Today we will make some preparations towards the application of Avila-Viana simplicity criterion to the case of the Kontsevich-Zorich cocycle over the -orbits of square-tiled surfaces (along the lines of a forthcoming article by C.M., M. Möller, and J.-C. Yoccoz). Of course, given that the simplicity criterion was stated in terms of locally constant cocycles over complete shifts on finitely or countably many symbols, we need to discuss how to “reduce”, or more precisely, to code, the Kontsevich-Zorich cocycle over the -orbits of square-tiled surfaces to the setting of complete shifts. In fact, the main purpose of this post (corresponding to J.-C. Yoccoz 7th lecture) is the presentation of an adequate coding of the Teichmüller flow and Kontsevich-Zorich cocycle over the -orbits of square-tiled surfaces.
Let be a reduced origami (i.e., an origami whose periods generate the lattice ). Let’s consider the Kontsevich-Zorich cocycle over the Teichmüller flow restricted to the unit tangent bundle of the Teichmüller surface (“curve”) associated to .
More concretely, we consider (the unit tangent bundle) , where is the Veech group of (that is, the stabilizer of under the action of on the moduli space of Abelian differentials). Recall that is a finite-index subgroup of (as is a reduced origami). In this language, the Teichmüller geodesic flow is the action of
We begin our discussion with the case of . In the sequel, we will think of as the space of normalized (i.e., unit covolume) lattices of , and we will select an appropriate fundamental domain. Here, it is worth to point out that we’re not going to consider the lift to of the “classical” fundamental domain of the action of on the hyperbolic plane . Indeed, as we will see below, our choice of fundamental domain is not -invariant, while any fundamental domain obtained by lifting to a fundamental domain of must be -invariant (as ).
Definition 1 A lattice is irrational if intersect the coordinate axis and precisely at the origin .
Equivalently, is irrational if and only if the orbit doesn’t diverge (neither in the past nor in the future) to the cusp of .
Our choice of fundamental domain will be guided by the following fact:
- “Top” case: and ;
- “Bottom” case: and .
Proof: Consider the following open unit area squares of the plane: and .
Observe that can’t contain two linearly independent vectors of : indeed, if , and are linearly independent, then , a contradiction with the fact that (as is normalized).
On the other hand, we claim that contains (at least) one vector of . In fact, since is normalized and irrational, one would have that is disjoint from some convex symmetric open set (strictly) containing the closure of , a contradiction with Minkowski theorem (that any convex symmetric set of volume intersects any normalized lattice of ).
In particular, we have three possibilities:
- (a) , ;
- (b) , ;
- (c) , .
Because the first two cases are similar, we’ll treat only items (b) and (c).
We start by item (b), that is, but . In this situation, we select a primitive , so that
Next, we select such that
- is a direct basis, i.e., ;
- is minimal.
Then, : otherwise we could replace by to contradict the minimality. Thus, (as is a direct basis, and forces ). Since , we have that , and hence is a basis of fitting the requirements of the top case. Now, we verify the uniqueness of such . Firstly, if fits the requirements and of the bottom case, then the relation
implies that , and, a fortiori, , a contradiction with our assumptions in item (b). Secondly, if fits the requirements and of the top case, then , and, therefore, (as can’t contain two linearly independent vectors of ). Now, we write , and we notice that, since , , and , one has , i.e., , and the analysis of item (b) is complete.
It remains only to analyze item (c). Take and primitive vectors. We have that
and, since , it follows that . Furthermore, because is irrational. Assume (the other case is analogous). Then, we set and where is the largest integer such that . We have that , verifies (as ), and (as was taken to be the largest possible). Furthermore, , as, otherwise, (recall that ) and would be linearly independent vectors of inside , a contradiction. In resume, is a basis of meeting the requirements of the bottom case. Now, we check the uniqueness of such . Here, since the argument is the same one used for item (b), we will illustrate only the bottom case , . In this situation, (as is normalized), so that . Then, we write , and we conclude that because , , and .
Using this proposition, we can describe the Teichmüller geodesic flow on the space of normalized lattices as follows. Let be a normalized irrational lattice, and let be the basis of given by the proposition above, i.e., the top, resp. bottom, condition. Then, we see that the basis of satisfies the top, resp. bottom condition for all , where in the top case, resp. in the bottom case.
However, at time , the basis of ceases to fit the requirements of the proposition above, but we can remedy this problem by changing the basis: for instance, if the basis of the initial lattice has top type, then it is not hard to check that
where is a basis of of bottom type.
Here, we observe that the quantity giving the ratios of the first coordinates of the vectors forming a top type basis of for any is related to the integer by the formula
Also, the new quantity giving the ratio of the first coordinates of the vectors forming a bottom type basis of is related to by the formula
where is the so-called Gauss map. In this way, we find the classical relationship between the geodesic flow on the modular surface and the continued fraction algorithm.
At this stage, we’re ready to code the Teichmüller flow over the unit tangent bundle of the Teichmüller surface associated to a reduced origami.
-Coding the geodesic flow on –
Let be the following graph: the set of its vertices is
and its arrows are
where , , , (resp. ) if (resp. ), and
Notice that this graph has finitely many vertices but countably many arrows. Using this graph, we can code irrational orbits of the flow on as follows. Given , let be the standard lattice and put . Also, let us denote .
By Proposition 2, there exists an unique such that , satisfying the conditions of the proposition (here, is the canonical basis of ). Denote by the type (top or bottom) of the basis of . We assign to the vertex .
For sake of concreteness, let’s assume that (top case). Recalling the notations introduced after the proof of Proposition 2, we notice that the lattice associated to has a basis of bottom type
In other words, starting from the vertex associated to the initial point , after running the geodesic flow for a time , we end up with the vertex where . Equivalently, the piece of trajectory from to is coded by the arrow
Evidently, we can iterate this procedure (by replacing by ) in order to code the entire orbit by a succession of arrows. However, this coding has the “inconvenient” (with respect to the setting of Avila-Viana simplicity criterion) that it is not associated to a complete shift but only a subshift (as we do not have the right to concatenate two arrows and unless the endpoint of coincides with the start of ).
Fortunately, this little difficulty is easy to overcome: in order to get a coding by a complete shift, it suffices to fix a vertex and consider exclusively concatenations of loops based at . Of course, we pay a price here: since there may be some orbits of whose coding is not a concatenation of loops based on , we’re throwing away some orbits in this new way of coding. But, it is not hard to see that the (unique, Haar) -invariant probability on gives zero weight to the orbits that we’re throwing away, so that this new coding still captures most orbits of (from the point of view of ). In any case, this allows to code by a complete shift whose (countable) alphabet is constituted of (minimal) loops based at .
Once we know how to code our flow by a complete shift, the next natural step (in view of Avila-Viana criterion) is the verification of the bounded distortion condition of the invariant measure induced by on the complete shift. This is the content of the next section.
-Verification of the bounded distortion condition-
As we saw above, the coding of the geodesic flow (and modulo the stable manifolds, that is, the “-coordinates” [vertical coordinates]) is the dynamical system
given by where is the Gauss map and with . In this language, becomes (up to normalization) the Gauss measure on each copy , , of the unit interval .
Now, for sake of concreteness, let us fix a vertex of top type. Given a loop based on , i.e., a word on the letters of the alphabet of the coding leading to a complete shift, we denote by the interval corresponding to , that is, the interval consisting of such that the concatenation of loops (based at ) coding the orbit of starts by the word .
In this setting, the measure induced by on the complete shift is easy to express: by definition, the measure of the cylinder corresponding to concatenations of loops (based at ) starting by is the Gauss measure of the interval up to normalization. Because the Gauss measure is equivalent to the Lebesgue measure (as its density satisfies in ), we conclude that the measure of is equal to
up to a multiplicative constant.
In resume, this reduces the bounded distortion condition to the problem of understanding the interval . Here, by the usual properties of the continued fraction, it is not hard to show that is a Farey interval
being t-reduced, i.e., .
Consequently, from this description, we recover the classical fact that
Given , and denoting by , resp. , resp. the matrices associated to , resp. , resp. , it is not hard to check that
so that . Because these matrices are t-reduced, we have that
Once we know that the basis dynamics (Teichmüller geodesic flow on ) is coded by a complete shift equipped with a probability measure with bounded distortion, we can pass to the study of the Kontsevich-Zorich cocycle in terms of the coding.
-Cocycle over the complete shift induced by –
Let be an arrow of and denote by an affine map of derivative , resp. . Of course, is only well-defined up to automorphisms of and/or . In terms of translation structures, given and a translation structure on , the identity map is an affine map of derivative .
Given a path in obtained by concatenation , and starting at and ending at , one has, by functoriality, an affine map given by .
Suppose now that is a loop based at . Then, by definition, the derivative . For our subsequent discussions, an important question is: what matrices of can be obtained in this way?
In this direction, we recall the following definition (already encountered in the previous section):
Definition 3 We say that is
- t-reduced if ;
- b-reduced if .
Observe that the product of two t-reduced (resp. b-reduced) matrices is also t-reduced (resp. b-reduced), i.e., these conditions are stable by products.
The following statement is the answer to the question above:
Corollary 4 The matrices associated to the loops based at the vertex are precisely the c-reduced matrices of .
Indeed, this is a corollary to the next proposition:
Furthermore, the decomposition above is unique.
Of course, one has similar statements for b-reduced matrices (by conjugation by the matrix ).
Actually, this proposition follows from the following slightly more general fact:
with for all , and if
Assuming the validity of Proposition 6, we can derive Proposition 5 as follows. As one can easily check, it suffices to rule out the possibilities or in the decomposition above. We treat only the case as is analogous. If , we would have
with (as ), that is, is not t-reduced.
Concerning the proof of Proposition 6, while it is not difficult (essentially an “Euclidean division algorithm”-like argument), we prefer to omit it in order to present the following related proposition:
Proposition 7 is conjugated (in ) to a t-reduced matrix if and only if its trace .
We close this post with the proof of this proposition.
has trace whenever , we have that if is conjugated to a t-reduced matrix, then, by Proposition 5, .
Conversely, given with , its eigenvalues satisfy . Let be a direct normalized basis with , .
There exists such that , with , . Geometrically, these conditions correspond to the following picture:
In this situation, the matrix has nonnegative coefficients. By Proposition 6, we have the following possibilities:
- (c) ,
- (d) ,
Evidently, the proof is complete in the cases (a) and (b). Also, the cases (c) and (d) are similar, so that the argument is finished once we treat (c): in this situation, we observe that
is conjugated to
a t-reduced matrix (by Proposition 5).