where and is the shift map equipped with the Bernoulli measure .
By Oseledets multiplicative ergodic theorem, the Lyapunov exponents of the random product of and (i.e., the linear cocycle ) are well-defined quantities (depending only on and ) describing the exponential growth of the singular values of the random products
for any -typical choice of .
Moreover, the fact that and are symplectic matrices implies that the Lyapunov exponents are symmetric with respect to the origin, i.e., for each . In other words, the Lyapunov exponents of the symplectic linear cocycle have the form:
In fact, this structure of the Lyapunov exponents of a symplectic linear cocycle reflects the fact that if is an eigenvalue of a symplectic matrix , then is also an eigenvalue of .
A natural qualitative question about Lyapunov exponents concerns their simplicity in the sense that there are no repeated numbers in the list above (i.e., for all ).
The simplicity property for Lyapunov exponents is the subject of several papers in the literature: see, e.g., the works of Furstenberg, Goldsheid-Margulis, Guivarch-Raugi, and Avila-Viana (among many others).
Very roughly speaking, the basic philosophy behind these papers is that the simplicity property holds whenever the monoid generated by and is rich. Of course, there are several ways to formalize the meaning of the word “rich”, for example:
- Goldsheid-Margulis and Guivarch-Raugi asked to be Zariski-dense in ;
- Avila-Viana required to be
Of course, these notions of “richness” of a monoid are “close” to each other, but they differ in a subtle detail: while the Zariski-density condition on is an algebraic requirement, the pinching and twisting condition on makes no reference to the algebraic structure of the linear group .
In particular, this leads us to the main point of this post:
How the Zariski-density and pinching and twisting conditions relate to each other?
The first half of this question has a positive answer: a Zariski-dense monoid is also pinching and twisting. Indeed:
- (a) a modification of the arguments in this blog post here (in Spanish) permits to prove that any Zariski-dense monoid contains a pinching matrix , and
- (b) the twisting condition on a matrix with respect to a pinching matrix can be phrased in terms of the non-vanishing of certain (isotropic) minors of the matrix of written in a basis of eigenvectors of ; thus, a Zariski-dense monoid contains a twisting matrix with respect to any given pinching matrix.
On the other hand, the second half of this question has a negative answer: we exhibit below a pinching and twisting monoid which is not Zariski dense.
Remark 1 The existence of such examples of monoids is certainly known among experts. Nevertheless, I’m recording it here because it partly “justifies” a forthcoming article joint with Artur Avila and Jean-Christophe Yoccoz in the following sense.
The celebrated paper of Avila-Viana quoted above (on Kontsevich-Zorich conjecture) shows that the so-called “Rauzy monoids” are pinching and twisting (and this is sufficient for their purposes of proving simplicity of Lyapunov exponents of the Kontsevich-Zorich cocycle for Masur-Veech measures).
On the other hand, since a pinching and twisting monoid is not necessarily Zariski dense (as we are going to see below), the results of Avila-Viana (per se) can not answer a question of Zorich (see also Remark 6.12 in Avila-Viana paper) about the Zariski density of Rauzy monoids.
In this direction, Artur, Jean-Christophe and I solve (in an article still in preparation) Zorich’s question about Zariski density of Rauzy monoids in the special case of hyperelliptic Rauzy diagrams, and the main example of this post (which will be included in our forthcoming article with Artur and Jean-Christophe) serves to indicate that the results obtained by Artur, Jean-Christophe and myself can not be deduced as “abstract consequences” of the arguments in Avila-Viana paper.
Remark 2 The main example of this post also shows that (a version of) Prasad-Rapinchuk’s criterion for Zariski density (cf. Theorem 9.10 of Prasad-Rapinchuk paper or Theorem 1.5 in Rivin’s paper) based on Galois-pinching (in the sense of this paper here) and twisting properties is “sharp”: indeed, an important feature of the main example of this post is the failure of the Galois-pinching property (cf. Remark 4 below for more comments).
1. A monoid of 4×4 symplectic matrices
Let be the third symmetric power of the standard representation of . In concrete terms, is constructed as follows. Consider the basis of the space of homogenous polynomials of degree on two variables and . By letting act on and as and , we get a linear map on whose matrix in the basis is
Remark 3 The faithful representation is the unique irreducible four-dimensional representation of .
The matrices preserve the symplectic structure on associated to the matrix
Indeed, a direct calculation shows that if , then
where stands for the transpose of .
Therefore, the image is a linear algebraic subgroup of the symplectic group , and the Zariski closure of the monoid generated by the matrices
is precisely .
Remark 4 Coming back to Remark 2, observe that does not contain Galois-pinching elements of in the sense of this paper here (i.e., pinching elements of with integral entries whose characteristic polynomial has the largest possible Galois group for a reciprocal polynomial [namely, the hyperoctahedral group]) because its rank is . Alternatively, a straightforward computation reveals that the characteristic polynomial of is
and, consequently, the eigenvalues of are
In particular, since the characteristic polynomial of always splits, it is never the case that is Galois-pinching.
On the other hand, the element is pinching because its eigenvalues are
Also, the matrix is twisting with respect to . Indeed, the columns of the matrix
consist of eigenvectors of . Thus, is the matrix of in the corresponding basis of eigenvectors of . Moreover, is twisting with respect to if and only if all entries of and all of its minors associated to isotropic planes are non-zero (cf. Lemma 4.8 in this paper here). Finally, this last fact is a consequence of the following exact calculation (see also the numerical approximations) for and its matrix of minors:
In summary, the monoid is pinching and twisting, but not Zariski dense in .