Posted by: matheuscmss | September 3, 2019

## 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, …, ArnoldHermanYoccoz, …) provide a very satisfactory picture of the dynamics of smooth circle diffeomorphisms:

• each ${C^r}$-diffeomorphism ${f\in\textrm{Diff}^r(\mathbb{T})}$ of the circle ${\mathbb{T}:=\mathbb{R}/\mathbb{Z}}$ has a well-defined rotation number ${\alpha=\rho(f)}$ (which can be defined using the cyclic order of its orbits, for instance);
• ${f\in\textrm{Diff}^r(\mathbb{T})}$ is topologically semi-conjugated to the rigid rotation ${R_{\alpha}(x)=x+\alpha}$ (i.e., ${h\circ f=R_{\alpha}\circ h}$ for a surjective continuous map ${h:\mathbb{T}\rightarrow \mathbb{T}}$) whenever its rotation number ${\alpha=\rho(f)}$ is irrational;
• if ${f\in\textrm{Diff}^2(\mathbb{T})}$ has irrational rotation number ${\alpha}$, then ${f}$ is topologically conjugated to ${R_{\alpha}}$ (i.e., there is an homeomorphism ${h:\mathbb{T}\rightarrow\mathbb{T}}$ such that ${h\circ f = R_{\alpha}\circ h}$);
• if ${f\in\textrm{Diff}^r(\mathbb{T})}$, ${r\geq 3}$, has an irrational rotation number ${\alpha}$ satisfying a Diophantine condition of the form ${|\alpha-p/q|\geq c/q^{2+\beta}}$ for some ${c>0}$, ${(r-1)/2>\beta\geq 0}$, and all ${p/q\in\mathbb{Q}}$, then there exists ${h\in\textrm{Diff}^{r-1-\beta-}(\mathbb{T}):= \bigcap\limits_{\varepsilon>0}\textrm{Diff}^{r-1-\beta-\varepsilon}(\mathbb{T})}$ conjugating ${f}$ and ${R_{\alpha}}$ (i.e., ${h\circ f = R_{\alpha}\circ h}$);
• etc.

In particular, if ${\alpha}$ has Roth type (i.e., for all ${\varepsilon>0}$, there exists ${c_{\varepsilon}>0}$ such that ${|\alpha-p/q|\geq c_{\varepsilon}/q^{2+\varepsilon}}$ for all ${p/q\in\mathbb{Q}}$), then any ${f\in\textrm{Diff}^r(\mathbb{T})}$ with rotation number ${\alpha}$ is ${C^{r-1-}}$ conjugated to ${R_{\alpha}}$ whenever ${r>3}$. (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 ${\mathbb{R}}$.)

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 ${R_{\alpha}}$ 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 ${R_{\alpha}}$ has restricted Roth type [when viewed as an i.e.t. on 2 intervals] if and only if ${\alpha}$ has Roth type) and proved that, for any ${r\geq 2}$, the ${C^{r+3}}$ g.i.e.t.s ${T}$ close to a standard i.e.t. ${T_0}$ of restricted Roth type such that ${T}$ is ${C^r}$-conjugated to ${T_0}$ form a ${C^1}$-submanifold of codimension ${(g-1)(2r+1)+s}$ where ${T_0}$ is the first return map to an interval transverse to a translation flow on a translation surface of genus ${g\geq 1}$ and ${T_0}$ is an i.e.t. on ${d=2g+s-1}$ 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 ${\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2}$, so that ${R_{\alpha}}$ has genus ${g=1}$ and also ${s=1}$. Hence, Marmi–Moussa–Yoccoz theorem says that its local conjugacy class of ${R_{\alpha}}$ with ${\alpha}$ of Roth type has codimension ${(g-1)(2r+1)+s=1}$ regardless of the differentiability scale ${r}$. 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 ${f}$ and ${R_{\alpha}}$ (with ${\alpha}$ of Roth type) is described by a single parameter, namely, the rotation number of ${f}$. On the other hand, Marmi–Moussa–Yoccoz theorem says that the codimension

$\displaystyle (g-1)(2r+1)+s$

of the local conjugacy class of an i.e.t. of restricted Roth type with genus ${g\geq 2}$ grows linearly with the differentiability scale ${r}$.

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 ${g\geq 2}$ 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 ${g\geq 2}$.

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 ${T_0}$, any ${C^4}$ g.i.e.t. ${T}$ with trivial conjugacy invariants (e.g., “simple deformations”) and ${C^0}$ conjugated to ${T_0}$ is also ${C^1}$ conjugated to ${T_0}$. In other words, the ${C^0}$ and ${C^1}$ conjugacy classes of a typical i.e.t. ${T_0}$ 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 ${T_0}$” 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 ${3}$ and ${4}$ are ${C^0}$ but not ${C^1}$ 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 ${C^r}$-g.i.e.t. ${f:I\rightarrow I}$ sending a finite partition (modulo zero) ${I=\bigcup\limits_{\alpha\in\mathcal{A}} I_{\alpha}^t}$ of ${I}$ into closed subintervals ${I_{\alpha}^t}$ disposed accordingly to a bijection ${\pi_t:\mathcal{A}\rightarrow\{1,\dots,d\}}$ to a finite partition (modulo zero) ${I=\bigcup\limits_{\alpha\in\mathcal{A}} I_{\alpha}^b}$ of ${I}$ into closed subintervals ${I_{\alpha}^b}$ disposed accordingly to a bijection ${\pi_b:\mathcal{A}\rightarrow\{1,\dots,d\}}$ (via ${C^r}$-diffeomorphisms ${f|_{I_{\alpha}^t}:I_{\alpha}^t\rightarrow I_{\alpha}^b}$), an elementary step of the Rauzy–Veech algorithm produces a new ${C^r}$-g.i.e.t. ${\mathcal{R}(f)}$ by taking the first return map of ${f}$ to the interval ${I\setminus J}$ where ${J=I_{\pi_t^{-1}(d)}^t}$, resp. ${I_{\pi_b^{-1}(d)}^b}$ whenever ${|I_{\pi_t^{-1}(d)}^t|<|I_{\pi_b^{-1}(d)}^b|}$, resp. ${|I_{\pi_t^{-1}(d)}^t|>|I_{\pi_b^{-1}(d)}^b|}$ (and ${\mathcal{R}(f)}$ is not defined when ${|I_{\pi_t^{-1}(d)}^t| = |I_{\pi_b^{-1}(d)}^b|}$).

We say that a ${C^r}$-g.i.e.t. ${f}$ has irrational rotation number whenever the Rauzy–Veech algorithm ${\mathcal{R}}$ 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 ${C^r}$-g.i.e.t. ${f}$ with irrational rotation number is topologically semi-conjugated to a standard, minimal i.e.t. ${T_0}$.

1.2. Denjoy counterexamples

Similarly to Denjoy’s theorem in the case of circle diffeomorphisms, the obstruction to promote topological semi-conjugations between ${f}$ and ${T_0}$ as above into ${C^0}$-conjugations is the presence of wandering intervals for ${f}$, i.e., non-trivial intervals ${A}$ whose iterates under ${f}$ are pairwise disjoint (i.e., ${f^i(A)\cap f^j(A)=\emptyset}$ for all ${i,j\in\mathbb{Z}}$, ${i\neq j}$).

Moreover, as it was also famously established by Denjoy, a little bit of smoothness (e.g., ${C^1}$ 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 ${C^1}$-diffeomorphisms with any prescribed irrational rotation number and possessing wandering intervals. Nevertheless, it was pointed out by several authors (including Camelier–GutierrezBressaud–Hubert–MaasMarmi–Moussa–Yoccoz, …), a high amount of smoothness is not enough to avoid wandering intervals for arbitrary ${C^r}$-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 ${C^r}$-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 ${T}$ be a piecewise affine i.e.t. with irrational rotation number, say ${T}$ is semi-conjugated to a standard i.e.t. ${T_0:\bigcup I_{\alpha}^t\rightarrow \bigcup I_{\alpha}^b}$. By definition, the logarithm ${\log DT}$ of the slope of ${T}$ is constant on the continuity intervals of ${T}$ and, hence, it allows to naturally define a function ${w}$ taking a constant value ${w_{\alpha}}$ on each continuity interval ${I_{\alpha}^t}$ of ${T_0}$. In this setting, it is possible to prove (see, e.g., the subsection 3.3.2 of Marmi–Moussa–Yoccoz paper) that ${T}$ has wandering intervals if and only if there exists a point ${x^*\in I=\bigcup I_{\alpha}^t}$ with bi-infinite ${T_0}$-orbit such that

$\displaystyle \sum\limits_{n\in\mathbb{Z}} \exp(S_n w(x^*))<\infty$

where the Birkhoff sum ${S_nw(x^*)}$ at a point ${x^*}$ with orbit ${T_0^j(x^*)\in \textrm{int}(I_{\alpha_j}^t)}$ for all ${j\in\mathbb{Z}}$ is defined as ${S_nw(x^*)=\sum\limits_{j=0}^{n-1}w_{\alpha_j}}$, resp. ${\sum\limits_{j=-1}^{n}w_{\alpha_j}}$ for ${n\geq 0}$, resp. ${n<0}$.

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 ${T}$ be a piecewise affine i.e.t. topologically semi-conjugated to a standard, minimal i.e.t. ${T_0}$. Denote by ${w}$ the piecewise constant function associated to the logarithms of the slopes of ${T}$.If ${\liminf\limits_{n\rightarrow\infty} |S_n w(y)|<\infty}$ for all ${y}$ with bi-infinite ${T_0}$-orbit, then ${T}$ is topologically conjugated to ${T_0}$ (i.e., ${T}$ is not a Denjoy counterexample).

1.3. Special Birkhoff sums and the Kontsevich–Zorich cocycle

An elementary step of the Rauzy–Veech algorithm ${\mathcal{R}}$ replaces a standard, minimal i.e.t. ${T_0}$ on an interval ${I=\bigcup\limits_{\alpha\in\mathcal{A}} I_{\alpha}^t}$ by a standard, minimal i.e.t. ${\mathcal{R}(T_0)}$ given by the first return map of ${T_0}$ on an appropriate subinterval ${J=\bigcup\limits_{\alpha\in\mathcal{A}} J_{\alpha}^t\subset I}$.

The special Birkhoff sum ${\mathcal{S}}$ associated to an elementary step ${\mathcal{R}}$ is the operator mapping a function ${\phi:I\rightarrow I}$ to a function ${\mathcal{S}\phi(x)=S_{r(x)}\phi(x):=\sum\limits_{j=0}^{r(x)-1}\phi(T_0^j(x))}$, ${x\in J}$, where ${r(x)}$ stands for the first return time to ${J}$.

The special Birkhoff sum operator ${S}$ preserves the space of piecewise constant functions in the sense that ${\mathcal{S}\phi}$ is constant on each ${J_{\alpha}^t}$ whenever ${\phi}$ is constant on each ${I_{\beta}^t}$. In particular, the restriction of ${\mathcal{S}}$ to the space of such piecewise constant functions gives rise to a matrix ${B:\mathbb{R}^{\mathcal{A}}\rightarrow \mathbb{R}^{\mathcal{A}}}$. 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 ${M_{EW}}$ and ${M_{O}}$ of genera ${3}$ and ${4}$ obtained from finite branched covers of the torus ${\mathbb{T}^2}$. Among their several curious features, we would like to point out that the following fact proved by Jean-Christophe Yoccoz and myself: if ${T_0}$ is a standard i.e.t. on ${\#\mathcal{A}=9}$ or ${10}$ intervals (resp.) associated to the first return map of the translation flow ${V}$ in a typical direction on ${M_{EW}}$ or ${M_{O}}$ (resp.), then there are vectors ${q_V}$, ${p_{T_0}}$ and a ${(\#\mathcal{A}-2)}$-dimensional vector subspace ${H}$ such that ${\mathbb{R}^{\mathcal{A}} = \mathbb{R} q_V\oplus H\oplus \mathbb{R} p_{T_0}}$ is an equivariant decomposition with respect to the matrices of the Kontsevich–Zorich cocycle with the following properties:

• (a) ${q_V}$ generates the Oseledets direction of the top Lyapunov exponent ${\theta_1>0}$;
• (b) ${p_{T_0}}$ generates the Oseledets direction of the smallest Lyapunov exponent ${-\theta_1}$;
• (c) the matrices of the Kontsevich–Zorich cocycle act on ${H}$ through a finite group.

In the literature, the Lyapunov exponents ${\pm\theta_1}$ 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 ${M_{EW}}$ and ${M_{O}}$ vanish.

In the next two sections, we will see that this curious behaviour of the Kontsevich–Zorich cocycle of ${M_{EW}}$ or ${M_{O}}$ along ${H}$ allows to construct plenty of piecewise affine i.e.t.s which are ${C^0}$ but not ${C^1}$ 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 ${M_{EW}}$ et ${M_{O}}$

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 ${T}$ is a piecewise affine i.e.t. topologically semi-conjugated to ${T_0}$ coming from (the first return map of the translation flow in the direction of a pseudo-Anosov homeomorphism of) ${M_{EW}}$ or ${M_{O}}$. It is well-known that the piecewise constant function ${w}$ associated to the logarithms of the slopes ${DT}$ of ${T}$ belongs to ${H\oplus \mathbb{R} p_{T_0}}$ (see, e.g., Section 3.4 of Marmi–Moussa–Yoccoz paper). In order to simplify the exposition, we assume that the “irrational rotation number” ${T_0}$ has “bounded type”, that is, ${T_0}$ is self-similar in the sense that some of its iterates ${\mathcal{R}^k(T_0)}$ under the Rauzy–Veech algorithm actually coincides with ${T_0}$ up to scaling.

If ${w\in H}$, then the item (c) from Subsection 1 above implies that all special Birkhoff sums of ${w}$ (in the future and in the past) are bounded. From this fact, we conclude that ${\liminf\limits_{n\rightarrow\infty} |S_nw(y)|\leq C}$ for all ${y}$ with bi-infinite ${T_0}$-orbit: indeed, as it is explained in details in Bressaud–Bufetov–Hubert article, if ${T_0}$ is self-similar, then the orbits of ${T_0}$ can be described by a substitution on a finite alphabet ${\mathcal{A}}$ and this allows to select a bounded subsequence of ${S_nw(y)}$ 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 ${T}$ topologically semi-conjugated to a self-similar standard i.e.t. ${T_0}$ coming from ${M_{EW}}$ or ${M_O}$ such that ${w\in H}$.

Remark 3 Actually, it is possible to explore the fact that ${p_{T_0}}$ is a stable vector (i.e., it generates the Oseledets space of a negative Lyapunov exponent) to remove the constraint “${w\in H}$” from the statement of the previous paragraph.

In other words, we showed that any ${w\in H}$ always provides a piecewise affine i.e.t. ${C^0}$-conjugated to ${T_0}$. Note that this is a relatively rich family of piecewise affine i.e.t.s because ${H}$ is a vector space of dimension ${7}$, resp. ${8}$, when ${T_0}$ is a self-similar standard i.e.t. coming from ${M_{EW}}$, resp. ${M_O}$.

3. Cohomological obstructions to ${C^1}$ conjugations

Closing this post, we will show that the elements ${w\in H\setminus\{0\}}$ always lead to piecewise affine i.e.t.s which are not ${C^1}$ conjugated to self-similar standard i.e.t.s of ${M_{EW}}$ or ${M_O}$. Of course, this shows that the ${C^0}$ and ${C^1}$ conjugacy classes of a self-similar standard i.e.t. of ${M_{EW}}$ or ${M_O}$ are distinct and, a fortiori, the Marmi–Moussa–Yoccoz conjecture about the coincidence of ${C^0}$ and ${C^1}$ 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 ${T}$ is a piecewise affine i.e.t. ${C^1}$-conjugated to a self-similar standard i.e.t. ${T_0}$ of ${M_{EW}}$ or ${M_O}$, say ${T\circ h = h\circ T_0}$ for some ${C^1}$-diffeomorphism ${h}$. By taking derivatives, we get

$\displaystyle (DT\circ h) \cdot h' = h'\circ T_0$

since ${T_0}$ is an isometry. Of course, we recognize the slope of ${T}$ on the left-hand side of the previous equation. So, by taking logarithms, we obtain

$\displaystyle w=\Psi\circ T_0 - \Psi$

where ${\Psi:=\log h'}$ is a ${C^0}$ function. In other terms, ${\Psi}$ is a solution of the cohomological equation and ${w}$ is a ${C^0}$-coboundary. Hence, the Birkhoff sums ${S_nw=\Psi\circ T_0^n-\Psi}$ are bounded and, by continuity of ${\Psi}$, the special Birkhoff sums ${\mathcal{S}w}$ of ${w}$ converge to zero. Equivalently, ${w\in\mathbb{R}^{\mathcal{A}}}$ 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 ${w\in H\setminus\{0\}}$ through a finite group of matrices and, thus, ${w\in H\setminus\{0\}}$ can not converge to zero under the Kontsevich–Zorich cocycle.

This contradiction proves that ${T}$ is not ${C^1}$-conjugated to ${T_0}$, as desired.

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