Fibered rotation vector and hypoellipticity for quasiperiodic cocycles in compact Lie groups

TL;DR

This paper introduces a fibered rotation vector for quasiperiodic cocycles in compact Lie groups and proves that Diophantine conditions on this vector imply smooth reducibility, linking dynamical systems and hypoellipticity.

Contribution

It defines a fibered rotation vector for quasiperiodic cocycles and establishes a hypoellipticity result under Diophantine conditions, extending the Greenfield-Wallach conjecture.

Findings

Fibered rotation vector is well-defined for almost reducible cocycles.

Diophantine rotation vectors imply smooth reducibility of cocycles.

Establishes a connection between dynamical properties and hypoellipticity in PDEs.

Abstract

Using weak solutions to the conjugation equation, we define a fibered rotation vector for almost reducible quasi-periodic cocycles in $\mathbb{T}^{d} \times G$, $G$ a compact Lie group, over a Diophantine rotation. We then prove that if this rotation vector is Diophantine with respect to the rotation in $\mathbb{T}^{d}$, the cocycle is smoothly reducible, thus establishing a hypoellipticity property in the spirit of the Greenfield-Wallach conjecture in PDEs.

Equations

$$\mathrm{H}\circ {\Phi }_1={\Phi }_2\circ \mathrm{H}$$

$$\Phi .(x,S)\mapsto (x+\alpha ,A(x).S)$$

$$S{W}_{\alpha }^{\infty }({\mathbb{T}}^d,G)={\pi }^{-1}(\{\alpha \})$$

$$A_1(\cdot )=H(\cdot +\alpha ).A_2(\cdot )H^{\ast }(\cdot )$$

$$(x,y)\mapsto (x+\alpha ,y+x)$$

$$\left(\begin{array}{c}\omega \\ F(\cdot )\end{array}\right)\in {\mathbb{T}}^d\times g,F(\cdot ):{\mathbb{T}}^d\to g$$

$$Ad(A).j_{\rho }=2i\pi \rho (a)j_{\rho },\forall \rho \in \Delta$$

$$\mathbb{R}{e}_{\rho }\oplus \mathbb{C}{j}_{\rho }\equiv \mathbb{R}\times \mathbb{C}\approx ({\mathbb{R}}^3,\cdot ,\times )\approx (su(2){)}_{\rho }$$

$$\exp (2i\pi \rho (a))=1,\forall \rho \in \Delta$$

$${\mathcal{D}}^{\prime }={\mathcal{D}}^{\prime }({\mathbb{T}}^d,G)={\cup }_{s\in \mathbb{R}}{H}^s({\mathbb{T}}^d,G)$$

$$P.v=f$$

$$|\alpha \cdot k{|}_{\mathbb{Z}}\ge \frac{{\gamma }^{-1}}{|k{|}^{\tau }}$$

$$|\beta -\alpha \cdot k{|}_{\mathbb{Z}}\ge \frac{{\tilde{\gamma }}^{-1}}{|k{|}^{\tilde{\tau }}}$$

$$\beta =k_r\cdot \alpha \mathrm{mod}\mathbb{Z}$$

$${\mathrm{Conj}}_{H_i(\cdot )}{\Phi }_1\stackrel{C^{\infty }}{\to }{\Phi }_2\text{ and }{H}_i(\cdot )\stackrel{H^s}{\to }H(\cdot )$$

$${\mathrm{Conj}}_{H_i(\cdot )}{\Phi }_1\stackrel{C^{\infty }}{\to }(\alpha ,A_{\infty })\text{ and }{H}_i(\cdot )\stackrel{H^s}{\to }H(\cdot )$$

$$({\Phi }_i{)}^{-1}\circ {\mathrm{Conj}}_{H_i(\cdot )}\Phi \stackrel{C^{\infty }}{\to }\mathrm{Id}\in S{W}^{\infty }({\mathbb{T}}^d,G)$$

$$A_i^{\ast }.H_i(\cdot +\alpha ).A(\cdot ).H_i^{\ast }(\cdot )\stackrel{C^{\infty }}{\to }\mathrm{Id}\in G$$

$$\Phi ={\mathrm{Conj}}_{\mathrm{H}}(\alpha ,A_{\infty })$$

$$d({\Lambda }_i,A_{n_i})<N_{n_i}^{-\nu }$$

$$\Vert Y_n(\cdot ){\Vert }_s=O(N_n^{-\infty }),\forall s\ge 0$$

$$B_{n_i}({\mathbb{Z}}^d)\in \mathcal{Z}\text{ and }\Vert B_{n_i}(\cdot ){\Vert }_s\simeq C_s{N}_{n_i}^{s+\lambda },\forall s\ge 0$$

$$B_{n_i}(\cdot +\alpha ).{\Lambda }_{n_i}.B_{n_i}^{-1}(\cdot )$$

$$\Vert F_n(\cdot ){\Vert }_s=O(N_n^{-\infty }),\forall s\ge 0$$

$$H_n(\cdot )=\{\begin{array}{l}{e}^{Y_n(\cdot )}{H}_{n-1}(\cdot )\text{ if }n\notin \{n_i\}\\ B_n(\cdot )e^{Y_n(\cdot )}{H}_{n-1}(\cdot )\text{ if }n\in \{n_i\}\end{array}$$

$$H_n(\cdot +\alpha )A{e}^{F(\cdot )}{H}_n^{-1}(\cdot )=A_n.e^{F_n(\cdot )}$$

$$B_i(x)=\exp (D_i).\exp (\sum _{\rho }2i\pi (k_i^{\rho }\cdot x)e_{\rho })\exp (-D_i)$$

$$H(\cdot )\mapsto H(\cdot +\alpha ).A(\cdot )H^{\ast }(\cdot )$$

$${\Lambda }_i\to A_{\infty }\in \mathcal{T}$$

$${\Lambda }_i^{\ast }{A}_i(\cdot )\stackrel{C^{\infty }}{\to }\mathrm{Id}$$

Full text

Keywords: Greenfield-Wallach conjecture, Hypoelliptic vector fields, KAM theory, Compact Lie groups, Cohomological equations.

AMS Classification: 37C55, 35H10

1. 1 Introduction

2. 2 Notation

3. 2.1 Lie groups

4. 2.2 Functional analysis

5. 2.3 Arithmetics

6. 2.4 Cocycles

7. 3 Statement of results

8. 4 The basic lemmas

9. 5 Existence and well-posedness of the rotation vector

10. 6 Proof of theorem 3.6

11. 7 Some comments

1 Introduction

In this note, we are interested in the regularity of the solutions $\mathrm{H}$ to the conjugation equation for dynamical systems,

[TABLE]

where the $\Phi_{i}$ are two $C^{\infty}$ smooth diffeomoprhisms of a compact manifold $\mathcal{M}$ and $\mathrm{H}$ can be a smooth or finitely differentiable diffeomorphism, a homeomorphism, or a measurable self-mapping well defined and invertible $a.e.$. We are interested in particular in definining weak solutions $\mathrm{H}$ in $\mathcal{D}^{\prime}$ and then imposing a condition on, say, $\Phi_{2}$ which guarantees $C^{\infty}$ smoothness for $\mathrm{H}$.

In our case, the manifold $\mathcal{M}$ will be the product space $\mathbb{T}^{d}\times G$, with $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ and $G$ a compact Lie group. We will not consider the full space of diffeomorphisms of this space, but work in the space of smooth quasi-periodic cocycles, denoted by $SW^{\infty}=SW^{\infty}(\mathbb{T}^{d},G)$. A cocycle $\Phi=(\alpha,A(\cdot))\in SW^{\infty}$, where $A(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$, acts on $\mathbb{T}^{d}\times G$ by

[TABLE]

The space $SW^{\infty}$ has the natural product topology, and is thus a Fréchet space. The dynamics of a cocycle fiber over $\mathbb{T}^{d}$. We define the projection $\pi$ of the cocycle on its frequency, $\pi(\Phi)=\alpha$, and introduce the notation

[TABLE]

As the rotation $\alpha$ will be fixed throughout the article, we take the liberty to make no distinction between the mapping $A(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ and the cocycle over $\alpha$ that it defines.

The relevant notion of conjugation in $SW^{\infty}$ is fibered conjugation, via the action of the subgroup $SW^{\infty}_{0}(\mathbb{T}^{d},G)$. The resulting conjugation equation, since conjugation does not act on frequences, is

[TABLE]

where we suppose that the cocycle $\mathrm{H}=(0,H(\cdot))$ conjugates the cocycles $\Phi_{j}=(\alpha,A_{j}(\cdot))$, and the unknown is the mapping $H(\cdot)$. We will use the notation $\Phi_{1}=\mathrm{Conj}_{H(\cdot)}\Phi_{2}=\mathrm{Conj}_{\mathrm{H}}\Phi_{2}$. In the particular case where $\Phi_{2}$ is a constant cocycle, i.e. where $A_{2}(\cdot)\equiv A_{2}\in G$ is a constant mapping, we say that $\Phi_{1}$ is reducible.

In our previous paper [Kar17b] we showed that a cocycle in an open subset of $SW_{\alpha}^{\infty}$ with $\alpha$ Diophantine, is $C^{\infty}$ reducible provided that it is measurably conjugate to a constant cocycle $(\alpha,A_{\infty})$, $A_{\infty}\in G$, whose eigenvalues $a_{\infty}$ satisfy a Diophantine condition with respect to $\alpha$. Our subsequent work, [Kar14] and [Kar18a], shows that the Diophantine condition on $A_{\infty}$ is optimal for this rigidity property.

In this note, we rid the main theorem of [Kar17b] of the ad-hoc hypothesis of the existence a measurable conjugation, by showing that any cocycle in that open subset can be assigned a fibered rotation vector. Subsequently, we show that any cocycle is reducible to a constant one with the same rotation vector, but by a transfer function with negative and finite Sobolev regularity (depending only on the dimension of the phase-space). We then use the hypoellipticity property implied by the same Diophantine hypothesis as in [Kar17b] in order to obtain a $C^{\infty}$ smooth conjugation.

The hypothesis of the existence of a measurable conjugation is, thus, totally unnatural, since for general reasons a distributional solution to the conjugation equation exists and it is sufficient in order to obtain the existence of a $C^{\infty}$ smooth solution. We remark that, by [Kar16], when $\alpha\in RCD\subset\mathbb{T}^{1}$, almost reducible cocycles form an open dense set in the space of cocycles222The group $G$ needs to be semisimple for that to hold., and that cocycles that are not almost reducible do not have a fibered rotation vector, because arbitrarily small perturbations of such cocycles admit the skew-shift mapping of $\mathbb{T}^{2}$

[TABLE]

as a factor.333Actually, use of the K.A.M. normal form and of the sequence of conjugations $H_{i}(\cdot)$, defined in eq. (4), in §10 of [Kar16] can show that such cocycles themselves admit the skew-shift as a factor. This technical issue is beyond the scope of the present note.

The results that we prove herein admit their continuous-time counter parts, where the corresponding objects are vector fields

[TABLE]

with $g$ a compact Lie algebra, which places the results directly in the context of hypoelliptic first order differential operators and the Greenfield-Wallach conjecture (problem 2 in [GW73]).

2 Notation

In order to keep this note short, we will follow the notation of our previous works that are cited herein, and whenever some deviaton from that notation is needed, it will be made explicit.

2.1 Lie groups

We denote by $G$ a compact Lie group, by $g$ its Lie algebra, and by $d^{\prime}$ its dimension. We fix $\mathcal{T}$, a maximal torus of $G$ and denote by $\mathfrak{t}$ the Lie albebra of $\mathcal{T}$. We will denote by $\mathcal{Z}$ the lattice of preimages of the center of $G$ in $\mathfrak{t}$.

The adjoint action of $\mathcal{T}$ on $g$ admits an eigenspace decomposition, known as root space decomposition (cf. [Die75]; we will follow the notation we used in [Kar16] and [Kar17b]). There exists a finite set $\Delta\subset\mathfrak{t}^{*}$ of non-zero $\mathbb{R}$-valued linear forms on $\mathfrak{t}$ and vectors $j_{\rho}\in\mathfrak{t}^{\perp}$, for $\rho\in\Delta$ for which the following holds.

If $A\in\mathcal{T}$ and $a\in\mathfrak{t}$ is such that $A=\exp(a)$, then

[TABLE]

Moreover, there exists a vector $e_{\rho}\in\mathfrak{t}$ such that

[TABLE]

is naturally isomorphic to $\mathbb{R}^{3}$ equipped with its scalar and vector product, and the Lie bracket of $g$ restricted to $\mathbb{R}e_{\rho}\oplus\mathbb{C}j_{\rho}$ is mapped to the vector product of $\mathbb{R}^{3}$. This Lie algebra is naturally isomorphic to $su(2)$, the Lie algebra of $SU(2)$, the group of special $2\times 2$ unitary matrices.

The vectors $e_{\rho}$ spam $\mathfrak{t}$, and $A\in Z_{G}$, the centre of $G$, iff

[TABLE]

This condition defines a lattice $\mathcal{Z}\subset\mathfrak{t}$ which is fixed along with $\mathfrak{t}$.

2.2 Functional analysis

We will denote by $\|\cdot\|_{s}$ the norm of the Sobolev space $H^{s}=H^{s}(\mathbb{T}^{d},G)$. The space of distributions will be denoted by

[TABLE]

The following definition is from [GW73].

Definition 2.1**.**

A differential operator $P$ is called Globally Hypoelliptic if

[TABLE]

with $v\in\mathcal{D}^{\prime}$ and $f\in C^{\infty}$ implies that $v\in C^{\infty}$.

In our context, the differential operator is replaced by the conjugation operator, the operator that defines the conjugation equation, eq. (1). This operator behaves like a differential operator, and its equivalent in the continuous time case, i.e. in the study of vector fields, is actually the Lie derivative with respect to a vector field.

2.3 Arithmetics

Definition 2.2**.**

We will denote by $DC(\gamma,\tau)$ the set of numbers $\alpha$ in $\mathbb{T}^{d}$ such that for any $k\in\mathbb{Z}^{d}\setminus\{0\}$,

[TABLE]

Numbers satisfying such a condition are called Diophantine.

The set $DC(\gamma,\tau)$, for $\tau>d+1$ fixed and $\gamma\in\mathbb{R}_{+}^{\ast}$ is of positive Haar measure in $\mathbb{T}^{d}$. If we fix $\tau$ and let $\gamma$ run through the positive real numbers, we obtain $CD(\tau)=\cup_{\gamma>0}DC(\gamma,\tau)$ which is of full Lebesgue measure. The numbers that do not satisfy any Diophantine condition are called Liouvillean. They form a residual set of [math] Lebesgue measure.

We now turn to definitions concerning arithmetics relative to a rotation $\alpha$.

Definition 2.3**.**

We will denote by $DC_{\alpha}(\tilde{\gamma},\tilde{\tau})$ the set of numbers $\beta$ in $\mathbb{T}$ such that for any $k\in\mathbb{Z}^{d}\setminus\{0\}$,

[TABLE]

Such numbers are called Diophantine with respect to $\alpha$.

In particular, $\alpha\in DC$ iff $0\in DC_{\alpha}$.

Definition 2.4**.**

We will denote by $Res_{\alpha}$ the set of numbers $\beta$ in $\mathbb{R}$ for which there exists $k_{r}\in\mathbb{Z}^{d}\setminus\{0\}$ such that

[TABLE]

Such numbers are called Resonant with respect to $\alpha$.

We end this block of definitions with some definitions concerning arithmetics of elements of $G$.

Definition 2.5**.**

We will denote by $DC_{\alpha}$ the set of elements $S\in G$ whose roots are Diophantine with respect to $\alpha$.

Definition 2.6**.**

We will denote by $Res_{\alpha}$, the set of elements $S\in G$ having at least one root that is resonant with respect to $\alpha$.

We will also use the terms Resonant or Diophantine roots, by obvious extension of the respective concepts.

We finally recall the notion of Recurrent Diophantine numbers, defined only when $d=1$.

Definition 2.7**.**

We will denote by $RDC(\gamma,\tau)$ the set of recurrent Diophantine numbers, i.e. the $\alpha$ in $\mathbb{T}\setminus\mathbb{Q}$ such that $\mathrm{G}^{n}(\alpha)\in DC(\gamma,\tau)$ for infinitely many $n$.

In purely local notation, $\mathrm{G}(\alpha)=\{\alpha^{-1}\}$ is the Gauss map ($\{\cdot\}$ stands for ”fractional part”). The set $RDC$ is also of full measure.

2.4 Cocycles

The dynamics and the notion of conjugation having already been introduced in §1, we give directly the following definition on conjugation.

Definition 2.8**.**

Two cocycles $\Phi_{j}\in SW^{\infty}_{\alpha}(\mathbb{T}^{d},G)$, $j=1,2$, are $H^{s}$ conjugate, with $s\in\mathbb{R}$, iff there exists a sequence $H_{i}(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ of conjugations such that

[TABLE]

This definition coincides with the classical one when $s\geq 0$, but is also meaningful for $s<0$.

Definition 2.9**.**

A cocycle $\Phi\in SW^{\infty}_{\alpha}(\mathbb{T}^{d},G)$, is $H^{s}$ reducible, with $s\in\mathbb{R}$, iff there exists a sequence $H_{i}(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ of conjugations and a constant $A_{\infty}$ such that

[TABLE]

Following [Eli02], we define Almost Reducibility.

Definition 2.10**.**

A cocycle $\Phi\in SW^{\infty}_{\alpha}(\mathbb{T}^{d},G)$, is almost reducible, iff there exists a sequence $H_{i}(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ of conjugations and a sequence of constant cocycles $\Phi_{i}$ such that

[TABLE]

Almost Reducibility thus amounts, with some obvious notation, to

[TABLE]

The first result of the present paper is the justification of the following definition.

Definition 2.11**.**

Let $\alpha\in DC$ and $\Phi\in SW^{\infty}_{\alpha}$ be almost reducible. Let also $H(\cdot)\in H^{s}$ for some $s\in\mathbb{R}$ and let $A_{\infty}=\exp(a_{\infty})\in\mathcal{T}$ with $a_{\infty}\in\mathfrak{t}$ be such that

[TABLE]

Then, we will say that $a_{\infty}$ is a rotation vector for $\Phi$ and write $a_{\infty}=\varrho(\Phi)$.444We point out the trouble in notation. We use $\rho$ for the roots of an element of $G$ and $\varrho$ for the rotation vector. Both quantities are customarily denoted by $\rho$, and we apologize for this notation.

We also point the reader to §6, theorem 6.1 of [Kar17a] for the following statement considering Almost Reducible cocycles.

Theorem 2.1** ([Eli02], [Kri99], [Kar16]).**

Let $G$ be a compact Lie group, and $\alpha\in DC(\gamma,\tau)$. Then, Almost Reducibility holds in an open subset of $SW^{\infty}(\mathbb{T}^{d},G)$.

More precisely, the K.A.M. scheme that proves Almost Reducibility produces:

a fast increasing sequence $N_{n}\in\mathbb{N}^{*}$, $N_{n+1}=N_{n}^{1+\sigma}$ for some $0<\sigma<1$ 2. 2.

a sequence of constants $A_{n}\in G$ 3. 3.

a number $M\in\mathbb{N}\cup\{\infty\}$ of resonant steps, a number $\nu>\tau$, and a subsequence $n_{i}$ of resonant steps where $A_{n_{i}}\notin DC_{\alpha}(1,\nu)$, $1\leq i<M$ 4. 4.

a sequence of resonant constants $\{\Lambda_{i}\}_{i=1}^{M}\subset Res_{\alpha}\subset G$ whose resonance $k_{n_{i}}^{\rho}\in\mathbb{Z}^{d}\setminus\{0\}$ in each resonant subalgebra $(su(2))_{\rho}$, satisfies $|k_{n_{i}}^{\rho}|\leq N_{n_{i}}$, as well as the following properties. The constant $\Lambda_{i}$ commutes with $A_{n_{i}}$ and

[TABLE]

if such an $\Lambda_{n_{i}}$ exists. The number $M\in\mathbb{N}\cup\{\infty\}$ counts the number of steps of the K.A.M. scheme for which $\Lambda_{n_{i}}$ is defined 5. 5.

a sequence of conjugations $\exp(Y_{n}(\cdot))\in C^{\infty}(\mathbb{T}^{d},G)$ satisfying

[TABLE] 6. 6.

a sequence of torus morphisms $B_{n_{i}}(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ satisfying

[TABLE]

for some constant $\lambda>0$. The $B_{n_{i}}(\cdot)$ commute with the respective $A_{n_{i}}$ and $\Lambda_{n_{i}}$, and the constant

[TABLE]

is $N_{n_{i}}^{-\tau}$-away from resonant constants. If $A_{n}$ is $N_{n}^{-\tau}$-away from resonant constants (i.e. if $n$ is not a resonant step) then $B_{n}(\cdot)$ is by convention defined as $\equiv\mathrm{Id}$ 7. 7.

a sequence of mappings $F_{n}(\cdot)\in C^{\infty}(\mathbb{T}^{d},g)$ satisfying

[TABLE]

and such that the conjugation constructed iteratively following $H_{0}=\mathrm{Id}$ and

[TABLE]

satisfies

[TABLE]

In this notation, the K.A.M. normal form of almost reducible cocycles (see [Kar17b], [Kar14] and [Kar18a] for a figure) is established as follows.

Lemma 2.2**.**

Under the assumptions of theorem 2.1, there exists a conjugation $D(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ such that the K.A.M. scheme applied to the cocycle $\mathrm{Conj}_{D(\cdot)}\Phi$ produces only resonant steps.

For a cocycle in normal form, we abbreviate $n_{i}$ to $i$. For a cocycle in normal form, all close-to-the-$\mathrm{Id}$ conjugations of theorem 2.1 are $\exp(Y_{n}(\cdot))\equiv\mathrm{Id}$. The conjugations $B_{i}(\cdot)$ are555Essentially, but the difference is cumbersome to write down and irrelevant to the goal of the note. The reader can consult §9 of [Kar16] for the details. of the form

[TABLE]

where $\exp(D_{i})$ conjugates a maximal torus passing by $A_{i}$ to a maximal torus passing by $A_{i+1}$.

We point out that if the K.A.M. normal form is finite, i.e. if $M<\infty$, then $\Phi$ is $C^{\infty}$ reducible, since the sequence $H_{n}(\cdot)$ converges in $C^{\infty}$.

3 Statement of results

Assumption 3.1**.**

In all following theorems, $\alpha$, the frequency of the cocycle $\Phi=(\alpha,A(\cdot))$, is Diophantine. The cocycle $\Phi\in SW^{\infty}_{\alpha}(\mathbb{T}^{d},G)$ is assumed to be almost reducible.

Our first theorem implies that definition 2.11 is not void.

Theorem 3.2**.**

For each cocycle $\Phi$ there exists a fibered rotation vector $\varrho=\varrho(\Phi)$ satisfying definition 2.11.

Concerning the invariance of the rotation vector under conjugations, we prove the following propositions.

Proposition 3.3**.**

The fibered rotation vector $\varrho$ is invariant under conjugation in $\mathcal{D}^{\prime}$, and well defined in $\mathfrak{t}\mod(\alpha\cdot\mathbb{Z}^{d})\mathcal{Z}$.

We remark that the same phenomenon of indeterminacy occurs for rotation vectors of diffeomorphisms of the torus $\mathbb{T}^{d}$, $d\geq 2$, when we allow non homotopic to the $\mathrm{Id}$ conjugations to act.

Proposition 3.4**.**

The fibered rotation vector $\varrho$ is invariant under conjugations $B(\cdot)$ for which there exists $b(\cdot):\mathbb{T}^{d}\rightarrow g$ such that $B(\cdot)=\exp(b(\cdot))$. Representatives of $\varrho(\Phi)$ depend continuously on $\Phi$ in the $C^{0}$ topology for fixed $\pi(\Phi)$.

We remark that a mapping $\mathbb{T}^{d}\rightarrow G$ may lift to a mapping $\mathbb{R}^{d}\rightarrow g$, for example the conjugations $B_{i}(\cdot)$ of theorem 2.1. Conjugations in the space of cocycles that admid a bounded lift in $g$ play the role of homotopic-to-the-$\mathrm{Id}$ conjugations for torus diffeomorphisms.

Theorem 3.5**.**

The fibered rotation vector $\varrho$ is a total invariant for $H^{-d-d^{\prime}}$ conjugation: every almost reducible cocycle is $H^{-d-d^{\prime}}$ conjugate to a constant one with the corresponding rotation vector.

We finally prove the following hypoellipticity theorem, stating that under some relevant arithmetic assumptions, the conjugation of theorem 3.5 is actually smooth.

Theorem 3.6**.**

If $\varrho=\varrho(\Phi)$ satisfies a Diophantine Condition with respect to $\alpha$, then $\Phi$ is smoothly conjugate to (any representative of) $(\alpha,\exp(\varrho))$.

In particular, we obtain the following affirmative answer to problem 2 of [GW73] in our context. We remind that the conjugation operator associated to a cocycle $\Phi=(\alpha,A(\cdot))$ is the operator acting on $C^{\infty}(\mathbb{T}^{d},G)$ by

[TABLE]

Theorem 3.6 is thus the non-linear analogue of the main theorem in [Kar14].

Corollary 3.7**.**

If the conjugation operator of an almost reducible cocycle (or quasi-periodic flow) in $\mathbb{T}^{d}\times G$, $G$ a compact Lie group, is Globally Hypoelliptic, then $G$ is a torus.

Some technicalities aside (that need to be settled), the same holds in the total space $SW^{\infty}_{\alpha}(\mathbb{T},G)$ of one-frequency cocycles over $RDC$ rotations (see the discussion preceeding eq. (2)), and we expect the same to hold for cocycles over all irrational rotations.

Acknowledgement: This work was supported by the LABEX CEMPI (ANR-11-LABX-0007-01).

4 The basic lemmas

The results proving the well-posedness of the definition of $\varrho$ are proved via the following sequence of lemmas. The reader not familiar with the structure of Lie groups can replace $G$ with $G=SU(2)$ in order to keep algebra simple. As in [Kar17b], this makes the arguments more transparent without harming generality.

Assumption 4.1**.**

In the lemmas throughout the section, we assume that the quantity $M$ of theorem 2.1 is infinite, in order to treat the difficult and interesting case.

Lemma 4.2**.**

Let $\Phi=(\alpha,A(\cdot))$ be an almost reducible cocycle. Then, there exists $H_{i}(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ a sequence of smooth conjugations converging in $H^{-d-d^{\prime}}$ and a sequence $(\alpha,\Lambda_{i})$ of resonant constant cocycles, $\Lambda_{i}\in\mathcal{T}$, such that

[TABLE]

and, if we call $(\alpha,A_{i}(\cdot))=\mathrm{Conj}_{H_{i}(\cdot)}\Phi$, then

[TABLE]

This is the form of theorem 2.1 that we will use in the proof of theorem 3.6.

Proof.

In the notation of [Kar18a], §3.4.3, or theorem 2.1, assume that the cocycle is in K.A.M. normal form. Then, the conjugations

[TABLE]

with notation as in item 3, §5.1 op.cit.666There is a typo in the reference, the correct index for the matrix $D_{i}$ is $i$ and not $i+1$. and eq. (3), satisfy

[TABLE]

and they have only the resonant Fourier coefficients, precisely $k_{i}^{\rho}$. This sequence is summable in, say, $H^{-d-d^{\prime}}$. This allows us to conclude, since the conjugations

[TABLE]

conjugate the cocycle $(\alpha,A(\cdot))$ to777Notice the abuse of notation. We also call the new sequence of cocycles $(\alpha,A_{i}(\cdot))$.

[TABLE]

with

[TABLE]

and

[TABLE]

This ends the proof. ∎

Lemma 4.3**.**

Let $(\alpha,A_{i})$, $j=1,2$ be two constant cocycles, $A_{i}\in\mathcal{T}$, that are conjugate in $\mathcal{D}^{\prime}$. Then, they are $C^{\infty}$ conjugate via a torus morphism $B(\cdot):\mathbb{T}^{d}\rightarrow\mathcal{T}$. Moreover,

[TABLE]

Proof.

It is §5.1 of [Kar17b] where the measurability assumption on the conjugation is unnecessary, as the Fourier transform is well defined for distributions. ∎

Lemma 4.4**.**

Let $H_{i}^{\prime}(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ converge in some $H^{s}$, and $\Lambda_{i}^{\prime}\in\mathcal{T}$ be a sequence of constants, both otherwise satisfying the same conclusions as in lemma 4.2 with $A_{\infty}^{\prime}$ replacing $A_{\infty}$. Then

[TABLE]

and there exists a $C^{\infty}$ smooth conjugation $D(\cdot)$ such that

[TABLE]

In particular, $H_{i}^{\prime}(\cdot)$ also converges in $H^{-d-d^{\prime}}$.

Proof.

The proof follows directly from §5.1 of [Kar18a]. Assume without loss of generality that both sequences of conjugations $H_{i}(\cdot)$ and $H_{i}^{\prime}(\cdot)$ correspond to K.A.M. normal forms of the cocycle, in which case the conjugations $H_{i}(\cdot)$ and $H_{i}^{\prime}(\cdot)$ are given by eq. (4).

If a product of admissible conjugations between K.A.M. normal forms converges in $\mathcal{D}^{\prime}$, then it contains a finite number of conjugations of the type denoted by $B_{i}^{\prime}(\cdot)$ as in eq. (3), for, otherwise, the product diverges in $\mathcal{D}^{\prime}$. Products of all other admissible conjugations converge in $H^{-d-d^{\prime}}$.

Since all other admissible conjugations do not modify the limit of the sequences $\Lambda_{i}\rightarrow A_{\infty}$ and $\Lambda_{i}^{\prime}\rightarrow A_{\infty}^{\prime}$, and the conjugation $D(\cdot)$ is given by the product of finitely many torus morphisms $\mathbb{T}^{d}\rightarrow\mathcal{T}$ and conjugations converging in $H^{-d-d^{\prime}}$, the proof is complete. ∎

Lemma 4.5**.**

Let $H_{i}(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$, $\Lambda_{i}\in\mathcal{T}$ and $A_{\infty}$ be as in lemma 4.2. If $H_{i}^{\prime}(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ and $\Lambda_{i}^{\prime}\in\mathcal{T}$ with $\Lambda_{i}^{\prime}\rightarrow A_{\infty}^{\prime}$ satisfy definition 2.10, but

[TABLE]

then $H_{i}^{\prime}(\cdot)$ does not converge in $\mathcal{D}^{\prime}$.

Proof.

Follows directly from the proof of lemma 4.4, since for eq. (5) to hold, infinitely many conjugations of the type $B_{i}^{\prime}(\cdot)$ must occur, in which case the product does not converge in any distribution space, and such conjugations have optimal Sobolev norms. ∎

Lemma 4.6**.**

If $H_{i}^{\prime}(\cdot)\in C^{\infty}(\mathbb{T}^{d},G)$ and $\Lambda_{i}^{\prime}\in\mathcal{T}$ satisfy definition 2.10, except that $\Lambda_{i}^{\prime}$ does not converge, then $H_{i}^{\prime}(\cdot)$ does not converge in $\mathcal{D}^{\prime}$.

Proof.

The same line of argument as in the proof of lemma 4.5. ∎

5 Existence and well-posedness of the rotation vector

In this section, we provide the proofs of theorem 3.2, propositions 3.3 and 3.4, and theorem 3.5. Taken altogether, they establish that the fibered rotation vector of an almost reducible cocycle, cf. definition 2.11, is a well-defined object and a total invariant of the dynamics, albeit in low regularity.

The well-posedness of the definition amounts to proving a weak form of rigidity for the conjugations produced by the K.A.M. scheme, and to providing a classification of the possible limits of the sequences of constants under this weak rigidity assumption.

Proof of theorem 3.2.

Lemma 4.2 proves the theorem, which in turn implies that definition 2.11 is not void. ∎

Proof of proposition 3.3.

The proof follows by combining lemmas 4.4, 4.5 and 4.6 ∎

Proof of proposition 3.4.

The proof follows by combining lemmas 4.3, 4.4 and 4.5.

The continuous $C^{0}$ dependence of representatives of $\varrho$ follows from the K.A.M. normal form. Fix two almost reducible cocycles $\Phi^{(j)}$ that are $C^{0}$ close, and assume without loss of generality that $\Phi^{(1)}$ is given in K.A.M. normal form. Denote by $\varrho_{j}$ their respective rotation vectors.

The K.A.M. normal form of $\Phi^{(1)}$ and the corresponding sequence of conjugations as in eq. (4) fixes a representative $a_{\infty}^{(1)}$ of $\varrho_{1}$. Then, by following the K.A.M. scheme applied to $\Phi^{(2)}$, and by constructing the corresponding sequence of conjugations as in eq. (4) one can obtain a representative $a_{\infty}^{(2)}$ of $\varrho_{2}$ whose distance to $\varrho_{1}$ is bounded by the $C^{0}$ distance of the cocycles. ∎

Proof of theorem 3.5.

It follows directly from lemma 4.2. ∎

6 Proof of theorem 3.6

The proof of theorem 1.1 of [Kar17b], applies verbatim. This is so, because the only function of the measurable conjugation is to assure that the K.A.M. scheme produces a finite number of resonances. This happens when $a_{\infty}$ ($a_{\infty}$ corresponds to $a_{d}$ of the reference) is polynomially away from resonances.

Proof of theorem 3.6.

Fix $a_{\infty}$, the representative of $\varrho(\Phi)$ obtained as the limit of cocycles constructed in the proof of lemma 4.2 assuming that $M$, the number of resonant steps, is infinite. Then, by assumption, $A_{\infty}=\exp(a_{\infty})$ satisfies

[TABLE]

We also have by construction that

[TABLE]

Assume, now, without loss of generality (cf. [Kar17b] for the same argument) that $\nu>\tilde{\tau}$. Then, we immediately get

[TABLE]

On the other hand, by construction

[TABLE]

which implies that

[TABLE]

This is a contradiction, which forces $M$ to be finite so that $\Phi$ is $C^{\infty}$ reducible. ∎

As we have already remarked, the Diophantine condition on the fibered rotation vector $\varrho$ is optimal for the hypoellipticity of the conjugation operator.

7 Some comments

Our proofs of the results needed in order to justify the definition of the rotation vector $\varrho$ rely heavily on the K.A.M. normal form, which is available only for almost reducible cocycles over Diophantine rotations. We expect, however, the rotation vector to be defined also for almost reducible cocycles over Liouvillean rotations.

The original definition of a rotation number was given by H. Poincaré for homeomorphisms of $\mathbb{T}^{1}$ (see, e.g. [KH96]), who also proved that a circle diffeomorphism with an irrational rotation number is semi-conjugate to the corresponding rotation. Subsequently, A. Denjoy proved that a twice differentiable diffeomorphism is actually conjugate to the rotation. The relevance of the arithmetics of the rotation number in questions of smoothness of the conjugation was showed by V.I. Arnol’d in [Arn61], where he proved that real analytic perturbations of a Diophantine rotation $\alpha\in\mathbb{T}$ whose rotation number is $\alpha$ are analytically conjugate to the rotation. The theory was pushed further by M. Herman and J.-C. Yoccoz, who established that the rotation number is a total invariant for smooth diffeomorphisms of the circle (cf. [Yoc95]).

Diffeomorphisms of tori of higher dimension can be assigned rotation sets rather than vectors, see [MZ91] for the definition. The analog of Arnol’d’s theorem for higher dimensional tori is proved in [Kar18b].

In the class of fibered dynamics, fibered rotation numbers can be defined for quasi-periodically forced circle maps. These are diffeomorphisms of $\mathbb{T}^{2}$ that are homotopic to the $\mathrm{Id}$ and of the form

[TABLE]

where for each $x\in\mathbb{T}$ the mapping $y\mapsto g(x,y)$ is a diffeomorphism of $\mathbb{T}^{1}$ In this case, the arithmetics of the rotation number in the $y$ component, the fibered rotation number, play an important role in the regularity of the reducing conjugation (see, e.g. [JS06] for a discussion).

For the same reason as for quasi-periodically forced circle maps, quasi-periodic cocycles in $\mathbb{T}^{d}\times SL(2,\mathbb{R})$ that are homotopic to the $\mathrm{Id}$ can be assigned a fibered rotation number by projectivization in the $SL(2,\mathbb{R})$ direction. The importance of the aritmetics of the fibered rotation number with respect to the rotation in the basis are exhibited by results like the one obtained by H. Eliasson in [Eli92].

For all of the above systems, the rotation number or set, fibered or not, is defined by dynamics alone. However, in the context of this note we do not dispose of combinatorial arguments, or, at least a priori, of any torus where the dynamics naturally lives.

The rotation vector can nonetheless be defined, because we manage to conjugate any given cocycle close to a converging sequence of constant cocycles, all belonging to the same torus, and without losing too much control of the sequence of conjugations. We think that this should also be the case when the rotation in the basis is Liouvillean, in which case a rotation vector would be defined for an open set of cocycles over all minimal rotations.