# Disquisitiones Mathematicae

### Serrapilheira Postdoctoral Fellowship at Universidade Federal do Ceará (Brazil)

Yuri Lima asked me to announce in this blog the following opportunity for a post-doctoral position in Dynamical Systems and Ergodic Theory at Universidade Federal do Ceará (located in Fortaleza, Brazil):

**Serrapilheira Postdoctoral Fellowship – UFC**

The Department of Mathematics at Universidade Federal do Ceará (UFC) invites applications for a Serrapilheira Postdoctoral Fellowship in dynamical systems and ergodic theory. The position is for one year with start date at any moment between March 2020 and September 2020, with possibility of extension for another year.

**Qualifications and expectations**

The position is part of the project “Jangada Dinâmica – boosting dynamical systems in Brazil’s Northeastern region”, which is funded by Instituto Serrapilheira and aims to boost dynamical systems and ergodic theory in the mathematical community of universities located in the Northeastern region of Brazil. The applicant must have completed a PhD and be qualified for conducting research in either dynamical systems and/or ergodic theory. There are NO teaching duties. As part of the program, and to foster interaction, the fellow shall visit another department of Mathematics in the Northeast for one month each semester or two months per year. Applications from underrepresented groups in Mathematics are highly encouraged.

**Salary**

The salary will range from 5000–6000 Brazilian Reais monthly, tax free, in a twelve month-base calendar, according to the applicant’s qualifications. There will be an extra 5000 Brazilian Reais for each of the two months of visits to another institution in the Northeast. The salary is more attractive than those offered by regular Brazilian funding agencies.

**Department of Mathematics at UFC**

The Department of Mathematics at UFC currently holds the highest rank among Brazilian Mathematics departments. Having a strong history in the field of differential geometry, during the last 15 years it has developed new research groups in analysis, graph optimization and, more recently, in dynamical systems. Currently, the group of dynamical systems has two members, with expertise on nonuniform hyperbolicity, partial hyperbolicity, and symbolic dynamics.

**Location**

UFC is located in the city of Fortaleza, which has approximately 2.5 million inhabitants and is the fifth largest city of Brazil. Located in the Northeastern region of Brazil, Fortaleza is becoming a common port of entry to the country, with many direct flights to the US and Europe. Historically known for touristic reasons, it is nearby beaches with warm water and white sand dunes, and its cost of living is cheaper than bigger cities like Rio de Janeiro and São Paulo, thus making the monthly stipend affordable.

**Documentation required**

– CV with publication list

– Research statement

– Two (or more) letters of recommendation.

All documents must be sent to jangadadinamica@gmail.com. The applicant must send the first two documents, and ask two (or more) professors to directly send their letters of recommendation.

**Deadline:** December 31, 2019.

**More information:** jangadadinamica@gmail.com.

### Wolpert’s examples of tiny Weil-Petersson sectional curvatures revisited

In this previous post here (from 2018), I described some “back of the envelope calculations” (based on private conversations with Scott Wolpert) indicating that some sectional curvatures of the Weil–Petersson (WP) metric could be at least exponentially small in terms of the distance to the boundary divisor of Deligne–Mumford compactification.

Very roughly speaking, this heuristic computation went as follows: the WP sectional curvature of any -plane can be written as the sum of three terms; for the -planes considered in the previous post, the main term among those three *seemed* to be a kind of -norm of Beltrami differentials with essentially disjoint supports; finally, this -type norm was shown to be really small once a certain Green propagator is *ignored*.

Last April 2019, I met Scott during an event at Simons Center for Geometry and Physics, and I took the opportunity to tell him that one could perhaps show that the measure of the set of -planes leading to tiny WP curvatures is very small using the real-analyticity of the WP metric.

More concretely, my idea was very simple: since the Grassmannian of -planes tangent to a point is a compact space, the WP sectional curvature defines a real-analytic function , and we dispose of good upper bounds for and all of its derivatives in terms of the distance of to the boundary (see this article here), we can hope to get reasonable estimates for the measure of the sets using the techniques of these articles here and here (which are close in spirit to the classical fact [explained in Lemma 3.2 of Kleinbock–Margulis paper, for instance] that the measure of the sets are small whenever is a polynomial function on whose degree and -norm are bounded).

As it turns out, Scott thought that this strategy made some sense and, in particular, he promised to use my suggestion as a motivation to review his arguments concerning WP sectional curvatures.

After several email exchanges with Howard Masur and I, Scott announced that there were some *mistakes* in the construction of tiny WP sectional curvature: in a nutshell, one should not restrict the analysis to a single “main term” in the formula for WP sectional curvatures as a sum of three expressions, and one can not ignore the effect of the Green propagator. More importantly, Scott made a detailed study of these mistakes which ultimately led him to establish polynomial upper bounds for WP sectional curvatures at the heart of his newest preprint available here.

In this post, we will follow closely Scott’s preprint in order to give an outline of the proof of a polynomial upper bound for WP sectional curvatures:

**Theorem 1 (Wolpert)** *Given two integers and with , there exists a constant with the following property.*If denotes the product of the lengths of the short geodesics of a hyperbolic surface of genus with cusps whose systole is sufficiently small, then the sectional curvatures of the Weil-Petersson metric at are at most

**Remark 1** *As it was pointed out by Scott in his preprint, it is likely that this estimate is not optimal: indeed, one expects that the best exponent should be rather than .*

In what follows, we’ll assume some familiarity with some basic aspects of the geometry of the Weil–Petersson metric (such as those described in these posts here and here).

**1. Weil–Petersson sectional curvatures**

Let be a hyperbolic surface of genus with . If we write , where is the usual hyperbolic plane and is a group of isometries of describing the fundamental group of , then the holomorphic *tangent* space at to the moduli space of Riemann surfaces of genus with punctures is naturally identified with the space of harmonic Beltrami differentials on (and the *cotangent* space is related to quadratic differentials).

In this setting, the Weil–Petersson metric is the Riemannian metric induced by the Hermitian inner product

where and is the hyperbolic area form on .

**Remark 2** *Note that is well-defined: if and are Beltrami differentials, then is a function on .*

The Riemann tensor of the Weil–Petersson metric was computed by Wolpert in 1986:

where and is an operator related to the Laplace–Beltrami operator on .

**Remark 3** *Our choice of notation here differs from Wolpert’s preprint! Indeed, he denotes the Laplace–Beltrami operator by and he writes .*

The Riemann tensor gives access to nice formulas for the sectional curvatures thanks to the work of Bochner. More concretely, given and span a -plane in the real tangent space to at , let us take Beltrami differentials and such that , , and is orthonormal. Then, Bochner showed that the sectional curvature of is

Hence, by Wolpert’s formula for the Riemann tensor of the WP metric, we see that

**2. Spectral theory of **

Wolpert’s formula for the Riemann tensor of the WP metric hints that the spectral theory of plays an important role in the study of the WP sectional curvatures.

For this reason, let us review some key properties of (and we refer to Section 3 of Wolpert’s preprint for more details and references). First, is a positive operator on whose norm is : these facts follow by integration by parts. Secondly, is essentially self-adjoint on , so that is self-adjoint on . Moreover, the maximum principle permits to show that is also a positive operator on with unit norm. Finally, has a positive symmetric integral kernel: indeed,

where the Green propagator is the Poincaré series

associated to an appropriate Legendre function . (Here, stands for the hyperbolic distance on .) For later reference, we recall that has a logarithmic singularity at and whenever is large.

**3. Negativity of the WP sectional curvatures**

Interestingly enough, as it was first noticed by Wolpert in 1986, the spectral features of described above are sufficient to derive the negativity of WP sectional curvatures from Cauchy-Schwarz inequality. More precisely, since is self-adjoint, i.e.,

and its integral kernel is a *real* function, a straightforward computation reveals that the equation (1) for the sectional curvature of a -plane can be rewritten as

If we decompose the function into its real and imaginary parts, say , then we see that

Since is a positive operator, we conclude that and, *a fortiori*,

The non-positivity of the right-hand side of (2) can be established in three steps. First, the positivity of also implies that

Secondly, the fact that has a positive integral kernel allows to apply the Cauchy–Schwarz inequality to get that . Therefore,

Finally, the Cauchy–Schwarz inequality also says that

In particular, , so that it follows from (2) that all sectional curvatures of the WP metric are non-positive, i.e., .

Actually, it is not hard to derive that at this stage: indeed, would force a case of equality in Cauchy-Schwarz inequality and this is not possible in our context because is orthonormal.

**Remark 4** *Philosophically speaking, the “analog” to this argument in the realm of Teichmüller dynamics is Forni’s proof of the spectral gap property for the Lyapunov exponents of the Teichmüller geodesic flow. In fact, after some computations with variational formulas for the so-called Hodge norm, Forni establishes that by ruling out an equality case in a certain Cauchy-Schwarz estimate.*

**4. Reduction of Theorem 1 to bounds on ‘s kernel**

The discussion in the previous section says that small WP sectional curvatures correspond to almost equalities in certain Cauchy-Schwarz inequalities.

Hence, a natural strategy towards the proof of Theorem 1 consists into showing that an almost equality in (3) is impossible. In this direction, Wolpert establishes the following result:

**Theorem 2 (Wolpert)** *There are two constants and with the following property. If we have an almost equality*

*between the terms and in (3), then and can not be almost equal:*

Of course, Theorem 1 is an immediate consequence of Theorem 2 (in view of (2) and the estimate [implied by (3)]).

Thus, it remains only to prove Theorem 2. For this sake, we need further spectral information on , namely, some lower bounds on its the kernel . In order to illustrate this point, let us now show Theorem 2 *assuming* the following statement.

**Proposition 3** *There exists a constant such that*

*whenever and do not belong to the cusp region of .*

**Remark 5** *We recall that the cusp region of is a finite union of portions of which are isometric to a punctured disk (equipped with the hyperbolic metric ).*

For the sake of exposition, let us first establish Theorem 2 when is compact, i.e., , before explaining the extra ingredient needed to treat the general case.

**4.1. Proof of Theorem 2 modulo Proposition 3 when **

Suppose that for a constant to be chosen later. In this regime, our goal is to show that is “big” and is “small”, so that is necessarily “big”.

We start by quickly showing that is “big”. Since and are unitary tangent vectors, it follows from Proposition 3 that

Let us now focus on proving that is “small”. Since (cf. (3)), if we write (where and are the positive and negative parts of the real part of , then we obtain that

Since is positive, we derive that . Thus, if is compact, i.e., , then Proposition 3 says that for all . It follows that

By orthogonality of , we have that , i.e., . By plugging this information into the previous inequality, we obtain the estimate

Next, we observe that (cf. (3)) in order to obtain that

On the other hand, Proposition 3 ensures that for . Since and are unitary tangent vectors, one has for . By inserting this inequality into the previous estimate, we derive that

whenever has a sufficiently small systole.

This bound on can be converted into a bound thanks to Cauchy integral formula. More concrentely, as it is explained in Section 2 of Wolpert’s preprint, after observing that and replacing Beltrami differentials and by the dual objects and (namely, quadratic differentials), we are led to study quartic differentials . By Cauchy integral formula on , one has

On the other hand, if has systole and the cusp region is empty, then the injectivity radius at any is . Thus, there exists an universal constant such that

for all . By plugging this inequality into (7), we conclude that

for all .

Since is a positive operator on with unit norm (cf. Section 2 above) and , we have that the previous inequality implies the following bound on :

for all . By combining this estimate with (7), we conclude that

In summary, (4) and (8) imply that

for the choice of constant . This proves Theorem 2 in the absence of cusp regions.

**4.2. Proof of Theorem 2 modulo Proposition 3 when **

The arguments above for the case also work in the case because the cusp regions carry only a tiny fraction of the mass of the relevant functions, Beltrami differentials, etc.

More precisely, as it is explained in Section 2 of Wolpert’s preprint, if the constant is chosen correctly, then the Cauchy integral formula and the Schwarz lemma can be used to prove that

for all holomorphic quartic differentials .

In particular, we do not lose too much information after truncating , , etc. to and this allows us to repeat the arguments of the case to the corresponding truncated objects , , etc. without any extra difficulty: see Section 5 of Wolpert’s preprint for more details.

**5. Proof of Proposition 3**

Closing this post, let us give an idea of the proof of Proposition 3 (and we refer the reader to Section 4 of Wolpert’s preprint for more details).

Since and (cf. Section 2 above), our task is reduced to give lower bounds on the Poincaré series

For this sake, let us first recall that a hyperbolic surface has thick-thin decomposition: the thick portion is the region where the injectivity radius is bounded away from zero by a uniform constant and the thin portion is the complement of the thick region. Geometrically, the thin region is the disjoint union of the cusp region and a finite number of *collars* around simple closed short geodesics: roughly speaking, a collar consisting of the points at distance of a short simple closed geodesic of length .

We can provide lower bounds on in terms of the behaviours of simple geodesic arcs connecting and on .

More concretely, let be the shortest geodesic connecting and . Since is simple, we have that, for certain adequate choices of the constants defining the collars, one has that can not “back track” after entering a collar, i.e., it must connect the boundaries (rather than going out via the same boundary component). Furthermore, can not go very high into a cusp. Thus, if we decompose according to its visits to the thick region, the collars and the cusps, then the fact that permits to check that it suffices to study the passages of through collars in order to get a lower bound on .

Next, if is a subarc of crossing a collar around a short closed geodesic , then we can apply Dehn twists to to get a family of simple arcs indexed by giving a “contribution” to of

for some constant depending only on the topology of . In this way, the desired result follows by putting all “contributions” together.

### On the low regularity conjugacy classes of self-similar interval exchange transformations of the Eierlegende Wollmilchsau and Ornithorynque

The celebrated works of several mathematicians (including Poincaré, Denjoy, …, Arnold, Herman, Yoccoz, …) provide a very satisfactory picture of the dynamics of smooth circle diffeomorphisms:

- each -diffeomorphism of the circle has a well-defined rotation number (which can be defined using the cyclic order of its orbits, for instance);
- is
*topologically semi-conjugated*to the rigid rotation (i.e., for a surjective continuous map ) whenever its rotation number is irrational; - if has irrational rotation number , then is
*topologically conjugated*to (i.e., there is an*homeomorphism*such that ); - if , , has an irrational rotation number satisfying a
*Diophantine condition*of the form for some , , and all , then there exists conjugating and (i.e., ); - etc.

In particular, if has *Roth type* (i.e., for all , there exists such that for all ), then any with rotation number is conjugated to whenever . (The nomenclature is motivated by Roth’s theorem saying that any irrational algebraic number has Roth type, and it is well-known that the set of Roth type numbers has full Lebesgue measure in .)

In the last twenty years, many authors gave important contributions towards the extension of this beautiful theory.

In this direction, a particularly successful line of research consists into thinking of circle rotations as standard interval exchange transformations on 2 intervals and trying to build smooth conjugations between generalized interval exchange transformations (g.i.e.t.) and standard interval exchange transformations. In fact, Marmi–Moussa–Yoccoz studied the notion of standard i.e.t. of *restricted Roth type* (a concept designed so that the circle rotation has restricted Roth type [when viewed as an i.e.t. on 2 intervals] if and only if has Roth type) and proved that, for any , the g.i.e.t.s close to a standard i.e.t. of restricted Roth type such that is -conjugated to form a -submanifold of codimension where is the first return map to an interval transverse to a translation flow on a translation surface of genus and is an i.e.t. on intervals.

An interesting consequence of this result of Marmi–Moussa–Yoccoz is the fact that local conjugacy classes behave differently for circle rotations and arbitrary i.e.t.s. Indeed, a circle rotation is an i.e.t. on 2 intervals associated to the first return map of a translation flow on the torus , so that has genus and also . Hence, Marmi–Moussa–Yoccoz theorem says that its local conjugacy class of with of Roth type has codimension *regardless* of the differentiability scale . Of course, this fact was previously known from the theory of circle diffeomorphisms: by the results of Herman and Yoccoz, the sole *obstruction* to obtain a smooth conjugation between and (with of Roth type) is described by a single parameter, namely, the rotation number of . On the other hand, Marmi–Moussa–Yoccoz theorem says that the codimension

of the local conjugacy class of an i.e.t. of restricted Roth type with genus *grows* linearly with the differentiability scale .

**Remark 1** *This indicates that KAM theoretical approaches to the study of the dynamics of g.i.e.t.s might be delicate because the “loss of regularity” in the usual KAM schemes forces the analysis of cohomological equations (linearized versions of the conjugacy problem) in several differentiability scales and Marmi–Moussa–Yoccoz theorem says that these changes of differentiabilty scale produce non-trivial effects on the numbers of obstructions (“codimensions”) to solve cohomological equations.*

In any case, this interesting phenomenon concerning the codimension of local conjugacy classes of i.e.t.s of genus led Marmi–Moussa–Yoccoz to make a series of conjectures (cf. Section 1.2 of their paper) in order to further compare the local conjugacy classes of circle rotations and i.e.t.s of genus .

Among these fascinating conjectures, the second open problem in Section 1.2 of Marmi–Moussa–Yoccoz paper asks whether, for *almost all* i.e.t.s , any g.i.e.t. with trivial conjugacy invariants (e.g., “simple deformations”) and conjugated to is also conjugated to . In other words, the and conjugacy classes of a *typical* i.e.t. coincide.

In this short post, I would like to transcript below some remarks made during recent conversations with Pascal Hubert showing that the hypothesis “for *almost all* i.e.t.s ” can *not* be removed from the conjecture above. In a nutshell, we will see in the sequel that the self-similar standard interval exchange transformations associated to two special translation surfaces (called *Eierlegende Wollmilchsau* and *Ornithorynque*) of genera and are but not conjugated to a rich family of piecewise affine interval exchange transformations. Of course, I think that these examples are probably well-known to experts (and Jean-Christophe Yoccoz was probably aware of them by the time Marmi–Moussa–Yoccoz wrote down their conjectures), but I’m including some details of the construction of these examples here mostly for my own benefit.

**Disclaimer:** As usual, even though the content of this post arose from conversations with Pascal, all mistakes/errors in the sequel are my sole responsibility.

**1. Preliminaries**

**1.1. Rauzy–Veech algorithm**

The notion of “irrational rotation number” for generalized interval exchange transformations relies on the so-called *Rauzy–Veech algorithm*.

More concretely, given a -g.i.e.t. sending a finite partition (modulo zero) of into closed subintervals disposed accordingly to a bijection to a finite partition (modulo zero) of into closed subintervals disposed accordingly to a bijection (via -diffeomorphisms ), an elementary step of the *Rauzy–Veech algorithm* produces a new -g.i.e.t. by taking the first return map of to the interval where , resp. whenever , resp. (and is not defined when ).

We say that a -g.i.e.t. has *irrational rotation number* whenever the Rauzy–Veech algorithm can be iterated indefinitely. This nomenclature is partly justified by the fact that Yoccoz generalized the proof of Poincaré’s theorem in order to establish that a -g.i.e.t. with irrational rotation number is *topologically semi-conjugated* to a standard, minimal i.e.t. .

**1.2. Denjoy counterexamples**

Similarly to Denjoy’s theorem in the case of circle diffeomorphisms, the *obstruction* to promote topological semi-conjugations between and as above into -conjugations is the presence of *wandering intervals* for , i.e., non-trivial intervals whose iterates under are pairwise disjoint (i.e., for all , ).

Moreover, as it was also famously established by Denjoy, a little bit of smoothness (e.g., with derivative of bounded variation) suffices to preclude the existence of wandering intervals for circle diffeomorphisms, and, actually, some smoothness is needed because there are several examples of -diffeomorphisms with any prescribed irrational rotation number and possessing wandering intervals. Nevertheless, it was pointed out by several authors (including Camelier–Gutierrez, Bressaud–Hubert–Maas, Marmi–Moussa–Yoccoz, …), a high amount of smoothness is *not* enough to avoid wandering intervals for arbitrary -g.i.e.t.: indeed, there are *many* examples of piecewise *affine* interval exchange transformations possessing wandering intervals.

**Remark 2** *The facts mentioned in the previous two paragraphs partly justifies the nomenclature Denjoy counterexample for a -g.i.e.t. with irrational rotation number possessing wandering intervals.*

In the context of piecewise affine i.e.t.s, the Denjoy counterexamples are also characterized by the behavior of certain *Birkhoff sums*. More concretely, let be a piecewise affine i.e.t. with irrational rotation number, say is semi-conjugated to a standard i.e.t. . By definition, the logarithm of the slope of is constant on the continuity intervals of and, hence, it allows to naturally define a function taking a constant value on each continuity interval of . In this setting, it is possible to prove (see, e.g., the subsection 3.3.2 of Marmi–Moussa–Yoccoz paper) that has wandering intervals if and only if there exists a point with bi-infinite -orbit such that

where the Birkhoff sum at a point with orbit for all is defined as , resp. for , resp. .

For our subsequent purposes, it is worth to record the following interesting (direct) consequence of this “Birkhoff sums” characterization of piecewise affine Denjoy counterexamples:

**Proposition 1** *Let be a piecewise affine i.e.t. topologically semi-conjugated to a standard, minimal i.e.t. . Denote by the piecewise constant function associated to the logarithms of the slopes of .**If for all with bi-infinite -orbit, then is topologically conjugated to (i.e., is not a Denjoy counterexample).*

**1.3. Special Birkhoff sums and the Kontsevich–Zorich cocycle**

An elementary step of the Rauzy–Veech algorithm replaces a standard, minimal i.e.t. on an interval by a standard, minimal i.e.t. given by the first return map of on an appropriate subinterval .

The *special Birkhoff sum* associated to an elementary step is the operator mapping a function to a function , , where stands for the first return time to .

The special Birkhoff sum operator preserves the space of piecewise constant functions in the sense that is constant on each whenever is constant on each . In particular, the restriction of to the space of such piecewise constant functions gives rise to a matrix . The family of matrices obtained from the successive iterates of the Rauzy–Veech algorithm provides a concrete description of the so-called *Kontsevich–Zorich cocycle*.

In summary, the behaviour of special Birkhoff sums (i.e., Birkhoff sums at certain “return” times) of piecewise constant functions is described by the Kontsevich–Zorich cocycle. Therefore, in view of Proposition 1, it is probably not surprising to the reader at this point that the Lyapunov exponents of the Kontsevich–Zorich cocycle will have something to do with the presence or absence of piecewise affine Denjoy counterexamples.

**1.4. Eierlegende Wollmilchsau and Ornithorynque**

The Eierlegende Wollmilchsau and Ornithorynque are two remarkable translation surfaces and of genera and obtained from finite branched covers of the torus . Among their several curious features, we would like to point out that the following fact proved by Jean-Christophe Yoccoz and myself: if is a standard i.e.t. on or intervals (resp.) associated to the first return map of the translation flow in a typical direction on or (resp.), then there are vectors , and a -dimensional vector subspace such that is an equivariant decomposition with respect to the matrices of the Kontsevich–Zorich cocycle with the following properties:

- (a) generates the Oseledets direction of the top Lyapunov exponent ;
- (b) generates the Oseledets direction of the smallest Lyapunov exponent ;
- (c) the matrices of the Kontsevich–Zorich cocycle act on through a
*finite*group.

In the literature, the Lyapunov exponents are usually called the *tautological* exponents of the Kontsevich–Zorich cocycle. In this terminology, the third item above is saying that all non-tautological Lyapunov exponents of the Kontsevich–Zorich associated to and vanish.

In the next two sections, we will see that this curious behaviour of the Kontsevich–Zorich cocycle of or along allows to construct plenty of piecewise affine i.e.t.s which are but not conjugated to standard (and uniquely ergodic) i.e.t.s.

**2. “Il n’y a pas de contre-exemple de Denjoy affine par morceaux issu de et ”**

In this section (whose title is an obvious reference to a famous article by Jean-Christophe Yoccoz), we will see that the Eierlegende Wollmilchsau and Ornithorynque never produce piecewise affine Denjoy counterexamples with irrational rotation number of “bounded type”.

More precisely, let us consider is a piecewise affine i.e.t. topologically semi-conjugated to coming from (the first return map of the translation flow in the direction of a pseudo-Anosov homeomorphism of) or . It is well-known that the piecewise constant function associated to the logarithms of the slopes of belongs to (see, e.g., Section 3.4 of Marmi–Moussa–Yoccoz paper). In order to simplify the exposition, we assume that the “irrational rotation number” has “bounded type”, that is, is self-similar in the sense that some of its iterates under the Rauzy–Veech algorithm actually coincides with up to scaling.

If , then the item (c) from Subsection 1 above implies that all special Birkhoff sums of (in the future and in the past) are bounded. From this fact, we conclude that for all with bi-infinite -orbit: indeed, as it is explained in details in Bressaud–Bufetov–Hubert article, if is self-similar, then the orbits of can be described by a *substitution* on a finite alphabet and this allows to select a bounded subsequence of thanks to the repetition of certain words in the *prefix-suffix decomposition*.

In particular, it follows from Proposition 1 above that there is *no* Denjoy counterexample among the piecewise affine i.e.t.s topologically semi-conjugated to a self-similar standard i.e.t. coming from or such that .

**Remark 3** *Actually, it is possible to explore the fact that is a stable vector (i.e., it generates the Oseledets space of a negative Lyapunov exponent) to remove the constraint “” from the statement of the previous paragraph.*

In other words, we showed that any *always* provides a piecewise affine i.e.t. -conjugated to . Note that this is a relatively rich family of piecewise affine i.e.t.s because is a vector space of dimension , resp. , when is a self-similar standard i.e.t. coming from , resp. .

**3. Cohomological obstructions to conjugations**

Closing this post, we will show that the elements always lead to piecewise affine i.e.t.s which are *not* conjugated to self-similar standard i.e.t.s of or . Of course, this shows that the and conjugacy classes of a self-similar standard i.e.t. of or are distinct and, *a fortiori*, the Marmi–Moussa–Yoccoz conjecture about the coincidence of and conjugacy classes of standard i.e.t.s becomes false if we remove “for almost all standard i.e.t.s” from its statement.

Suppose that is a piecewise affine i.e.t. -conjugated to a self-similar standard i.e.t. of or , say for some -diffeomorphism . By taking derivatives, we get

since is an isometry. Of course, we recognize the slope of on the left-hand side of the previous equation. So, by taking logarithms, we obtain

where is a function. In other terms, is a solution of the cohomological equation and is a -coboundary. Hence, the Birkhoff sums are bounded and, by continuity of , the special Birkhoff sums of converge to zero. Equivalently, belongs to the *weak stable space* of the Kontsevich–Zorich cocycle (compare with Remark 3.9 of Marmi–Moussa–Yoccoz paper).

However, the item (c) from Subsection 1.4 above tells that the Kontsevich–Zorich cocycle acts on through a finite group of matrices and, thus, can *not* converge to zero under the Kontsevich–Zorich cocycle.

This contradiction proves that is not -conjugated to , as desired.