multiplicity ) of Lyapunov exponents.
Definition 2 We say that is not strongly irreducible whenever there exists a finite collection of subspaces of such that
for all .
Definition 3 We say that is proximal if there exists such that has distinct eigenvalues. (Here, is the Zariski closure of the group generated by .)
Remark 4 If is Zariski dense in , then is strongly irreducible and proximal.
Theorem 4 (Guivarch-Raugi, Goldstein-Margulis)
- 1) If is strongly irreducible and proximal, then (i.e., the top Lyapunov exponent is simple/has multiplicity );
- 2) If is Zariski dense in , then .
2. Statement of the main result
In their work, Avila, Eskin and Viana consider how the Lyapunov exponents change when the probability distribution varies. Among the results that they will prove in their forthcoming article is:
Theorem 5 (Avila-Eskin-Viana) Suppose is a fixed probability vector, and consider the probability measures
whose support varies. Then, for each , the Lyapunov exponent is a continuous function of .
Remark 5 This statement looks innocent, but it is known that Lyapunov exponents do not vary continuously (only upper semi-continuously) “in general”. See, e.g., this article of Bochi (and the references therein) for more details.
3. Previous works and related results
The theorem of Avila-Eskin-Viana generalizes to any dimension the work of Bocker-Neto and Viana in dimension :
Theorem 6 (Bocker-Neto-Viana) For a fixed probability vector , the two Lyapunov exponents of
depend continuously on .
On the other hand, if one decides to fix the support and to vary the vector of probabilities, then Peres showed in 1991 that:
Theorem 7 (Peres) Let us fix the support . Then, the simple Lyapunov exponents of
are locally real-analytic function of . More precisely, given and a probability vector such that the th Lyapunov exponent is simple (i.e., multiplicity ), then the th Lyapunov exponent is a real-analytic function of near .
The formula for -a.e. for the top Lyapunov exponent is not very useful to study how Lyapunov exponents vary with because the notion of “-a.e. ” changes radically with .
A slightly more useful formula was found by Furstenberg:
where and is a -stationary measure (i.e., is invariant in average with respect to , that is, ) on the projective space of lines in .
Of course, the cocycle depends nicely on and , but the dependence on of the stationary measure in Furstenberg’s formula is not obvious to determine. In particular, one needs to “feed” Furstenberg’s formula with extra information in order to deduce continuity of the top Lyapunov exponent in a given setting.
For example, if one feeds the following remark
Remark 6 If is strongly irreducible and proximal, then the stationary measure on is unique.
to Furstenberg’s formula, then one can deduce:
Proposition 8 Suppose (in the weak-* topology), is proximal and strongly irreducible. Then, .
Proof: Denote by the sequence of stationary measures associated to in Furstenberg’s formula. It is not hard to check that any accumulation of the sequence is -stationary. By the previous remark, has an unique stationary measure on , so that any accumulation of coincides with the stationary measure in Furstenberg’s formula for . In other words, , and the desired proposition now follows immediately from Furstenberg’s formula.
Remark 7 Le Page showed that the conclusion of the previous proposition can be improved from continuity to real-analyticity. However, in general (without strong irreducibility and proximality of ), one can not expect anything better than Hölder continuity.
4. Some ideas of the proof of Avila-Eskin-Viana theorem
Let us simplify the exposition by considering the following toy case: we are given two sequences of matrices
and we want to show that the top Lyapunov exponents of the probabilities
The projective actions of the matrices and on the projective circle are of “north pole–south pole type”: there are two fixed points and corresponding to the directions of the coordinate axes and of and the points of are either attracted or repelled towards and under the actions of and . In particular, one can infer from this that an arbitrary -stationary measure on has the form
Therefore, if we denote by the -stationary measures coming from Furstenberg’s formula, then
and our goal is to show that . However, there is not so easy as it seems (in the sense that naive methods don’t work well) and one has to look for appropriate tools.
In this direction, the notion of Margulis function comes at hand. Given a probability measure on a group acting on a space , let
be the Markov operator associated to . We say that is a Margulis function if:
- 2) on a “negligible set”
- 3) there are constants and such that , i.e., when is large at a point (a step of a -random walk approaches ), the value of at the -images of this point decrease in average (the next step of a -random walk tend to get far from ).
Coming back to toy case, it is possible to show that for the function given by
is a Margulis function for .
This type of information is useful to show simplicity of the Lyapunov exponents of , but it does not help us to show the continuity statement or . In fact, the difficulty comes from the fact that is not a Margulis function of because the south pole of is changing location (even though they are close to ), so that a single Margulis function is not capable of assigning the value to all of the south poles of without being trivial.
Here, one can try to overcome the technical obstacle of the moving south poles of by considering the diagonal action of on and by introducing the function
for and close to . As it turns out, this function is a good candidate of Margulis function for in the sense that the inequality in item 3) involving the Markov operator is satisfied near , and it seems that we are doing some progress.
Unfortunately, we made no progress at all with the idea in the previous paragraph: indeed, the technology of Margulis functions requires globally defined functions and so far we were able only to exhibit locally defined functions (in a neighborhood of ).
At this point, the basic idea of Avila-Eskin-Viana is the introduction of measure-theoretical analogs of Margulis functions. In other terms, they want to replace “functions” by “measures” to get objects that are slightly more flexible but still capable of doing the same job than Margulis functions.
The measure-theoretical analog of Margulis functions are called couplings with finite costs. Concretely, we say that a probability measure on is a coupling of to itself if the projection of to both factors is . Given a coupling of to itself, we define its cost as:
where is an adequate small neighborhood of .
In this setting, we can see that the task of showing is reduced to find a large constant and a sequence of couplings of to itself such that
for all . Indeed, this is so because the cost of coupling to itself is and thus has finite cost only when .
At this point, the time of Alex Eskin was essentially out and he concluded by saying that the main point is that finding couplings with finite costs is easier than building globally defined Margulis functions, and the desired couplings with uniformly bounded costs could be found by analyzing the analog of item 3) in the definition of Margulis functions for couplings of to itself with optimal (minimal possible) costs.
Let us try to give more explanations to the discussion in the previous 6 paragraphs above (following my conversation with Alex Eskin [or what I can remember of it…]).
We start by selecting the small neighborhood of so that the limit stationary measure gives mass
Then, we restrict our measures to and we change the dynamics so that these restrictions are stationary: formally, we replace the Markov operator by an adequate “local transfer operator” such that is -stationary.
In these terms, the “local version”
of the “usual candidate to Margulis function” seems to be a Margulis function at first sight, but unfortunately it does not satisfies item 3). Indeed, the pointwise estimates of the form
with and do not hold always because there are some couples of points that are pushed together towards despite the fact that the probability of this event is small.
For this reason, Avila-Eskin-Viana replace “functions” by “measures” with the idea that this probabilistic tendency felt by most couples of getting away from is better expressed as estimates for measures than pointwise estimates for functions.
More concretely, by selecting an appropriate subinterval , one can see that the -measure of the set
of elements of pushing a point towards is . From this information, it is not difficult to construct some measures on such that projects to on both factors and . From the measures , one obtains some couplings with finite costs.
However, this is not quite the end of the history: we need couplings whose costs are uniformly bounded for all . Here, the trick is to study couplings with optimal costs (i.e. with smallest possible costs). In fact, by applying the “dynamics” to , one has the following analogue of item 3) in the definition of Margulis functions:
for some universal constants and (thanks to the probabilistic tendency of most couples of points to get pushed away from ). On the other hand, since has optimal (smallest) cost, we conclude that
In other terms, the analog for measures of item 3) in the definition of Margulis functions allows to check that the costs of the sequence optimal cost couplings are uniformly bounded by , as desired.