Local rigidity for periodic generalised interval exchange transformations

Abstract

In this article we study local rigidity properties of generalised interval exchange maps using renormalisation methods. We study the dynamics of the renormalisation operator $\mathcal {R}$ acting on the space of $\mathcal {C}^{3}$-generalised interval exchange transformations at fixed points (which are standard periodic type IETs). We show that $\mathcal {R}$ is hyperbolic and that the number of unstable direction is exactly that predicted by the ergodic theory of IETs and the work of Forni and Marmi–Moussa–Yoccoz. As a consequence we prove that the local $\mathcal {C}^1$-conjugacy class of a periodic interval exchange transformation, with d intervals, whose associated surface has genus g and whose Lyapounoff exponents are all non zero is a codimension $g-1 +d-1$ $\mathcal {C}^1$-submanifold of the space of $\mathcal {C}^{3}$-generalised interval exchange transformations. This solves a conjecture analogous to that of Marmi–Moussa–Yoccoz, stated for almost all IETs, in the special case of self-similar IETs.

Appendices

Appendix A: Properties of the renormalisation operator

1.1 The Banach structure on $\mathcal {X}^r$

Let r be an integer greater or equal to 1. Recall that $\mathcal {X}^r_{\sigma } = \mathcal {X}^r$ is the space of GIETs with permutation $\sigma$ on d intervals of class $\mathcal {C}^r$. In Sect. 2.2 we explained how $\mathcal {X}^r$ naturally identifies with

$$\begin{aligned} \mathcal {A} \times \mathcal {P} \end{aligned}$$

where $\mathcal {A}$ is the space of affine IETs with permutation $\sigma$ and $\mathcal {P}$ is the product of d copies of $\mathrm {Diff}^r_+([0,1])$ the set of orientation preserving $\mathcal {C}^r$ diffeomorphism of the interval. The set $\mathrm {Diff}^r_+([0,1])$ can be seen as a subset of the affine space of $\mathcal {C}^r$-maps of the interval taking value 0 in 0 and value 1 in 1. The latter can be endowed with the $\mathcal {C}^r$-norm to give the structure of a Banach affine space modelled on the vector space $\mathcal {C}^r_0([0,1], \mathbb {R})$ of $\mathcal {C}^r$-maps vanishing at both 0 and 1. $\mathrm {Diff}^r_+([0,1])$ is easily seen to be an open subset of this Banach affine space with respect to the topology induced by the $\mathcal {C}^r$-norm, this naturally endows $\mathrm {Diff}^r_+([0,1])$ with the structure of a Banach manifold whose tangent space at any point naturally identifies with $\mathcal {C}^r_0([0,1], \mathbb {R})$.

On the other hand, $\mathcal {A}$ naturally identifies with an open subset of the projective space $\mathbb {RP}^{2d-2}$ and by that mean is naturally endowed with a structure of finite dimensional smooth manifold which specialises into a structure of Banach manifolds. In turn, $\mathcal {X}^r$ seen as the product $\mathcal {A} \times \mathcal {P}$ is naturally endowed with the structure of a Banach manifold as a product of Banach manifold.

1.2 An easy lemma on smooth functions

Lemma 35

Let $I \subset \mathbb {R}$ be an open connected interval. The map

$$\begin{aligned} \begin{array}{ccc} \mathcal {C}^r(I, \mathbb {R}) \times I &{} \longrightarrow &{} \mathbb {R} \\ ( \varphi , p) &{} \longmapsto &{} \varphi (p) \end{array} \end{aligned}$$

is of class $\mathcal {C}^1$.

Proof

We compute

$$\begin{aligned} (\varphi +h)(p+\epsilon )= & {} \varphi (p+\epsilon ) + h(p+\epsilon ) = \varphi (p) + \varphi '(p)\epsilon \\&+ o(\epsilon ) + h(p) + h'(p)\epsilon + o(\epsilon ). \end{aligned}$$

But $h'(p)\epsilon$ is a $o(\sup (\epsilon , ||h||_{\mathcal {C}^1})$ therefore

$$\begin{aligned} \begin{array}{ccc} \mathcal {C}^r(I, \mathbb {R}) \times I &{} \longrightarrow &{} \mathbb {R} \\ ( \varphi , p) &{} \longmapsto &{} \varphi (p) \end{array} \end{aligned}$$

is of class $\mathcal {C}^1$ with derivative at $(\varphi ,p)$ equal to

$$\begin{aligned} (h, \epsilon ) \longmapsto \varphi '(p)\epsilon + h(p) \end{aligned}$$

$\square$

An easy but important for our purpose consequence of this lemma is that if $f_1, \ldots , f_n$ are $\mathcal {C}^1$ maps then

$$\begin{aligned} (f_1, \ldots , f_n, p) \mapsto f_n \circ f_{n-1} \circ \cdots \circ f_1 (p) \end{aligned}$$

is of class $\mathcal {C}^1$, provided the for all k the range of $f_k$ belongs to the interval of definition of $f_{k+1}$.

1.3 Analytic properties of the renormalisation operator

Recall the following definitions and notation from Sect. 5. We can identify a neighbourhood of $\mathcal {X}^r$ with an open neighbourhood of 0 in the Banach space upon which $\mathcal {X} = \mathcal {A} \times \mathcal {P}$ is modelled. In these coordinates, we will use the notation $T_0 = (0_{\mathcal {A}}, 0_{\mathcal {P}})$ where $0_{\mathcal {P}}$ represents the point $(\mathrm {Id}, \mathrm {Id}, \ldots , \mathrm {Id}) \in \mathrm {Diff}^r_+([0,1])$. Here $\mathcal {P}$ abusively denotes (a neighbourhood of 0 in) the Banach space upon which $(\mathrm {Diff}^r_+([0,1]))^d$ is modelled.

In these coordinates, we write

$$\begin{aligned} \mathcal {R} = (\mathcal {R}_{\mathcal {A}}, \mathcal {R}_{\mathcal {P}}). \end{aligned}$$

Finally, we denote by $\pi _{\mathcal {A}}$ and $\pi _{\mathcal {P}}$ the projection from $\mathcal {X}$ onto $\mathcal {A}$ and $\mathcal {P}$ respectively.

Proposition 36

$\mathcal {R}$ is continuous in a neighbourhood of $T_0$ for the $\mathcal {C}^0$-topology.

Proof

This results from the continuity of the following functions, with respect to the $\mathcal {C}^0$-topology

• (1) restriction of a function to an interval;

• (2) evaluation of a function at a given point;

• (3) composition of functions.

$\square$

We now move to proving that $\mathcal {R}_{\mathcal {A}}$ is differentiable. To achieve this we need a set of coordinates on $\mathcal {A}$. Recall that $\mathcal {A}$ is the set of affine interval exchange maps on d intervals with permutation d. A point in $\mathcal {A}$ is completely determined by the its discontinuity points $0< u^t_1< \cdots< u^t_k< \cdots< u^t_{d-1} < 1$ at the ”top” and their images $0< u^b_1< \cdots< u^b_k< \cdots< u^b_{d-1} < 1$ at the ”bottom”. These $2d-2$ parameters provide a set coordinates compatible with the smooth structure of $\mathcal {A}$.

Proposition 37

There exists a neighbourhood of $T_0$ in $\mathcal {X}^r$ such that $\mathcal {R}_{\mathcal {A}}$ is of class $\mathcal {C}^1$ in this neighbourhood for the $\mathcal {C}^r$-norm.

Proof

As indicated in the above discussion above, $\mathcal {R}_{\mathcal {A}}(T)$ is entirely determined by the positions of finitely many iterates of T on finitely many points. We explain how positions can all be expressed as a finite combination of the functions from Lemma 35 applied to coordinates of $\pi _{\mathcal {A}}(T)= (u^t_1(T), \ldots , u^t_{d-1}(T), u^b_1(T), \ldots , u^b_{d-1}(T)$ and $\pi _{\mathcal {P}}(T) = (\varphi _1(T), \ldots , \varphi _d(T))$ which will give the result.

$\mathcal {R}(T)$ is the (rescaled) first return map of T on an interval of the form $[0,T^k(u_i^t(T))]$ for a certain $k \in \mathbb {Z}$ and a certain $i \le d-1$. Moreover, the discontinuities of $\mathcal {R}(T)$ are also of the form $T^k(u_i^t(T))$ and therefore it is enough to show that for any k and i there exists a neighbourhood of $T_0$ in $\mathcal {X}^r$ for which the function $T \mapsto T^k(u_i^t(T))$ is of class $\mathcal {C}^1$. Now denote by $l_k^t = u^t_{k} - u^t_{k-1}$ the length of the kth interval of continuity of T at the top and $l_k^b(T) = u^b_{k}(T) - u^b_{k-1}(T)$ the length of the kth interval of continuity of T at the bottom. These maps (depending upon T) are smooth (by Lemma 35). The restriction of T to the interval $]u_{i-1}^t, u_i^t[$ is of the following form

$$\begin{aligned} x \mapsto l^b_j \cdot \varphi _i(\frac{x - u^t_{i-1}}{l_i}) + u^b_j \end{aligned}$$

for a certain $j \le d-1$.

Now, $T^k(u_i^t(T))$ can be expressed as finitely many compositions of the function of Lemma 35 applied to the $\varphi _i(T)$ and affine maps depending smoothly upon the $u_i^t(T)$s and $u_i^b(T)$s. This implies (by Lemma 35) that $T \mapsto T^k(u_i^t(T))$ is of class $\mathcal {C}^1$. This concludes the proof.

$\square$

Appendix B: Fine grids and $\mathcal {C}^{1+\delta }$ homeomorphisms

This appendix is devoted to the proof of Theorem 29. We reproduce here some material from [4] and apply it to the special case of periodic GIETs.

1.1 Fine grids

A fine grid is a sequence of finite partitions $(\mathcal {Q}_n)_{n\in \mathbb {N}}$ of [0, 1] such that

• $\forall n \in \mathbb {N}$, $\mathcal {Q}_{n+1}$ is a refinement of $\mathcal {Q}_n$;

• there exists an integer $a>0$ such that for all n, each atom of $\mathcal {Q}_n$ is the union of at most a atoms of $\mathcal {Q}_{n+1}$;

• there exists $c >0$ such that for all I, J adjacent atoms of $\mathcal {Q}_n$ we have

  $$\begin{aligned} c^{-1}|I| \le |J| \le c |I|. \end{aligned}$$

One easily checks the following fact

Proposition 38

Let T be a GIET which is $\mathcal {C}^1$ conjugate to $T_0$. Then the dynamical partition $(\mathcal {P}_n)_{n\in \mathbb {N}}$ of T form a fine grid.

The main technical tool of [4] is the following proposition

Proposition 39

[de Faria–de Melo, [4]] Let $h : [0,1] \longrightarrow [0,1]$ be a homeomorphism and assume that $(\mathcal {Q}_{n})$ is a fine grid. Assume furthermore that there exist positive constants $C>0$ and $\lambda <1$ such that for every I, J adjacent atoms in $\mathcal {Q}_n$ we have

$$\begin{aligned} | \frac{I}{J} - \frac{h(I)}{h(J)} | \le C \lambda ^n. \end{aligned}$$

Then

• (1) There exists $\delta$ such that h is of class $\mathcal {C}^{1+\delta }$.

• (2) $\sup _{x,y \in [0,1]}{\frac{|h'(x) - h'(y)|}{|x-y|^{\delta }}} \le C$.

This Proposition is not exactly stated as such in [4]: the second point is implicit and one will find it in the proof of Proposition 4.3, p358.

1.2 Application to the conjugating map

In this Section we apply the above material to our context. We prove the following

Proposition 40

There exists a uniform $\delta$ depending only on $T_0$ and a continuous positive function $A : (0,\nu ) \rightarrow \mathbb {R}_+$ such that $\lim _{\epsilon \rightarrow 0}{A(\epsilon )} = 0$ such that the following holds. Assume $T_1$ and $T_2$ belong to $\mathcal {K}_0$. Then the map conjugating $T_1$ to $T_2$ is $A(d_{\mathcal {C}^1}(T_1,T_2))$-close to the identity in the $\mathcal {C}^{1+\delta }$-topology.

The fact that the conjugating is $A(d_{\mathcal {C}^3}(T_1,T_2)$ close to the identity in the $\mathcal {C}^1$-topology was already implicit in Sect. 7. Indeed, we have the following fact :

there exists $\kappa < 1$ such that for $T_1$ and $T_2$ as in the Proposition above the following holds

$$\begin{aligned} d_{\mathcal {C}^1}(\mathcal {R}^n(T_1), R^n(T_2)) \le A(d_{\mathcal {C}^1}(T_1,T_2)) \kappa ^n. \end{aligned}$$

for a certain function A whose limit in 0 is 0. This is, as in the proof of Proposition 28, because the $\mathcal {C}^0$-norm the solution to the cohomological equation depend linearly on the norm of the Birkhoff sums of the variable. In the case we are studying, the conjugating map is given by integrating the solution to the cohomological equation for the difference

$$\begin{aligned} \log \mathrm {D}T_1 - \log \mathrm {D}T_2. \end{aligned}$$

Because $d_{\mathcal {C}^1}(\mathcal {R}^n(T_1), R^n(T_2)) \le A(d_{\mathcal {C}^1}(T_1,T_2)) \kappa ^n$, Birkhoff sums of this difference are never any bigger that $D \cdot A(d_{\mathcal {C}^3}(T_1,T_2))$ where D is a uniform constant depending only on $T_0$.

Thus the only bit missing to prove Proposition 40 is the fact that the derivative of the conjugating map is $\delta$-Hölder and that its Hölder-norm is controlled by $A(d_{\mathcal {C}^3}(T_1,T_2))$. This will be a consequence of the following

Proposition 41

Let $T_1$ and $T_2$ be as in Proposition 40, let h be the map conjugating $T_1$ to $T_2$ and let $(\mathcal {P}_n)$ be the sequence of dynamical partitions of $T_1$. Then there exists $\kappa '<1$ such that for all $n\in \mathbb {N}$ and adjacent I, J in $\mathcal {P}_n$ we have

$$\begin{aligned} |\frac{I}{J} - \frac{h(I)}{h(J)}| \le A(d_{\mathcal {C}^3}(T_1,T_2)) \cdot \kappa '^n. \end{aligned}$$

We explain below why this Proposition is a direct consequence of [12][Section 9, p. 113–121] for circle diffeomorphisms with on break point. Precisely, we explain how methods from [12] straightforwardly adapt to our context. First we comment on how the two contexts compare at a formal level.

• [12] is concerned with GIETs with two intervals, which is a particular case of the one we treat here.

• Their renormalisation operator is defined by taking successive first return maps on carefully chosen nested intervals, exactly as it is here.

• They are working at a fixed point of the renormalisation operator.

• Finally, they work with dynamical partitions defined in the exact same way as it is in the present article. Their notation for the dynamical partition of level n is $\xi _n$

Hence, the only obstruction for the proof of [12] to extend to our context is to check that the number of intervals of the GIET does not play a particular role in the proof. The only place in [12][Section 9, p. 113–121] where the fact that their GIET have only two intervals is used is in the proof of Lemma 35. They use the fact that any end point of an interval of the level n partition is an image under the GIET of a single singular point. The conclusion of this Lemma (and its proof) still hold if this unique singular point is replaced by the finitely many discontinuity points, which is sufficient for our purpose. We add a sketch of the proof explaining in a more informal way the content of [12][Section 9, p. 113–121] for the interested reader who does not want to read all the detail of the proofs from [12].

Sketch of proof

Recall a bit of notation and facts demonstrated in this article.

• (1) $\mathcal {P}_n(T_1)$ and $\mathcal {P}_n(T_2)$ are the dynamical partition of [0, 1] associated with the nth renormalisations of $T_1$ and $T_2$.

• (2) The base of the partitions $\mathcal {P}_n(T_1)$ and $\mathcal {P}_n(T_1)$ are the discontinuity intervals of $\mathcal {R}^n(T_1)$ and $\mathcal {R}^n(T_2)$ (pre-rescaling).

• (3) There exists universal constants $K >0$ and $\lambda < 1$ such that for any $I \in \mathcal {P}_n(T_i)$, $| I | \le K \lambda ^n$.

• (4) The map h conjugating $T_1$ to $T_2$ maps corresponding intervals of the respective dynamical partition to one another.

• Because of the exponential convergence of renormalisations of $T_1$ and $T_2$, the estimate obviously holds for adjacent intervals that are in the base of the dynamical partitions $\mathcal {P}_n(T_1)$ and $\mathcal {P}_n(T_1)$. Precisely for adjacent I and J in the base of $\mathcal {P}_n(T_1)$ we have

  $$\begin{aligned} | \frac{I}{J} - \frac{h(I)}{h(J)} | \le A(d_{\mathcal {C}^3}(T_1,T_2)) \cdot \kappa _1^n \end{aligned}$$

  for a certain constant $\kappa _1 < 1$.

• Now assume that I and J belong to the partition of level $(p+1)n$ for a certain integer p whose value we will set afterwards. There exists an integer k such that $T_1^k(I\cup J)$ belongs to a base interval for the partition $\mathcal {P}_n$, and $T_1^k$ is continuous restricted to $I\cup J$. We have $|k| \le q^n$ for q the largest return time used to define $\mathcal {R}$. The length of $\cup _{i \le k}{T_1^i(I\cup J)}$ is less than

  $$\begin{aligned} 2 \sup _{i \in \mathbb {Z}}{ ||D(T_1^i)|| } q^n L \alpha ^{(p+1)n} \end{aligned}$$

  for constant $\alpha < 1$ and $L > 0$. We thus see that if p is taken to be large enough (such that $\kappa _2 = q \alpha ^{p+1} < 1$) we have

  $$\begin{aligned} |\bigcup _{i \le k}{T_1^i(I\cup J)}| \le K \sup _{i \in \mathbb {Z}}{ ||D(T_1^i)|| } \kappa _2^n \end{aligned}$$

• Thus the error induced when initially bringing back I and J to the base of $\mathcal {P}_n$ can be controlled by the fact that the distortion of $T_1^k$ and $T_2^k$ are proportional to the measure of $\bigcup _{i\le k}{ \mathcal {T}^i(I \cup J)}$ (by applying the standard distortion Lemma 5) which is exponentially small. Precisely we get this way that $| \frac{|I|}{|J|} - \frac{|T_1^k(I)|}{|T_1^k(J)|}$ and $| \frac{|h(I)|}{|h(J)|} - \frac{|T_2^k(h(I))|}{|T_2^k(h(J))|}$ are exponentially small. (This step corresponds to Lemma 37 in [12].)

• The previous step was about taking two corresponding pairs of adjacent intervals (I, J) and h(I), h(J) in $\mathcal {P}_{(p+1)n}(T_1)$ and $\mathcal {P}_{(p+1)n}(T_2)$ and bringing them back to the base intervals of $\mathcal {P}_{n}(T_1)$ and $\mathcal {P}_{n}(T_2)$ and showing that little is lost in the process. We now take these intervals I, J and h(I), h(J) back to the base of the partitions $\mathcal {P}_{(p+1)n}(T_1)$ and $\mathcal {P}_{(p+1)n}(T_2)$ using $\mathcal {R}^{n}(T_1)$ and $\mathcal {R}^{n}(T_2)$ respectively. Because $\mathcal {R}^{n}(T_1)$ and $\mathcal {R}^{n}(T_2)$ are $(A(d_{\mathcal {C}^3}(T_1,T_2)) \cdot \kappa _1^n)$-close, the error made we comparing them to an image by successive iterations of $\mathcal {R}^{n}(T_1), \mathcal {R}^{2n}(T_1), \ldots , \mathcal {R}^{np}(T_1)$ and$\mathcal {R}^{n}(T_1), \mathcal {R}^{2n}(T_1), \ldots , \mathcal {R}^{np}(T_1)$ in the basis of $\mathcal {P}_{(p+1)n}$ is comparable to $A(d_{\mathcal {C}^3}(T_1,T_2)) \cdot \kappa _1^n$. This is the content of Lemmas 35 and 36 in [12].

• Interpolating the previous two steps, to get two arbitrary pairs of corresponding adjacent intervals

  • (1) first in the base of $\mathcal {P}_n(T_1)$ and $\mathcal {P}_n(T_2)$ respectively;

  • (2) then to the base of $\mathcal {P}_{n(p+1)}(T_1)$ and $\mathcal {P}_{n(p+1)}(T_2)$ respectively;

  and utilising the two mechanisms described above to control the error made in estimating the ratios $\frac{|I|}{|J|}$ and $\frac{|h(I)|}{|h(J)|}$ on the way; we get the expected result.

$\square$