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 1This 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 2The 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 1Let 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 3Actually, 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.

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