Last Wednesday, Jean-Christophe Yoccoz gave a talk (in French) entitled “Regularité holdérienne des solutions de l’équation cohomologique pour les échanges d’intervalle”.
This was the second talk of a new “flat surfaces” seminar organised by himself, Anton Zorich and myself at Instut Henri Poincare (IHP) in Paris. The details about this seminar (such as current schedule, previous and next talks, abstracts, etc.) can be found at this website here.
For the time being, this seminar is an experiment in the sense that IHP allows us to use their rooms from March to June 2014. Of course, if the experiment is a success (i.e., if it manages to gather a non-trivial number of participants interested in flat surfaces and Teichmueller dynamics), then we plan to continue it.
Below the fold, I will reproduce my notes of Jean-Christophe’s talk about a new result together with Stefano Marmi on the cohomological equation for interval exchange transformations of restricted Roth type. Logically, it goes without saying that any errors/mistakes are my entire responsibility.
A classical method to study the properties of (“quasi-periodic”) dynamical system consists into finding an adequate linearization, i.e., one seeks a (“smooth”) change of coordinates so that the new dynamical system is “linear”/“algebraic” in some sense (e.g., a rigid rotation on a circle, a translation on a torus, etc.).
Of course, given and a “good candidate” for a linear model of , the problem of finding is non-linear (because the conjugation equation is non-linear in ). For this reason, it is often the case that before attacking the conjugation equation one studies the following linear version
called cohomological equation for (where is given and we want to solve for ). In fact, the relationship between the cohomological equation and the conjugation equation was already discussed in this blog (see, e.g., this post), where we emphasized Herman’s Schwartzian derivative trick to convert solutions of the cohomological equation into solutions of the conjugation equation in the context of circle diffeomorphisms.
Today, we will discuss exclusively the existence and regularity of solutions of the cohomological equation for interval exchange transformations (but we will not study the conjugation equation).
In order to motivate the main results in this post, let us recall some of the known theorems about the existence and regularity of solutions of the cohomological equation for rotations on the circle of angle (or, equivalently, an interval exchange transformation of two intervals of lengths and ).
Definition 1 We say that an irrational number is of Roth type whenever for all there exists such that
for all . Here, means the distance to the closest integer.
Remark 1 The nomenclature “Roth type” is motivated by Roth’s theorem stating that any irrational algebraic integer is of Roth type.
Proposition 2 (Russmann, Herman, …) Let be of Roth type. Given and (i.e., is a function on with zero mean), there exists a solution of the cohomological equation
for the rotation of angle on with the property that for all .
In other terms, this result says that we can solve the cohomological equation for circle rotations of Roth type with a loss of ()-derivative for all .
Remark 2 The analog of this result in the Sobolev scale (i.e., when and belong to standard Sobolev spaces ) follows from an elementary Fourier analysis (cf. this post). On the other hand, the statement above (in Hölder scale ) requires some extra work, but it is still within the framework of Harmonic Analysis in the sense that one uses Littlewood-Paley decomposition and interpolation inequalities (cf. Herman’s article for more details).
For the sake of comparison, let us give the following statement (where the boldface terms will introduced later):
for every .
Of course, since a rotation of the circle is an interval exchange transformation in two intervals, the theorem of Marmi-Yoccoz extends the previous proposition (of Russmann, Herman, …) to a larger important class of quasi-periodic dynamical systems.
Remark 3 It is possible to check that a circle rotation of Roth type is an interval exchange transformation of two intervals of restricted Roth type. Furthermore, in this particular context one also can show that .
Remark 4 The theorem of Marmi-Yoccoz applies to almost every interval exchange transformation: in fact, the restricted Roth type condition has full measure in the space of interval exchange transformations (with respect to the natural Lebesgue measure obtained by parametrizing interval exchange transformations with fixed combinatorics via the lengths of the permuted intervals).
On the other hand, the loss of derivative in Marmi-Yoccoz theorem is not very good compared to the previous proposition: in fact, the quantity depends on and the definition of restricted Roth type in a highly non-trivial way, so that is usually very small and, a fortiori, is not close (in general) to the optimal loss in the previous proposition.
Remark 5 Here, Jean-Christophe said that it is likely that the definition of restriction Roth type must be changed if one desires an optimal loss of derivatives.
Before explaining the terms in boldface in Marmi-Yoccoz theorem, let us recall some previous related results on cohomological equations for interval exchange transformations and translation flows (the“continuous time analogs” of interval exchange transformations).
First, Forni considered in these two papers here (1997) and here (2007) the cohomological equation for translation flows on translation surfaces (i.e., Forni studied the “continuous time analog” of the cohomological equation for interval exchange transformations). Using several tools from Harmonic Analysis (including weighted Sobolev spaces) on compact surfaces, Forni managed to construct solutions of the cohomological equation for “almost all” (choice of direction for) translation flows in any given translation surface with an optimal loss of derivative (in a weighted Sobolev scale).
Secondly, Marmi, Moussa and Yoccoz considered in these two papers here (2005) and here (2012) showed the existence of continuous solutions of the cohomological equation for interval exchange transformations of restricted Roth type. In particular, the new result of Marmi-Yoccoz improves these previous results by asserting that the existence of Hölder continuous () solutions for the cohomological equations studied in their previous papers with Moussa.
As the reader can see, the results of Forni and Marmi-Moussa-Yoccoz have both strong and weak points. On one hand, Forni’s result gives solutions to the cohomological equation for “almost all” translation flows with optimal loss of derivative in Sobolev scale, but the “Diophantine condition” (i.e., the subset of full measure of translation flows) in his theorem is not explicit. On the other hand, the results of Marmi-Moussa-Yoccoz result and Marmi-Yoccoz give solutions to the cohomological equation for interval exchange transformations with a poor gain of regularity in Hölder scale, but their “Diophantine condition” (restricted Roth type) on the interval exchange transformation is “relatively explicit”. Also, it is not easy to compare “directly” these results: even though there is a “natural” notion of restricted Roth type translation flow (in the sense that the return map of the translation flow to an appropriate transverse section is an interval exchange transformation of restricted Roth type) in this paper of Marmi-Moussa-Yoccoz, it is not clear that a restricted Roth type translation flow fits the “Diophantine condition” of Forni.
Remark 6 Very roughly speaking, one of the (several) difficulties in relating the Diophantine conditions of Forni and Marmi-Moussa-Yoccoz is related to the application of Oseledets theorem for the Kontsevich-Zorich cocycle: indeed, Oseldets theorem provides a non-explicit set of full measure of points such that the Kontsevich-Zorich cocycle along the Teichmüller flow orbit of these points have a particularly nice behavior (see, e.g., the introduction of this paper of Forni for more explanations). Nevertheless, it is worth to point out that the recent results of Chaika-Eskin give some hope towards relating the Diophantine conditions of Forni and Marmi-Moussa-Yoccoz.
After these comments on Theorem 3 (and some related results), it is time to define the objects involved in the statement of this theorem.
2. Interval exchange transformations
Recall that an interval exchange transformation is determined by the following data. Given a finite alphabet with letters, an interval and two partitions into subintervals with
for every , the interval exchange transformation is the piecewise translation sending to . Here, , resp. stands for top, resp. bottom subintervals, that is, the subintervals of the partition one sees before, resp. after applying . The figure below gives some examples of interval exchange transformations.
We denote by , resp. , the extremities of the subintervals , resp. (), so that and are the extremities of the interval . In particular, , resp. , are the discontinuities of , resp. .
Using these notations, we are ready to introduce the first term marked boldface in Theorem 3:
and is the subspace of zero mean functions in . In concrete terms, is the space of piecewise -functions on that are on the intervals admitting natural extensions to the intervals (but these extensions might disagree at the points ‘s).
Next, we introduce the constants and attached to an interval exchange transformation . An interval exchange transformation can be naturally seen (in many ways) as the first return map of a translation flow on a translation surface (by means of Masur’s suspension construction or Veech’s zippered rectangles construction): the reader can find more details in this post here (for instance). The translations surfaces obtained from in this way have a genus and a number of conical singularities depending only on . Alternatively, one can define and by combinatorial means (in terms of the cycles of the permutation on induced by the way permutes the subintervals and ).
At this point, the sole undefined term in boldface in the statement of Theorem 3 is “restricted Roth type”. In order to do so, we have to introduce the Rauzy-Veech algorithm and the (discrete version of the) Kontsevich-Zorich cocycle (using this survey of Jean-Christophe as a basic reference).
3. Rauzy-Veech algorithm, Kontsevich-Zorich cocycle and restricted Roth type
We say that an interval exchange transformation has a connection if there are , such that
Since , resp. , is a discontinuity of , resp. (so that the future orbit of , resp. past orbit of is ill-defined), we see that has a connection whenever it has an orbit that is “blocked” (can not be extended) in the future and in the past.
An interval exchange transformation without connections are very similar to irrational rotations of the circle: by a result of Keane, any without connections has a minimal dynamics.
Starting with without connections, we denote by the first return map of to the subinterval
It is not hard to check that is also an interval exchange transformation permuting a finite collection , of subintervals of naturally indexed by the alphabet . Furthermore, also has no connections. In the literature, the map is called an elementary step of the Rauzy-Veech algorithm.
Of course, the two facts described in the previous paragraph imply that we can iterate this procedure: starting with without connections, by successively applying the elementary steps of the Rauzy-Veech algorithm, one obtains a sequence of interval exchange transformations acting on a decreasing sequence of subintervals . Moreover, it is possible to show that the lengths of the intervals tend to zero as .
For later use, let us observe that, by definition, for any , is the first return map of to .
In terms of the Rauzy-Veech algorithm, the Kontsevich-Zorich cocycle can be described as follows. Given and , we consider the special Birkhoff sum
where is the first return time of (under iterates).
It is possible to check that . In particular, denoting by
we see that
is a linear operator inducing a matrix whose entries have the following dynamical interpretation: is the number of visits of to under -iterates before its return to . The matrices form a linear cocycle (i.e., ) called (discrete) Kontsevich-Zorich cocycle.
The restricted Roth type for an interval exchange transformation is defined in terms of the features of the Kontsevich-Zorich cocycle .
More precisely, we define inductively and is the smallest integer such that for all . It is possible to show that this definition leads to a sequence with as .
We say that has restricted Roth type whenever the following four conditions are fulfilled.
- (a) Roth type condition: for each , one has
for all .
Remark 7 The fact that the Roth type condition is satisfied for almost all interval exchange transformations (i.e., for Lebesgue almost all choices of lengths of the intervals ) was checked in this paper of Marmi-Moussa-Yoccoz (see also the paper of Avila-Gouezel-Yoccoz).
Remark 8 For sake of comparison, in the case of the rotation of angle on the circle (i.e., interval exchange transformation permuting two intervals of lengths and ), one can check that and where are the entries of the continued fraction expansion of and are the denominators of the best rational approximations of . In particular, the Roth type condition is equivalent to
for all , i.e., is of Roth type.
- (b) Spectral gap: there exists such that
where is the subspace of functions with zero mean.
Remark 9 The spectral gap property is also satisfied by almost all interval exchange transformations thanks to the work of Veech. In fact, this property is closely related to the non-uniform hyperbolicity of the Teichmueller flow (and the constant is the second Lyapunov exponent of the Kontsevich-Zorich cocycle over the Teichmueller flow).
- (d) Hyperbolicity: the stable space
of the Kontsevich-Zorich cocycle has dimension .
Remark 10 The hyperbolicity property is verified for almost all interval exchange transformations thanks to the work of Forni.
- (c) Coherence property: denoting by the restriction of the Kontsevich-Zorich cocycle to the stable space , and by the action of the Kontsevich-Zorich cocycle on the “center-unstable spaces” , then for each , one has
Remark 11 The coherence property is also verified for almost all interval exchange transformations: indeed, this is a consequence of Oseledets theorem applied to the Kontsevich-Zorich cocycle (see, e.g., Marmi-Moussa-Yoccoz paper for more details).
Remark 12 We called “item (d)” the hyperbolicity property and “item (c)” the coherence property just to keep the same notations of this paper of Marmi-Moussa-Yoccoz (and also because Jean-Christophe did the same during the talk 🙂 ).
At this point, all boldfaced terms in Theorem 3 were defined and now it is time to discuss some points of the proof of this result.
4. Some steps of the proof of Theorem 3
Recall that the quantity is the number of marked points of a translation surface obtained by suspension of the interval exchange transformation . Combinatorially, these marked points can be seen as cycles of a permutation on keeping track of the ‘s one sees when turning counterclockwise around the conical singularities in the translation surface . (See, e.g., this survey of Jean-Christophe for more details)
The permutation allows to define a boundary operator ,
where is the set of cycles of , and , resp. , means , resp. .
It is not hard to see that the boundary operator has the following properties:
- the restriction of to is the usual boundary operator in relative homology
after appropriate (natural) identifications and .
In this setting, Marmi-Yoccoz deduce Theorem 3 as an immediate consequence of the following more precise statement:
Theorem 4 (Marmi-Yoccoz) Let be an interval exchange transformation of restricted Roth type.Denote by the kernel of the (restriction of the) boundary operator (to ) and consider an arbitrary supplement of in (i.e., ).
Then, there are two bounded operators and such that
where and .
Remark 13 This theorem says that we can solve the cohomological equation whenever and (i.e., there is no obstruction coming from the boundary operator and the operator ). In the literature, these conditions (or “obstructions”) are called Forni’s distributions.
Closing today’s post, let us give some steps of the proof of Theorem 4.
The first three steps are the following:
- (1) there exists such that for all with (zero mean) one has
- (2) there exists and a bounded operator such that for all one has
- (3) by Gottschalk-Hedlund theorem applied to the homeomorphism of the compact space obtained after blowing up the orbits of the discontinuities of and (and by keeping track of what happens once one undo the blowup), one can use the previous steps to write for some continuous function .
In fact, these three steps were performed in this paper of Marmi-Moussa-Yoccoz from 2012 (where they are explained in details).
Remark 14 As it turns out, the “non-optimal loss of derivatives” in Marmi-Yoccoz is already present here: in fact, it seems that for an optimal loss of derivatives for the solutions of the cohomological equation in the setting of Marmi-Yoccoz one has to improve the information on the constant appearing in the first step above.
At this stage, it is clear that the proof of Theorem 4 is reduced to show that the function appearing in Step 3 above is Hölder continuous. Here, the main novelty introduced by Marmi-Yoccoz is the following fourth step:
- (4) Given an interval , denote by ,
for each , and
Then, there exists such that one has the following “almost recurrence relation”
for the sequence of vectors .
At this point, Jean-Christophe was running out of time so that he decided to skip the proof of this almost recurrence relation (based, of course, on the definition of restricted Roth type) in order to explain how this new step allows to conclude Theorem 4.
First, one uses the fact that is continuous to check that (this is not difficult since we are not asking for moduli of continuity/rate of convergence).
Finally, the proof of the estimate (1) itself consists into three steps.
One constructs first a vector where is the intersection of with the kernel of and is a natural Kontsevich-Zorich cocycle invariant symplectic form on (related to the intersection form on ).
After this, Marmi-Yoccoz introduce the vector .
Then, Marmi-Yoccoz use the coherence property (c) in the definition of restricted Roth type (among several other facts) to show the analog of (1) for . Moreover, they can check that this estimate on can be transferred to . Furthermore, by exploiting the symplecticity of the Kontsevich-Zorich cocycle (among several other facts), Marmi-Yoccoz show the analog of (1) for the vectors imply the estimate (1) for the vectors , so that the argument is complete.