A Rigidity Property of Perturbations of n Identical Harmonic Oscillators

Abstract

Let $X_{\varepsilon } : S^{2n-1} \rightarrow T S^{2n-1}$ be a smooth perturbation of $X_0$, the vector field associated to the dynamical system defined by n identical uncoupled harmonic oscillators constrained to their 1-energy level. We are dealing with the case when any orbit of every $X_{\varepsilon }$ is closed: while in general is false that the vector fields of the perturbation are orbitally equivalent to the unperturbed $X_0$ (Villarini in Ergod Theory Dyn Syst 39:1–32, 2019), we prove that this rigidity behaviour is indeed true if each $X_{\varepsilon }$ restricted to a codimension 2 sphere in $S^{2n-1}$ is orbitally conjugated to a subsystem of $X_0$ made by $n-1$ harmonic oscillators. In other words: to have a non-rigid, or truly non-linear, perturbation of $X_0$ at least two harmonic oscillators must be destroyed by the perturbation. We use this rigidity result to prove a linearization theorem for real analytic multicentres. Finally we give an example of a real analytic perturbation of $X_0$ showing discontinuous changing of integer invariants of the vector fields of the perturbation.

4. Appendix A

A standard tool in several fields of mathematics, e.g. in algebraic geometry, see [7], the blowing up construction will be here described in the context of blowing up a compact submanifold $\Gamma$, embedded in the total space P of a circle bundle

$$\begin{aligned} \xi _0 : S^1 \hookrightarrow P \rightarrow ^{\pi } M \end{aligned}$$

and fibered by the bundle’s fibers. We will pay special attention to lifting vector fields $\varepsilon \rightarrow X_{\varepsilon } : P \rightarrow TP$ such that $X_0$ is the infinitesimal generator of the $S^1$ action associated to $\xi _0$ and $\Gamma$ is a tangency manifold of this perturbation i.e.

$$\begin{aligned} {X_{\varepsilon }}_{\vert \Gamma } \equiv {X_0}_{\vert \Gamma } \end{aligned}$$

We will put special emphasis on the smooth, and even $C^k$-case, $k\ge 3$: the real analytic case is settled by considering the holomorphic version of the blow up applied to complex analytic extension of the real analytic objects (manifolds, vector fields), and using the Riemann Extension Theorem, see e.g. [7].

Topics to be considered below can be found, in the case of blowing up a point, in [11] or in Dumortier’s article in [5]: the generalization of these results to the blowing up along a submanifold, though direct, is essential for our article and is therefore presented here.

On the other hand, the results and methods of proof, being of local nature, are not changed from the general case to the case when P is a 3-manifold and $\Gamma \simeq S^1$, and therefore we restrict our study to that case, which is the one we are interested in the application of the present article.

Blowing up $\xi _0$ along $\Gamma$ means the definition of the triple $({\tilde{P}} , \sigma , \tilde{X}_0)$ where

• (i) ${\tilde{P}}$ is a smooth manifold

• (ii) $\sigma : {\tilde{P}} \rightarrow P$ is a smooth map, $\sigma _{\vert } : {\tilde{P}} - {\mathcal {E}} \rightarrow P - \Gamma$ is a diffeomorphism, ${\mathcal {E}}=\sigma ^{-1}(\Gamma )$, is the total space of the projectivized normal bundle to $\Gamma$ (see below)

• (iii) $\tilde{X}_0$ defines a smooth foliation by circles of ${\tilde{P}}_{\gamma _0}$ such that $\sigma _{\vert } (\tilde{{\mathcal {F}}}_0) = {\mathcal {F}}_0$ on ${\tilde{P}}_{\gamma _0} - {\mathcal {E}}$.

We recall that the normal bundle to $\Gamma$ in P is

$$\begin{aligned} {\left\{ \begin{array}{ll} \rho : &{} N_{ \Gamma } \rightarrow \Gamma \\ N_{ \Gamma } &{}= \frac{(TP)_{\vert \Gamma }}{T\Gamma } \end{array}\right. } \end{aligned}$$

and its projectivization is the projective bundle

$$\begin{aligned} \mathbb {P}(N \Gamma ) \rightarrow ^{\pi _{\Gamma }} \Gamma . \end{aligned}$$

${\mathcal {E}} = {\pi _{\Gamma }}^{-1} (\Gamma )$ is called the divisor of the blow up.

The construction of $\tilde{P_{\gamma }}$ calls for the definition of a smooth atlas $\tilde{{\mathcal {A}}}$, definition which is based on the atlas ${\mathcal {A}}$ of P adapted to the bundle structure of $\xi _0$. To clarify this point, we consider the case when $\xi _0$ is trivialized over two sets $U_0 \times S^1$, $U_1 \times S^1$ where $U_0 = \mathbb {D}_{2 r} (q_0)$, i.e. $U_0$ is a small disk centred at $q_0 = \pi (\Gamma )$ and $U_1 = M - \{ q_0 \}$, M base surface of the bundle. This hypothesis is made for the sake of simplicty and has no influence on the generality of our study. The atlas ${\mathcal {A}}$ is defined by the two trivializing charts $(U_j \times S^1 , \psi _j)$, $j=0,1$. A coordinate function is chosen in $U_0$ in the form of a complex coordinate $w=x+iy$, $\vert w \vert < 2r$, $w(q_0)=0$, and we intentionally will identify $U_0$ with the disc $\mathbb {D}_{2 r} (q_0)$ in $\mathbb {C}$, and $U_0 \cap U_1$ with the punctured disc $\dot{\mathbb {D}_{2 r} (q_0)}$. Moreover we will always consider a coordinate along the fiber $S^1$ as a complex number of modulus 1, $z=e^{i \varphi }$, or equivalently as a $mod \, 2 \pi$ real variable $\varphi$.

The atlas $\tilde{{\mathcal {A}}}$ is obtained by ${\mathcal {A}}$ removing the local chart

$$\begin{aligned} (w,\varphi ) : U_0 \times S^1 \rightarrow \mathbb {D} \times S^1 \end{aligned}$$

and substituting it with two charts

$$\begin{aligned} {\left\{ \begin{array}{ll} ((x,u),\varphi ) :&{} \tilde{U_{0x}} \times S^1 \rightarrow \{((x,u),\varphi ): x^2 (1+u^2)<1\} \times S^1 \\ ((v,y),\varphi ) :&{} \tilde{U_{0y}} \times S^1 \rightarrow \{((v,y),\varphi ): y^2 (1+v^2)<1\} \times S^1 \end{array}\right. } \end{aligned}$$

the coordinate change between these two charts being

$$\begin{aligned} {\left\{ \begin{array}{ll} v=\frac{1}{u} \\ y=xu \\ \varphi &{}= \varphi . \end{array}\right. } \end{aligned}$$

All the other coordinate changes are obtained from the atlas ${\mathcal {A}}$ and the laws (projections obtained by restriction of $\sigma ^{-1}$ to the base space)

$$\begin{aligned} {\left\{ \begin{array}{ll} x =x \\ y =xu \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} x =yv \\ y =y . \end{array}\right. } \end{aligned}$$

The manifold $\tilde{P_{\Gamma }}$ has the following geometric description, see [7] $\S 4.6$. Firstly, we observe that the blow up is locally isomorphic to the blow up of the product $\Delta = \mathbb {D} \times I$ where I is an interval parametrizing locally the fiber $S^1$: still naming $\varphi$ the coordinate along I we choose on $\Delta$ local coordinates $(x,y,\varphi )$, and we define

$$\begin{aligned} {{\tilde{\Delta }}} \hookrightarrow \Delta \times \mathbb {RP} \end{aligned}$$

as the smooth manifold

$$\begin{aligned} {{\tilde{\Delta }}} = \{((x,y,\varphi ),l): l=[l_1 , l_2] , x l_2 = y l_1 \} \end{aligned}$$

covered by the two coordinate charts $\tilde{U_{0x}} \times S^1$, $\tilde{U_{0y}} \times S^1$, togheter with the smooth projection

$$\begin{aligned} \sigma : {{\tilde{\Delta }}} \rightarrow \mathbb {D} \times S^1 \end{aligned}$$

defined in $\tilde{U_{0x}} \times S^1$ by

$$\begin{aligned} ((x,u,\varphi ),[x,y]) \rightarrow ((x,xu,\varphi )) \end{aligned}$$

this map satisfying point (ii) in (4). In other words, each point along $\Gamma$ is blown up as a point of the corresponding fiber in the normal bundle to $\Gamma$ in P. The change of coordinates between $(x,u,\varphi )$ and $(v,y,\varphi )$ proves that ${\mathcal {E}} = \{ x=0 \}$, resp. ${\mathcal {E}} = \{ y=0 \}$, in each of these coordinates .Hence ${\mathcal {E}}$ is diffeomorphic to a Klein bottle in the particular case considered. This remark ends the description of the first two terms in the triple $(\tilde{P_{\Gamma }} , \sigma , \tilde{X_0})$ which forms the blow up of $\xi _0$ along $\gamma _0$.

To lift $X_0$ to $\tilde{X_0}$ on $\tilde{P}$ one could use $\sigma ^{-1}$ outside $\gamma _0$ and then try to extend smoothly the obtained foliation by circles on $\tilde{P_{\gamma _0}} - {\mathcal {E}}$ to the divisor ${\mathcal {E}}$: in spite of being straightforward, this approach is difficult to be carried out. We prefer to lift the infinitesimal law describing $X_0$: of course, we can work in a neighbourhood of $\Gamma$.

In local trivializing coordinates $(x,y, \varphi ) \in U_0 \times S^1$

$$\begin{aligned} X_0 : {\left\{ \begin{array}{ll} \dot{x} = 0 \\ \dot{y} = 0 \\ {\dot{\varphi }} = 1 \end{array}\right. } \end{aligned}$$

which changes according to the blowing up transformation

$$\begin{aligned} \sigma : {\left\{ \begin{array}{ll} x= x \\ y= ux \\ \varphi = \varphi \end{array}\right. } \end{aligned}$$

to

$$\begin{aligned} \left( \begin{array}{c} \dot{x} \\ \dot{u} \\ {\dot{\varphi }} \end{array} \right) = \frac{\partial (x,u,\varphi )}{\partial (x,y,\varphi )} \left( \begin{array}{c} \dot{x} \\ \dot{y} \\ {\dot{\varphi }} \end{array} \right) = \left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0 \\ -\frac{u}{x} &{}\quad \frac{1}{x} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) \end{aligned}$$

This vector field, a priori defined in $\tilde{P} - {\mathcal {E}}$, extends as a real analytic vector field to the divisor, represented in local coordinates by ${\mathcal {E}} = \{ x=0 \}$ as

$$\begin{aligned} \tilde{X_0}_{\vert {\mathcal {E}}} : {\left\{ \begin{array}{ll} \dot{u} = 0 \\ {\dot{\varphi }} = 1 \end{array}\right. } \end{aligned}$$

(3.7)

and therefore define a foliation by circles of $\tilde{P}$ which extends to the divisor the lift through $\sigma ^{-1}$ of $X_0$.

Summarizing, we started by $X_0 : P \rightarrow TP$ defining a circle bundle $\xi _0$ and we lifted these mathematical objects to the circle bundle $\tilde{\xi _0}$, the blow up of $\xi _0$, which we find useful to identify with $(\tilde{P_{\gamma _0}},\sigma , \tilde{X_0})$, where $\tilde{{\mathcal {F}}_0}$ is the foliation whose leaves are the integral curves of $\tilde{X_0} : \tilde{P_{\gamma _0}} \rightarrow T \tilde{P_{\gamma _0}}$.

In the next proposition we show that an analogous extension to the divisor holds for the vector fields $\tilde{X}_{\varepsilon }$.

Proposition 4.1

Let $\varepsilon \rightarrow X_{\varepsilon }$ be a perturbation of the infinitesimal generator $X_0$ of $\xi _0$ which is tangent to $X_0$ along the fiber $\Gamma$. Then each

$$\begin{aligned} X_{\varepsilon } :P \rightarrow TP \end{aligned}$$

lifts to a vector field

$$\begin{aligned} {\tilde{X}}_{\varepsilon } :{\tilde{P}} \rightarrow T {\tilde{P}} \end{aligned}$$

such that

$$\begin{aligned} \tilde{X_{\varepsilon }}_{\vert {\mathcal {E}}} = \tilde{X_{0}}_{\vert {\mathcal {E}}} + {\mathcal {O}}(\varepsilon ) \end{aligned}$$

Proof

Let

$$\begin{aligned} {X_{\varepsilon }} : \left( \begin{array}{c} \dot{x} \\ \dot{y} \\ {\dot{\varphi }} \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) + {\underline{G}} (x,y,\varphi , \varepsilon ) \end{aligned}$$

(4.2)

where G is smooth, $2 \pi$-periodic in $\varphi$, satisfying $G={\mathcal {O}}(\vert w \vert ^2)$ uniformly with respect to the rest of variables, $w = x + i y$. The condition of tangency along $\Gamma$ and the normalization of the periods implies

$$\begin{aligned} G(0,0,\varphi , \varepsilon ) \equiv 0 . \end{aligned}$$

(4.3)

The flow of $X_{\varepsilon }$ is of the form

$$\begin{aligned} h_t (x,y,\varphi , \varepsilon ) = \left( \begin{array}{c} x \\ y \\ \varphi \end{array} \right) + t X_0 (x,y,\varphi ) + {\underline{F}} (t,x,y,\varphi , \varepsilon ) \end{aligned}$$

(4.4)

where ${\underline{F}}$ is smooth, $2 \pi$-periodic with respect to $\varphi$, ${\underline{F}} = {\mathcal {O}}(t^2)$, ${\underline{F}} = o(1)$ with respect to $\varepsilon$ and from (4.3)

$$\begin{aligned} {\underline{F}} (t,0,0,\varphi , \varepsilon ) \equiv 0. \end{aligned}$$

(4.5)

In order to lift $X_{\varepsilon }$ to a vector field $\tilde{X_{\varepsilon }}$ we will lift the flow $h_t :P \rightarrow P$ to a flow $\tilde{h_t} : \tilde{P} \rightarrow \tilde{P}$ according to

$$\begin{aligned} \sigma \circ \tilde{h_t} = h_t \circ \sigma . \end{aligned}$$

(4.6)

or

$$\begin{aligned} \sigma \circ \tilde{h} = h \circ \sigma . \end{aligned}$$

(4.7)

where t-variable is kept fixed.

As

$$\begin{aligned} \sigma \circ {\tilde{h}} =({\tilde{h}}_1 , {\tilde{h}}_1 {\tilde{h}}_2 , \tilde{h}_3) \end{aligned}$$

(4.6) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} {\tilde{h}}_1 (x,u,\varphi ) = h_1(x,xu,\varphi ) \\ {\tilde{h}}_2 (x,u,\varphi ) = \frac{h_2 (x,xu,\varphi )}{h_1 (x,xu,\varphi )} \\ {\tilde{h}}_3 (x,u,\varphi ) = h_3 (x,xu,\varphi ) \end{array}\right. } \end{aligned}$$

(4.8)

so we are left to prove the $C^{k-1}$ extension of the definition in the second equation in (4.8) when $x \rightarrow 0$. Using that

$$\begin{aligned} {\left\{ \begin{array}{ll} h_1 (x,y,\varphi ) = x + {\mathcal {O}}(\vert w \vert ^2 , \varphi ) \\ h_2 (x,y,\varphi ) = y + {\mathcal {O}}(\vert w \vert ^2 , \varphi ) \\ h_3 (x,y,\varphi ) = \varphi + {\mathcal {O}}(\vert w \vert ^2 , \varphi ) \end{array}\right. } \end{aligned}$$

and Taylor’s formula with Peano’s form of the remainder term, we get that the second equation in (4.8) becomes

$$\begin{aligned} {\tilde{h}}_2 (x,u,\varphi ) = \frac{u + x g_2(u, \varphi ) + \cdots +x^{k-1} g_k (u,\varphi ) + \frac{G_k (x,xu,\varphi )}{x}}{1 + x f_2 (u,\varphi ) + \cdots + x^{k-1} f_k (u,\varphi ) + \frac{F_k (x,xu,\varphi )}{x}} \end{aligned}$$

(4.9)

where $g_j , f_j$ are polynomials in u of degree nothigher thant j and coefficients smoothly depending on $\varphi$, and $G_k , F_k$ are $C^k$-functions, $k \ge 2$ such that $G_k , F_k = {\mathcal {O}}(\vert w \vert ^k)$ , i.e. are k-flat at $\Gamma$ uniformly with respect to $\varphi$. The fact that the function in (4.9) extends as a $C^{k-1}$-function $(x,u,\varphi ) \rightarrow u + x (\cdots )$ when $x=0$, i.e. along the divisor, follows from the following simple

Lemma 4.2

If $G(x,y,\varphi ) = {\mathcal {O}}(\vert w \vert ^k)$, and G is of class $C^k$, $k\ge 3$, then the function $\frac{G(x,xu,\varphi )}{x}$ extends as a function of class $C^{k-2}$ to $\{ x=0 \}$, and is $k-2$-flat at $\{ x=0 \}$.

Proof

Continuos extension of $H(x,u,\varphi )=\frac{G(x,xu,\varphi )}{x}$ to $\{ x=0 \}$ as an identically 0 function is obvious. The worst (in the sense of the behaviour of the limit as $x \rightarrow 0$) j-th derivatives of H are the $\frac{\partial ^j H}{\partial x^j}$, and we focus on them. When $j=1$

$$\begin{aligned} \frac{\partial H}{\partial x} = \frac{G_x + G_y u}{x} - \frac{G}{x^2}. \end{aligned}$$

(4.10)

The term $\frac{G}{x^2} = o(1)$ for $k \ge 3$, moreover from uniqueness of the Taylor formula with Peano’s remainder, $G_x , G_y = {\mathcal {O}}(\vert w \vert ^{k-1})$ are $C^{k-1}$ functions, hence also $\frac{G_x + G_y u}{x} = o(1)$ therefore proving that $\frac{\partial H}{\partial x}$ smoothly extends to 0 as $x \rightarrow 0$. A straightforward iteration of this argument proves the lemma. $\square$

Applying this lemma we get that, using (4.5), $\tilde{h}_t (x,u,\varphi )$ is a $C^{k-2}$-regular 1-parameter group of diffeomorphisms of $\tilde{P}$ and therefore defines a $C^{k-2}$ vector field $\tilde{X_{\varepsilon }}$. The analytic form of $\tilde{X_{\varepsilon }}$ on the divisor follows from (4.2) .$\square$