Posted by: matheuscmss | March 21, 2010

The local linearization problem of generalized interval exchange maps

Two weeks ago Stefano Marmi posted on Arxiv his joint paper with Pierre Moussa and Jean-Christophe Yoccoz about a local conjugation theorem for generalized interval exchange transformations (see this link for the preprint). Morally speaking, the main goal of the paper is the extension of the theory of smooth linearization of circle diffeomorphisms (of V. Arnold, M. Herman, H. Russmann, J.-C. Yoccoz, etc.) to the case of interval exchange transformations (i.e.t. for short), i.e., they show that, for almost every standard i.e.t. {T_0} (i.e., {T_0} is locally a translation), the local {C^r}-conjugacy class of {T_0} amongst {C^{r+3}} generalized i.e.t. {T} {C^{r+3}} close to {T_0} with trivial {C^{r+3}} conjugacy invariant (i.e., no non-trivial obstructions to conjugation at the level of the first {r+3} derivatives) is a {C^1} submanifold of explicitly computable codimension.

The basic idea of S. Marmi, P. Moussa and J.-C. Yoccoz is the following: usually, the conjugation problem can be understood from the corresponding cohomological equation, i.e., the linearized version of the conjugation equation; in this respect, the cohomological equations related to i.e.t.’s were already studied (by e.g. G. Forni and S. Marmi, P. Moussa, J.-C. Yoccoz), so that we dispose of nice criteria for its solvability; in particular, if we can convert a solution of the linearized (cohomological) equation into a solution of the conjugacy equation, we are done.

Of course, the previous strategy should be worked in details: for instance, the results of Marmi, Moussa and Yoccoz about the cohomological equations of i.e.t.’s don’t apply directly to the situation at hand and, even after obtaining the necessary results, it is not clear how to convert solutions of cohomological equations into the desired conjugations. Because of the limitation of space, I’ll focus today on the discussion of the second problem (namely, the conversion of solutions of cohomological equations into true conjugations) in the more simple context of circle diffeomorphisms. More precisely, we are going to study M. Herman’s simple proof of a local conjugacy theorem for circle diffeomorphisms using the Schwartzian derivative trick. The reason why we’re restricting our discussion to this particular topic is two-fold: besides the fact that Herman’s trick gives a simple method to exhibit conjugations once we can solve cohomological equations, it is so nice that it can be generalized to the case of i.e.t. (and this one of the crucial remarks of Marmi, Moussa and Yoccoz). My basic references for the material below are M. Herman’s original article, Appendix B of Marmi, Moussa, Yoccoz preprint and my notes of Yoccoz’s 2009-2010 course (at Collège de France) on this paper.

Introduction

Let’s warm up with the following local conjugacy problem on the circle {S^1=\mathbb{R}/\mathbb{Z}}: given a smooth diffeomorphism {f:S^1\rightarrow S^1} close to the rotation {R_\alpha} of irrational angle {\alpha\in\mathbb{R}-\mathbb{Q}}, we want to know when {f} is smoothly conjugated to {R_\alpha}, i.e., we’re searching for a circle diffeomorphism {h} satisfying the conjugacy equation{h\circ R_\alpha = f\circ h}. Of course, the nonlinear nature of the conjugacy equation indicates that we shouldn’t try to attack it directly: in fact, we’ll pursue the standard strategy of linearizing the conjugacy equation in order to get an idea of how to solve the original problem. More precisely, we consider the following ansatz: since {f} is close to {R_\alpha}, we’ll restrict ourselves to smooth conjugacies {h} close to the identity. In other words, we write {f=R_\alpha+\Delta\phi} and {h=id+\Delta\psi}. In this case, {h\circ R_\alpha = f\circ h} becomes {R_\alpha+\Delta\psi\circ R_\alpha = R_\alpha+\Delta\psi+\Delta\phi\circ (id+\Delta\psi)}, i.e.,

\displaystyle \Delta\psi\circ R_\alpha = \Delta\psi+\Delta\phi\circ(id+\Delta\psi).

If we think of {\Delta\phi} and {\Delta\psi} as perturbation terms, the first order approximation of {\Delta\phi\circ(id+\Delta\psi)} is {\Delta\phi}, so that the linearized version of previous equation is the so-called cohomological equation

\displaystyle \Delta\psi\circ R_\alpha(x)=\Delta\psi(x) +\Delta\phi(x).

Here {\Delta\phi} is our initial data and we’re searching a solution {\Delta\psi} of this linear equation. The discussion of solutions of cohomological equations is a recurrent theme in Dynamical Systems (with several applications such as Furstenberg’s example of a minimal non-ergodic area-preserving analytic diffeomorphism of the two-dimensional torus) and the curious reader can look at Hasselblat and Katok’s book (and references therein) for more details.

In the present case, the cohomological equation can be solved in Sobolev scale by Fourier analysis. By taking the Fourier transform in the cohomological equation, we obtain

\displaystyle \sum\limits_{n\in\mathbb{Z}}\widehat{\Delta\psi}(n)\cdot e^{2\pi in(x+\alpha)} = \sum\limits_{n\in\mathbb{Z}}\widehat{\Delta\psi}(n)\cdot e^{2\pi i nx}+\sum\limits_{n\in\mathbb{Z}}\widehat{\Delta\phi}(n)\cdot e^{2\pi i nx}.

By comparison of the Fourier coefficients, we get

\displaystyle \widehat{\Delta\psi}(n) = \frac{\widehat{\Delta\phi}(n)}{e^{2\pi i n\alpha}-1} \ \ \ \ \ (1)

for every {n\in\mathbb{Z}-\{0\}}. Observe that the zero-th Fourier mode gives the normalization condition {\int_{S^1}\Delta\phi(x)\,dx=\widehat{\Delta\phi}(0)=0} (which is necessary condition for the solvability of the cohomological equation). For sake of simplicity, we take {\widehat{\Delta\psi}(0)=0}. Observe that there is no loss of generality here since {\Delta\psi-c} (where {c} is a constant) solves the cohomological equation whenever {\Delta\psi} is a solution of this equation.

From the previous formula, we see that the Sobolev regularity of {\Delta\psi} depends on the sizes of the so-called small divisors {1/(e^{2\pi i n\alpha}-1)}, i.e., on the Diophantine properties of {\alpha}.

More precisely, we consider the Sobolev spaces

\displaystyle H^s(S^1):=\{a\in L^2(S^1): \|a\|_{H^s}^2:=\sum\limits_{n\in\mathbb{Z}}(1+n^2)^s|\widehat{a}(n)|^2<\infty\}

where {s\geq 0} for sake of definiteness, and the Diophantine conditions

\displaystyle DC(\gamma,\tau)=\{\alpha\in S^1: \|n\alpha\|_{\mathbb{Z}}\geq \gamma n^{-1-\tau}\, \forall\, n\geq 1\}

where {\gamma,\tau>0} and {\|x\|_{\mathbb{Z}}:=\inf\limits_{p\in\mathbb{Z}}|x-p|}.

Since {4\|n\alpha\|_{\mathbb{Z}}\leq |e^{2\pi in\alpha}-1|\leq 2\pi\|n\alpha\|_{\mathbb{Z}}}, we can derive from (1) the following proposition:

Proposition 1 Let {\alpha\in DC(\gamma,\tau)} and {\Delta\phi\in H^s(S^1)}. Suppose that {s\geq 1+\tau}. Then, the solution {\Delta\psi} of the cohomological equation {\Delta\psi\circ R_\alpha = \Delta\psi+\Delta\phi} obtained from (1) verifies {\Delta\psi\in H^{s-1-\tau}(S^1)} and\displaystyle \|\Delta\psi\|_{H^{s-1-\tau}}\leq C(s)\gamma^{-1}\|\Delta\phi\|_{H^s}.

Before proceeding further, let’s make a few remarks about this proposition.

Remark 1 In other words, we are able to solve the cohomological equation (in Sobolev scale) with a controlled loss of derivative depending on the strength of the Diophantine properties of {\alpha} (i.e., we start with {\Delta\phi\in H^s} and we end up with {\Delta\psi\in H^{s-1-\tau}}). This loss of derivatives phenomenon is well-studied in Dynamical Systems and it can’t be avoided in general.

Remark 2 While the Sobolev scales are useful for many purposes (e.g., in Harmonic Analysis and PDEs), it is less handful when dealing with Dynamical Systems by the following simple reason. Generally speaking, the nonlinear terms of several important PDEs have a polynomial nature (e.g., they are obtained by taking powers of our functions), so that Sobolev spaces can be handful because e.g. they form an algebra with respect to the multiplication when the regularity index {s} is sufficiently high. However, the nonlinear terms of important equations related to Dynamical Systems (e.g., the conjugation equation) are obtained by composition of our functions, and unfortunately Sobolev spaces are bad-behaved with respect to composition.For this reason, it is pretty common to find the Sobolev scale in PDE problems and the {C^k} (and/or Hölder) scale in Dynamics problems. For instance, a major problem related to this difficulty is the extension of Dolgopyat-type estimate (for exponential mixing) from the hyperbolic {C^2} case (where Sobolev-like scales can be used) to the {C^{1+\alpha}} case: indeed, although this seems a technical (minor) regularity problem, it is one of the main obstacles to study the rate of mixing of the Lorenz attractor.

In the light of the previous remark, we state the following version of the previous proposition to the Hölder scale:

Theorem 2 (Russmann, Herman, …) Let {\Delta\phi\in C^r(S^1)} and {\alpha\in DC(\gamma,\tau)}. Suppose that {r>1+\tau} and {s=r-1-\tau\notin\mathbb{Z}}. Then, the solution {\Delta\psi} of  (1) verifies {\Delta\psi\in C^s(S^1)} and {\|\Delta\psi\|_{C^s}\leq C(r)\gamma^{-1}\|\Delta\phi\|_{C^r}}.

The proof of this result is similar in spirit to the Sobolev case: one considers Littlewood-Paley decomposition and apply Hadamard’s interpolation inequalities to handle the Hölder norms. The details can be found in the fourth chapter of M. Herman’s article.

At this stage, our understanding of the cohomological (i.e., linearized conjugacy) equation on the circle is sufficiently developed and we can pass to the study of the initial nonlinear (conjugacy) problem.

Herman’s Schwartzian derivative trick

The main result of this post is:

Theorem 3 (M. Herman) Let {f} be a {C^{r+3}} circle diffeomorphism {C^{r+3}}-close to the irrational rotation {R_\alpha}. Suppose that {\alpha\in DC(\gamma,\tau)} with {0\leq\tau<1}. Then, {f=R_t\circ h\circ R_\alpha\circ h^{-1}} for a unique {h} {C^r} circle diffeomorphism {C^r}-close to {id} and {t\in S^1} close to {0}. Furthermore, the map {f\mapsto (h,t)} is {C^1}.

Remark 3 By direct inspection of the statement, the reader can see that it is not optimal in several senses: we loose {3} derivatives to solve the conjugacy equation, we don’t treat all Diophantine conditions (since we assume {0\leq\tau<1}) although this covers a full Lebesgue measure set of angles {\alpha}, etc. However, the relevance of this result consists into its flexible proof.

Remark 4 At a first sight, the appearance of the extra rotation {R_t} seems strange, but it is necessary to adjust the rotation number of {f}: in fact, we know (from H. Poincaré’s work) that, if {f} is conjugated to a rotation {R_\alpha}, then its rotation number must be {\alpha}. In other words, we can’t hope to find a conjugation between a diffeomorphism {f} close to {R_\alpha} unless we use {R_t} to match the rotation numbers.

The basic idea of M. Herman consists into a slight change of linearization operator: instead of taking usual derivative {D} to analyze the conjugacy equation, he “linearizes” it with the mildly nonlinear Schwartzian derivative (which has a good behavior under composition). We recall that the Schwartzian derivative {Sh} of {h\in C^3} is

\displaystyle Sh:=D^2\log Df - \frac{1}{2}(D\log Df)^2

Amongst its main properties, we can quote:

  • (a) {Sh=0} if and only if {h(x) = (ax+b)/(cx+d)} with {a,b,c,d\in\mathbb{R}} and {ad-bc=1};
  • (b) {S(f\circ g)= (Sf\circ g)\cdot (Dg)^2 + Sg}.

The geometrical meaning of the Schwartzian derivative is explained by (a): it measures how far {h} is fractional linear transformations. Also, the fact that Schwartzian derivative is adapted to Dynamics problems is explained by (b): it interacts well with the composition operation.

Coming back to Herman’s theorem, let’s analyze the conjugation equation {f=R_t\circ h\circ R_\alpha\circ h^{-1}} with the aid of the Schwartzian derivative: we rewrite this equation as {f\circ h = R_t\circ h\circ R_\alpha} and we apply the Schwartzian derivative to get:

\displaystyle S(f\circ h)(Dh)^2 + Sh = Sh\circ R_\alpha,

i.e., we obtain the following “cohomological equation”:

\displaystyle S(f\circ h)(Dh)^2 = Sh\circ R_\alpha - Sh \ \ \ \ \ (2)

This linear difference equation on {Sh} resembles (1) except for the fact that the left-hand side {S(f\circ h)(Dh)^2} depends on {h}. Nevertheless, this suggests a fixed-point approach to find our solution {h}: we introduce the operator

\displaystyle \Phi(f,h):=Sf\circ h(Dh)^2

and we seek for a solution {h} of {\Phi(f,h)=Sh\circ R_\alpha-Sh}.

As we learn in ODE courses, we need good (Banach) functional spaces to perform fixed-point arguments. In this direction, we consider the spaces {\textrm{Diff}^r_0(S^1)} of {C^r} circle diffeomorphisms {h} with {\int_{S^1}(h-id)=0} and {C^r_0(S^1)} of {C^r} functions on {S^1} with zero mean (in addition to the spaces {\textrm{Diff}^r(S^1)} of {C^r} circle diffeomorphisms and {C^r(S^1)} of {C^r} functions on {S^1}).

We begin with two simple exercises:

Exercise 1 Show that {\Phi:\textrm{Diff}^{r+3}(S^1)\times\textrm{Diff}^r_0(S^1)\rightarrow C^{r-1}(S^1)}, {\Phi(f,h)=(Sf\circ h)(Dh)^2}, is a {C^1} map whose differential at {(R_\alpha,id)} is\displaystyle D\Phi(R_\alpha,id)(\delta f,\delta h) = D^3\delta f.

Exercise 2 Show that {\mathcal{S}:\textrm{Diff}^{r}_0(S^1)\rightarrow C^{r-3}_0(S^1)} (defined by {\mathcal{S}h:= Sh-\int_{S^1}Sh}) is a {C^{\infty}} map and {\mathcal{S}} is a diffeomorphism near the identity. (Hint: {\mathcal{S}} is {C^{\infty}} because {\log} is a {C^{\infty}} function. Furthermore, the derivative of {\mathcal{S}} at the identity is {D\mathcal{S}(id)(\delta h) = D^3\delta h}, so that the inverse function theorem guarantees that {\mathcal{S}} is a local diffeomorphism near the identity).

In the sequel, the local inverse of {\mathcal{S}} near identity is denoted by {\mathcal{P}} (the letter P stands for “primitive”) and its derivative at {0} is denoted by {P}.

To reinforce our arsenal of operators, we use the theorem 2 to construct {L:C^{r-1}(S^1)\rightarrow C_0^{r-3}(S^1)} such that, for every {\phi\in C^{r-1}(S^1)},

\displaystyle \phi = \int_{S^1}\phi + L(\phi)\circ R_\alpha - L(\phi)

that is, {L(\phi)} is the solution of the cohomological equation with initial data {\phi}. Observe that the theorem 2 ensures that {L} is a bounded operator because, by hypothesis, {\alpha\in DC(\gamma,\tau)}, {0\leq\tau<1}.

In this notation, a fixed point {h} of the operator {\mathcal{P}L\Phi(f,.):\textrm{Diff}^r_0(S^1)\rightarrow\textrm{Diff}_0^r(S^1)} solves the cohomological equation (2) modulo the constant {c=\int_{S^1}\Phi(f,h)}, i.e., {h} verifies

\displaystyle (Sf\circ h)(Dh)^2 = c + Sh\circ R_\alpha - Sh,

that is,

\displaystyle S(f\circ h) = c + S(h\circ R_\alpha). \ \ \ \ \ (3)

By choosing {t',t''\in\mathbb{R}} conveniently, we have the normalization {R_{t'}\circ f\circ h\in\textrm{Diff}_0^{r}(S^1)} and {R_{t''}\circ h\circ R_\alpha\in\textrm{Diff}_0^{r}} (to kill off the averages). On the other hand, the equation (3) says that

\displaystyle \mathcal{S}(R_{t'}\circ f\circ h) = \mathcal{S}(R_{t''}\circ h\circ R_\alpha).

It follows from the exercise 2 that {R_{t'}\circ f\circ h=R_{t''}\circ h\circ R_\alpha}, i.e.,

\displaystyle R_t\circ f\circ h= h\circ R_\alpha,

where {t=t'-t''}.

Hence, the proof of the main theorem will be complete once we can find fixed points of the operator {\mathcal{P}L\Phi(f,.)}. Keeping this goal in mind, we note that the differential of {\mathcal{P}L\Phi} at {(R_\alpha,id)} is

\displaystyle (\delta f,\delta h)\mapsto PL(D^3\delta f).

Because this linear map is a (super) contraction on the variable {\delta h}, it follows from the implicit function theorem that {\mathcal{P}L\Phi(f,.)} has a unique fixed point {h=h(f)} close to the identity for every {f} sufficiently close to {R_\alpha} (and the map {f\mapsto h(f)} is {C^1}). This ends the post.

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