As usual, we’ll start with a quick revision of SPCS 3. Let be an origami and denote by the (finite) group of its automorphisms. Given a -irreducible -representation, let be the -module associated to . In this way, we wrote the interesting part of the (first absolute) homology of as
where is the isotypical component associated to . Then, we saw that could have three types (real, complex or quaternionic), and we studied the nature of the group of automorphisms of (as -module) preserving the restriction to of the natural (symplectic) intersection form on . This is illustrated in the following table:
where . Also, in the case of quaternionic, is the multiplicity of the unique complex representation with .
After this quick revision, we are ready to discuss (below the fold) the content of J.-C. Yoccoz’s 4th lecture, that is, how to use the information above on the structure of the group can be used to discuss Lyapunov exponents of the Kontsevich-Zorich cocycle on the -orbit of origamis.
-Moduli spaces of translation surfaces-
Let be a topological surface, and let a finite subset of points, say , . Consider a list of integers such that
Here, we will think that is “attached” to , and sometimes we denote .
Let be the group of homeomorphisms of preserving and , that is, for each , and let be the connected component of the identity element in . The quotient group is the so-called modular group.
We denote by the set of translation structures on , and we introduce
the Teichmuller space of translation structures on , and
the moduli space of translation structures on .
-Action of on and –
Given , and a translation structure associated to a translation atlas on , we define the translation structure associated to the translation atlas . This gives a -action on commuting with the action of (as acts by post-composition with translation charts while acts by pre-composition with translation charts). Hence, it induces -actions on and .
In this language, the Teichmuller geodesic flow is given by the action of the diagonal matrices .
Given , its Veech group is the stabilizer of under the action of . Observe that implies (as our translation surfaces have finite area). This makes it natural to introduce .
Given a translation structure , we denote by
the affine group of .
Note that we have a well-defined map obtained by looking at the linear part of in the charts of . This map is part of an exact sequence
Observe also that induces an isomorphism where .
Finally, it is possible to check that the natural map is injective and its image is exactly the set of such that .
Definition. is a Veech surface whenever is a lattice in .
Example. Let be a reduced origami (i.e., whose periods generate ). Then, is a finite index subgroup of , and hence any origami is a Veech surface.
By a result of J. Smillie, is closed if and only if is a Veech surface.
Observe that, in the case of Veech surfaces, is naturally identified with the unit tangent bundle of the hyperbolic surface . Here, denotes Poincare’s upper-half plane (we avoided the more standard notation as we’re currently using this symbol for Hamilton’s quaternions). In the literature, is called Teichmuller surface (or Teichmuller curve).
In this language, the restriction of the Teichmuller flow to is the geodesic flow on .
We begin with the following trivial bundle over (whose fibers are -dimensional):
Observe that the modular group acts on both factors of the total space of this trivial bundle: the action on was already discussed, while the action on the fibers is by pull-back (as usual). In particular, by using this action we can consider the quotient bundle
over . This quotient bundle is a non-trivial bundle known in the literature as Hodge bundle. The Hodge bundle is the phase space of a cocycle (in the sense of Dynamical Systems)
called Kontsevich-Zorich cocycle. At first sight this cocycle seems to be trivial, but this is far from being true because the Hodge bundle itself is non-trivial!
Observe that is a symplectic cocycle (as the natural intersection form on is preserved).
Furthermore, this cocycle admits plenty of invariant measures. Below we review the construction of some of them.
a. Masur-Veech measures
The so-called period map given by
(where is the natural holomorphic 1-form associated to by pull-back from the plane through the translation charts) is a local homeomorphism. Since is a complex vector space of dimension containing a canonical lattice (the image under of origamis), we can induce a Lebesgue measure on by using to pull-back the Lebesgue measure on normalized so that the canonical lattice has covolume 1.
This Lebesgue measure on is -invariant, but it has infinite mass. On the other hand, the induced Lebesgue measure on the moduli space of translation structures leading to unit area surfaces has better properties (as it was shown by H. Masur and W. Veech independently):
Theorem (H. Masur, W. Veech). The measure has finite mass. The Teichmuller flow is ergodic (and mixing) on each connected component of .
In the literature, the -invariant measure is called Masur-Veech measure. Of course, the Masur-Veech measure is invariant under the action of , it is Teichmuller flow invariant (in particular). So, this provides a class of Teichmuller flow invariant measures.
b. Haar measures on Teichmuller surfaces
Let be a Veech surface. The unit tangent bundle of the Teichmuller surface has a natural finite and -invariant Haar measure. The Teichmuller flow is ergodic (and mixing) with respect to this measure because we are in the setting of geodesic flows on the unit tangent bundle of a negatively curved surface.
Once we fix an ergodic Teichmuller flow invariant finite measure , we can start the statistical study of the Kontsevich-Zorich cocycle with the aid of Oseledets theorem.
Roughly speaking, Oseledets theorem ensures (in our setting) the existence of real numbers (Lyapunov exponents) such that for -almost every one has a decomposition
(and denotes the projection into the fibers, i.e., second factor, of Hodge bundle).
Here is any norm on the fibers of the Hodge bundle such that verifies the following integrability condition: . In general this integrability condition is a crucial issue for the application of Oseledets theorem, but in the context of the Kontsevich-Zorich cocycle it is possible to prove that it is automatically satisfied: for instance, by the works of G. Forni, for a particular norm known as the Hodge norm, so that the integrability conditions holds (with respect to the Hodge norm) as soon as has finite mass.
By the symplecticity of one can show that and .
Remark. It is not hard to check from definitions that and .
It was shown by W. Veech that in general.
On the other hand, for the Lebesgue/Masur-Veech measure on a connected component of , G. Forni proved that is hyperbolic in the sense that for all , and A. Avila and M. Viana improved this result (and solved the so-called Kontsevich-Zorich conjecture) by proving that for all .
More recently, A. Eskin, M. Kontsevich and A. Zorich developed explicit formulas for the sum of the non-negative Lyapunov exponents with respect to any “good ”-invariant finite measure.
Our discussion so far concerned the Kontsevich-Zorich cocycle and its Lyapunov exponents with respect to general Teichmuller flow/ invariant ergodic measures. From now on, we will specialize to the case of origamis.
-Lyapunov exponents associated to origamis-
Let be an origami. The decomposition
- constant along ;
- invariant under the action of the affine group ;
- preserved by .
The factor corresponds to the Lyapunov exponents and , and the action of the group of orientation-preserving elements of factors through where acts in the standard way on . This justifies the superscript “st” on as it is the factor of homology where we act in a standard way.
Next, we recall the notation from previous lectures. Given , we define an automorphism of by
In this way, we obtain a homomorphism . Since and, a fortiori, are finite, we have that
is a finite-index subgroup of .
By definition, any commutes with the action of , so that the induced map
is a homomorphism of -modules preserving the natural intersection form on .
Consider the decomposition
into isotypical irreducible representations.
It follows that, for , one has , and thus . In other words, we have a natural homomorphism
The decomposition (1) above is constant along the orbit of , so that it is invariant by (as and any character is invariant under conjugation). This allows us to fix once and for all , and consider and its Lyapunov exponents denoted by by a slight abuse of notation.
Let be a generic point (in the sense that Oseledets theorem can be applied to it). We can write
Since is a finite index subgroup of and a generic point is recurrent, we can take a sequence of times as such that is very close to in such a way that by “closing up” the segment with a bounded path between and , the corresponding closed path leads to an affine diffeomorphism . See the figure below.
In this way, since (as the largest exponent of the Kontsevich-Zorich cocycle is ) we have
Proposition. is a -module.
Proof. Because is a finite group (and the definition of Lyapunov exponents doesn’t depend on the norm), we may assume that the norm is -invariant (by the usual averaging argument). Since , it commutes with the action . It follows that, for any ,
and thus whenever .
This proposition implies that . Furthermore,
- (as ), and
- (by symplecticity).
It is worth to note that to the best of our knowledge, the following question is open:
Question. Is there an example where for some .
In general, we have that , and
- when is complex;
- when is quaternionic.
In fact, while this general discussion is the best one can do concerning the case real, it is possible to say a little more about Lyapunov exponents in the complex and quaternionic cases. This is the content of the last section of today’s post.
-Lyapunov exponents when is complex or quaternionic-
Recall that or when is complex or quaternionic.
Lemma 1. Let , where . Then, , where is the Hermitian form invariant by of signature with and . The real part of considered as a real quadratic form has signature .
Proof. Without loss of generality we may assume (as the other case is symmetric). This implies that
so that the proof is complete.
Lemma 2. Let be a non-degenerate quadratic form of signature and let be a vector subspace such that . Then, .
Proof. Without loss of generality we may assume (as the other case is symmetric), and by appropriately choosing a basis,
If , we notice that there exists such that : indeed, the algebraic operation of annihilating a given coordinate reduces the dimension of by at most, so that we can find a non-trivial vector whose first coordinates vanish inside any subspace of dimension (at least). By the expression of above, this would mean that , a contradiction with the assumption .
In the setting of origamis, we take and
By Lemma 1, , so that, by Lemma 2, we have the following corollary
Of course, a similar statement is true for
In particular, we conclude the following estimate concerning zero Lyapunov exponents:
Partly motivated by this discussion, we have the following questions:
Question. Is there an example where the inequality is strict?
Question. When is real, is there an example with some zero Lyapunov exponent in ?
Question. In the case of hyperelliptic origamis (i.e., origamis whose underlying Riemann surface is hyperelliptic), is there a difference in our discussions so far if we replace by
Closing today’s post, we mention that the plan for the next lecture is to apply the discussions of SPCS 1, 2, 3 and 4 to concrete examples (especially regular and quasi-regular origamis).