Alex Eskin and I have just uploaded to ArXiv our paper “Semisimplicity of the Lyapunov spectrum for irreducible cocycles”.
In this article, we are mainly interested in the constraints on the Lyapunov spectrum of certain linear cocycles acting in a irreducible way on the fibers.
More concretely, we consider the following setting. Let be a semisimple Lie group acting on a space . Denote by a compactly supported probability measure on and a -stationary probability on , i.e., (that is, is “invariant on -average” under push-forwards by elements of ).
A linear cocycle is where is a real finite-dimensional vector space. For our purposes, we will assume that the matrices are bounded for any in the support of .
From the point of view of Dynamical Systems, we think of a (linear) cocycle as the “fiber dynamics” of the following system modeling random products of the matrices while following a forward random walk on .
Let and denote by be the natural shift map where . Observe that, by -stationarity of , the product probability measure is -invariant where . For the sake of simplicity, we will assume that the -stationary measure is ergodic in the sense that is -ergodic.
In this language, the orbits of the map given by
are modeling random products of the matrices along random walks in . Indeed, this is clearly seen through the formula:
In this context, Oseledets theorem applied to ensures the existence of a collection of numbers with multiplicities called Lyapunov exponents and, at -almost every point , a Lyapunov flag
such that is a subspace of dimension and
The collection of Lyapunov exponents and Lyapunov flags of (or, more precisely, ) is called the Lyapunov spectrum of .
Evidently, the Lyapunov spectrum of an “arbitrary” cocycle can exhibit “any” wild behavior. However, concerning “specific” cocycles, a general question of great interest in Dynamical Systems/Ergodic Theory is the following: what can one say about the Lyapunov spectrum of a cocycle satisfying certain geometrical and/or algebraic constraints?
Of course, this question is somewhat vague and, for this reason, there is no unique answer to it. On the other, it is precisely the vagueness of this question that makes it so appealing and this explains the vast literature providing answers for several formulations of this question.
For instance, the seminal works of Furstenberg, Goldsheid-Margulis and Guivarc’h-Raugi gave geometrical (“contraction”/“proximality” and “strong irreducibility”) and algebraic conditions (full Zariski closure of the monoid of matrices generated by the cocycle in the special linear group or in the symplectic group ) ensuring the simplicity of the Lyapunov spectrum (that is, the multiplicities are all equal to ). More recently, Avila-Viana gave an alternative set of geometrical conditions (“pinching” and “twisting”) ensuring simplicity for cocycles taking values in both and , and it was observed by Möller, Yoccoz and myself in this paper here that the arguments of Avila-Viana can be further extended to cocycles taking values in the classical groups , and (resp.) of real, complex and quaternionic matrices (resp.) preserving an indefinite form of signature on .
Actually, a careful inspection of these papers (in particular the ones by Golsheid-Margulis and Guivarc’h-Raugi) reveals that one can get partial constraints on the Lyapunov spectrum by assuming only some parts of the geometrical conditions required in these articles, and, as it turns out, this fact is already important in some applications.
More precisely, during the proof of their profound Ratner-type theorem for the -action on the moduli space of translation surfaces, Eskin-Mirzakhani needed to know a certain property of semisimplicity of the so-called Kontsevich-Zorich cocycle (in order to apply the “exponential drift” idea of Benoist-Quint via appropriate “time-changes”). Here, by semisimplicity we mean that, up to conjugating (or, equivalently, replacing the cocycle by for some adequate measurable map ), the cocycle is block-conformal, i.e.,
where ‘s are conformal (that is, belongs to an orthogonal group after multiplication by an adequate constant). Equivalently, has semisimple Lyapunov spectrum if we can write the quotients between consecutive subspaces of the Lyapunov flag as
where each admits a non-degenerate quadratic form such that for all and for all one has
with satisfying the cocycle relation
In this setting, Eskin-Mirzakhani needs the following fact essentially contained in the works of Goldsheid-Margulis and Guivarc’h-Raugi: a cocycle as above has semisimple Lyapunov spectrum whenever it is strongly irreducible, i.e., no finite cover of (or rather the induced cocycle in a finite cover of ) preserves a measurable family of proper subspaces of in the sense that for -almost every and -almost every (where is the natural measure induced in the corresponding finite cover of ).
Unfortunately, even though the experts in the subject know that this statement really follows from the ideas of Goldsheid-Margulis and Guivarc’h-Raugi, it is hard to deduce this fact directly from the statements in these papers.
The proof of this fact in the case of a strongly irreducible cocycle whose algebraic hull is is explained in Appendix C of Eskin-Mirzakhani’s paper. Here, by algebraic hull we mean the smallest -algebraic subgroup such that for some measurable (conjugation) map (by a result of Zimmer, algebraic hulls always exist and they are unique up to conjugation). Nevertheless, for their application to the Kontsevich-Zorich cocycle, they need the semisimplicity statement for the general case, i.e., without assumptions on the algebraic hull (because the Kontsevich-Zorich takes values in some other smaller classical groups in certain examples), and this is precisely one of the main purposes of our preprint with Alex Eskin.
In other terms, one of our main objectives is to adapt the ideas of Goldsheid-Margulis and Guivarc’h-Raugi to show that a strongly irreducible cocycle has semisimple Lyapunov spectrum. Also, we show in the same vein that the top Lyapunov exponent is associated to a single conformal block in the sense that the decomposition in (1) is trivial (i.e., ) for .
In a sense, some of the ideas of proof of the statement in the previous paragraph were previously discussed in this blog (see this post here) in some particular cases.
For this reason, we will not give here a detailed discussion of our preprint with Alex Eskin. Instead, we strongly encourage the reader that is not used to these types of arguments/ideas to replace by (or , ) in our paper with Alex Eskin and then to compare the statements there with the ones in this previous blog post here and/or in Appendix C of Eskin-Mirzakhani’s paper. By doing so, the reader will be convinced that the basic ideas are the same up to replacing some linear algebra statements by the analogous facts for the action of elements of on stationary measures supported on a flag variety (where is a certain parabolic subgroup corresponding to some subset of simple roots), etc.
Closing this post, let us just to try to summarize in a couple of words the proof of the semisimplicity of the Lyapunov spectrum of strongly irreducible cocycles . Firstly, by analyzing (in Section 2 of our preprint) the action of elements of the algebraic hull on adequate flag varieties (for some choices of subsets of simple roots), we show that can be conjugated to take its values in a certain parabolic subgroup (cf. Proposition 3.2 of our preprint). In a certain sense, this information seems of little value because, very roughly speaking, the fact that takes values in a parabolic subgroup essentially amounts to say that we can put in the form
and this is certainly not the desired block conformality property (as this last property means that all entries above are all zero). Nonetheless, we combine the information takes values in a certain parabolic subgroup with the analogous statement for the backward cocycle (i.e., the cocycle obtained by following the bacwards random walk on ) to deduce that is Schmidt-bounded, i.e., it is uniformly bounded on large compact sets of almost full measure. By a result of Schmidt (discussed in this previous blog post), up to conjugation, any Schmidt-bounded cocycle takes values in a compact subgroup, and from this last fact one can establish that is block-conformal (i.e., all ‘s are zero in the equation above). Finally, the statement that the top Lyapunov exponent corresponds to a single conformal block essentially follows from the well-known fact that the highest weight of the irreducible action of the algebraic hull of the strongly irreducible cocycle on the real finite-dimensional vector space has multiplicity .