# My Life with Augustin *

Marc Chaperon

► To cite this version:

Marc Chaperon. My Life with Augustin *. African Diaspora Journal of Mathematics, 2012, Special Volume in Honor of Prof. Augustin Banyaga, 14 (12), pp.90-97. hal-04876908

HAL Id: hal-04876908

https://hal.science/hal-04876908v1

Submitted on 9 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# MY LIFE WITH AUGUSTIN*

MARC CHAPERON†

Université Paris Diderot,
75205 Paris cedex 13, France

## Abstract

This is a very brief and partial account of Augustin's mathematics and our common interests. In the end is presented some recent work of mine whose birth he encouraged.

**AMS Subject Classification:** 01-02; 01A80; 34C20; 34C23; 34C45; 37C15; 37C20; 37D10; 37G05; 37G10; 37G15; 37J10; 37J55; 53D35.

**Keywords:** Contact or symplectic geometry, dynamical systems, bifurcations, catastrophes.

In August 2004, after visiting the ethnographic museum in Porto Novo (Benin), I wanted to write a few words in the visitors' book... and found out that Augustin Banyaga had been there five hours before. This might be a symbol of our relationship, even though the inter-twining of our destinies owes little to coincidence: indeed, we are mathematical cousins, our masters André Haefliger and René Thom being two sons of Charles Ehresmann.

## Symplectic geometry

I heard of Augustin for the first time in 1982, during the weekly meetings on symplectic geometry organized by Daniel Bennequin at the École Normale Supérieure: among the results discussed was Banyaga's theorem<fnref n="1" /> that a symplectic transformation of a compact symplectic manifold (M,ω) is Hamiltonian isotopic to the identity<fnref n="2" /> if and only if it belongs to the subgroup [G<sub>ω</sub>, G<sub>ω</sub>] of commutators of the component G<sub>ω</sub> of the identity in the group of symplectic transformations of M.

<footnote>*This paper was written while the author was a member of the Institut de Mathématiques de Jussieu - Paris Rive Gauche, UMR 7586 du CNRS, one of the beneficiaries of the grant "ANR-10-BLAN 0102" of the Agence Nationale de la Recherche and an affiliate of the IMCCE, Observatoire de Paris, CNRS, UPMC, Université Lille1, 77 avenue Denfert-Rochereau, F-75014 Paris, France.</footnote>

<footnote>†E-mail address: chaperon@math.jussieu.fr</footnote>

<footnote n="1">¹In his thesis [3] under the supervision of André Haefliger.</footnote>

<footnote n="2">²That is, obtained by integrating a time-dependent Hamiltonian vector field; in other words, its Calabi invariant is zero.</footnote>
These meetings were very fruitful<fnref n="3" />, as they coincided with a breakthrough in what is now called symplectic topology, the proof by Conley and Zehnder [20] of an Arnold conjecture [2] that had been defying topologists for almost twenty years: when $\omega$ is a standard symplectic structure on the $2n$-torus, every $h \in [G_\omega, G_\omega]$ has at least $2n + 1$ fixed points, and at least $2^{2n}$ if all of them are non-degenerate<fnref n="4" />.

I was lucky enough to see at once that a more general conjecture of Arnold [1] could be proven along the same lines [9]: every symplectic transformation of the cotangent bundle $T^*T^n$, Hamiltonian isotopic to the identity, sends the zero section $d0$ to a submanifold $L$ intersecting $d0$ at least at $n+1$ points, and at least $2^n$ if all these intersections are transversal<fnref n="5" />.

One year later, in order to extend this result from $T^n$ to an arbitrary closed manifold $M$, I gave a much simpler proof of the same theorem [10], constructing by purely elementary means a generating phase for $L$. The extension of the theorem to an arbitrary $M$ was obtained by Hofer [21] via “hard analysis” following the idea of [9] but, very little later, Laudenbach and Sikorav [23] found a simpler proof using my new method.

The method was immediately taught all around the world, in particular by Augustin.

## Smooth local conjugacies

Even though I may be essentially known for those two little contributions to symplectic geometry, they constitute a parenthesis in my work, whose first years had been devoted to various aspects of the smooth local classification of dynamical systems [11, 12]. With Rafael de la Llave and Gene Wayne, Augustin contributed to the subject [4, 5], using the so-called *path method* introduced by Samuel and Thom in singularity theory and applied by Moser to the conjugacy problem for volume or symplectic forms. The device having been the key of Roussarie’s work [25] on the local classification of vector fields (and differential forms), Banyaga, de la Llave and Wayne deal chiefly with the case of diffeomorphisms.

You wish to prove that two local diffeomorphisms $h_0, h : (\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ are conjugate near 0. Setting $R := h - h_0$ the idea is to consider the path $h_t := h_0 + tR$ from $h_0$ to $h_1 = h$ and solve the seemingly more difficult problem: find a smooth path $g_t$ with $g_0 = \text{Id}$ in the space of local diffeomorphisms $(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ such that, in a neighbourhood of the origin, $g_t^*h_t = h_0$ for all $t \in [0, 1]$, where $g_t^*h_t := g_t^{-1} \circ h_t \circ g_t$; as $g_0^*h_0 = h_0$ by definition, this is equivalent to solving the equation $\frac{d}{dt}g_t^*h_t = 0$ near $0 \in \mathbb{R}^n$ for all $t \in [0, 1]$. Now, the local isotopy $g_t$ will determine (and be determined by) the time-dependent local vector field $X_t := (\frac{d}{dt}g_t) \circ g_t^{-1}$, called its *infinitesimal generator*<fnref n="6" />, which must satisfy $X_t(0) = 0$ for $0 \le t \le 1$; near $0 \in \mathbb{R}^n$, setting $Y_t := (\frac{d}{dt}h_t) \circ h_t^{-1} = R \circ h_t^{-1}$, the equation $\frac{d}{dt}g_t^*h_t = 0$ writes<fnref n="7" /> as the “cohomological equation”

$$X_t = h_{t*}X_t + Y_t$$

<footnote n="3">³Other participants were Michèle Audin, Alain Chenciner, Ivar Ekeland, Albert Fathi, Misha Gromov, Michael R. Herman, François Laudenbach, Jean-Claude Sikorav, Claude Viterbo...</footnote>

<footnote n="4">⁴This implies for example [9] the smooth case of the Poincaré-Birkhoff theorem (“Poincaré’s last geometric theorem”) on the number of fixed points of area-preserving twist transformations of the annulus.</footnote>

<footnote n="5">⁵If $L$ projects diffeomorphically onto $T^n$, it is (the image of) the differential $d\varphi$ of a function $\varphi$ on $T^n$; thus, the result generalizes the classical Lyusternik-Schnirelmann/Morse estimates for the number of fixed points of a $C^2$ function on $T^n$.</footnote>

<footnote n="6">⁶For each $x$ near $0 \in \mathbb{R}^n$, the path $t \mapsto g_t(x)$ is the solution of the equation $\frac{dy}{dt} = X_t(y)$ equal to $x$ for $t = 0$.</footnote>

<footnote n="7">⁷If $g_t^*Y(x) = Dg_t(x)^{-1}Y(g_t(x))$ for each vector field $Y$ defined near $0 \in \mathbb{R}^n$ and $h_{t*} := (h_t^{-1})^*$.</footnote>
in the unknown $X_t$. This affine fixed point problem is solved as usual when $h_t$ is a strict dilation, yielding

$$X_t = \sum_{k \ge 0} h_{t*}^k Y_t,$$

which converges near $0 \in \mathbb{R}^n$ provided $Y_t$ vanishes to a high enough order at the origin$^8$. Similar arguments work in the general hyperbolic case.

In [8, 12] such problems had been solved more directly. Since then, I have treated the conjugacy problem (and related questions like the solution of cohomological equations) as a part of invariant manifold theory [13, 17, 14, 15]; the starting point is the remark that $h: X \to Y$ is a semi-conjugacy between $f: X \to X$ and $g: Y \to Y$ (i.e. $g \circ h = h \circ f$) if and only if its graph is invariant by $f \times g: (x, y) \mapsto (f(x), g(y))$.

## Contact geometry and Weinstein's conjecture

I first heard this story from Albert Fathi, then a professor at the University of Florida, Gainesville. The result [6] is the following: let $M$ be a $(2n+1)$-dimensional oriented closed manifold with a 1-form $\alpha$ such that the 2-form $\omega = d\alpha$ has rank $2n$ everywhere. If there exists a locally free circle action $\rho$ on $M$ which preserves $\omega$, then the characteristic foliation of $\omega$ has at least two closed leaves for $n \ge 1$.

The wonderfully simple proof should be contrasted with the very hard prize-winning work of Cliff Taubes on the Weinstein conjecture in dimension three$^9$; it goes as follows: if we view the circle as $T := \mathbb{R}/\mathbb{Z}$, the 1-form $\alpha_0 := \int_T \rho^{\theta*} \alpha d\theta$ is $\rho$-invariant and satisfies

$$ \begin{aligned} d\alpha_0 &= d \int_T \rho^{\theta*} \alpha d\theta = \int_T d(\rho^{\theta*} \alpha) d\theta = \int_T (\rho^{\theta*} d\alpha) d\theta = \int_T \rho^{\theta*} \omega d\theta = \int_T \omega d\theta \\ &= \omega \end{aligned} $$

because $\omega$ is $\rho$-invariant; denoting by $Z := \frac{d}{d\theta}\rho^{\theta}|_{\theta=0}$ the infinitesimal generator of $\rho$, the invariance of $\alpha_0$ under $\rho$ reads

$$ \mathcal{L}_Z \alpha_0 := \frac{d}{d\theta} \rho^{\theta*} \alpha_0 |_{\theta=0} = 0, $$

which, by the Cartans' formula$^{10}$ $\mathcal{L}_Z \alpha_0 = (d\alpha_0)Z + d(\alpha_0 Z) = \omega Z + d(\alpha_0 Z)$, can be rewritten

$$ dS = \omega Z \quad \text{if} \quad S := -\alpha_0 Z. $$

It follows that the critical points of $S$ are those $x \in M$ such that $Z(x)$ belongs to the characteristic direction $ker \omega(x)$ of $\omega$ at $x$; now, since $Z$ and $\alpha_0$ are $\rho$-invariant, so is $S$, hence the whole orbit of the critical point $x$ under $\rho$ consists of critical points $y$ of $S$, at which $Z(y)$ belongs to the characteristic direction of $\omega$; thus, the orbit of $x$ under $\rho$ is a closed characteristic of $\omega$. As $M$ is compact, the function $S$ has a maximum and a minimum, which are critical values because $M$ has no boundary; if they are distinct, the corresponding critical points yield distinct closed characteristics of $\omega$; if they coincide, every point of $M$ is a critical point of $S$ and therefore every characteristic of $\omega$ is closed.

<footnote>$^8$Which means that $R$ does. Convergence occurs in the same open neighbourhood of 0 for all $t \in [0, 1]$; hence, as $X_t(0) = 0$, the time-dependent vector field $X_t$ does generate a local isotopy.</footnote>

<footnote>$^9$As far as I know, the general case, on which I spent quite a lot of infructuous time in the early nineties, is still open.</footnote>

<footnote>$^{10}$Where $\beta X$ denotes the interior product of a differential form $\beta$ and a vector field $X$.</footnote>
**Generalized Hopf bifurcations**

In January 2010, for my sixtieth birthday, a conference took place at the Institut Henri Poincaré in Paris. Among the speakers were Augustin<sup>11</sup>, whose talk began with a minute of silence for the victims of the earthquake in Haiti, and Mark Levi, who invited me to spend some time at Penn State in October and November the same year.

This visit, during which I enjoyed the hospitality of Augustin, Mark and Aissa Wade, played a key role in my work on generalized Hopf bifurcations, as the (polite?) interest it arose<sup>12</sup> gave me the courage to complete a substantial part of its redaction [16].

**Overview of the results.** They are about the birth of dynamics out of statics (or the non-linear coupling of oscillators). In generic smooth one-parameter families of vector fields, simple examples are *the Hopf bifurcation*, in which an attracting equilibrium point becomes unstable while giving rise to an attracting periodic orbit, and the *Sacker-Naimark bifurcation*, in which an attracting periodic orbit becomes unstable while giving rise to an attracting invariant 2-torus.

These $T^0 \to T^1$ and $T^1 \to T^2$ bifurcations are *not* paralleled by a $T^2 \to T^3$ bifurcation which, far from being generic, requires infinitely many conditions [19]: in generic one-parameter families, the invariant 2-torus will break down when it loses attractivity and chaos (“turbulence” [26]) will develop. Thus, to study the birth of *n*-tori with $n > 2$, it is best to follow René Thom’s advice: “Look for the organising center or phenomena” and consider families depending on more parameters.

Loosely speaking, the result is that $T^0 \to T^n$ and $T^1 \to T^{n+1}$ bifurcations (among others) occur smoothly in generic families depending on at least *n* parameters.

More precisely, attracting<sup>13</sup> invariant *n*-tori—and more suprising invariant submanifolds—are born smoothly at partially elliptic stationary points in generic families of vector fields (resp., transformations) depending on at least *n* parameters. Here, “partially elliptic” means that the eigenvalues of the linearised dynamics which lie on the imaginary axis (resp., unit circle) are simple and consist of *n* pairs of conjugate complex numbers<sup>14</sup>.

The corresponding values $u_0$ of the parameter $u$ form a submanifold of codimension $n$, but we shall see soon that the set $\mathcal{V}$ of those $u$ for which the attracting invariant submanifold exists (and depends differentiably on $u$) contains an open subset with nonempty *open* tangent cone at $u_0$, implying that the phenomenon is not negligible<sup>15</sup>.

**The birth lemma.** Under a mild nonresonance condition, taking a suitable chart and restricting the dynamics to a central manifold, one may assume that, near $u_0$ and the partially elliptic stationary point considered in phase space, the dynamics under study form a local

<footnote n="11"><sup>11</sup>In the nineties, he and his wife Judith had become our friends when they had spent six months in Paris.</footnote>

<footnote n="12"><sup>12</sup>At the colloquium of the mathematics department and the Fall meeting of the Penn State/Maryland Workshop in Dynamical Systems and Related Topics.</footnote>

<footnote n="13"><sup>13</sup>More generally, normally hyperbolic.</footnote>

<footnote n="14"><sup>14</sup>For maps, [16] treats similarly the case where there are $n-1$ pairs, plus $-1$ but not $+1$.</footnote>

<footnote n="15"><sup>15</sup>Here, Thom’s advice is spectacularly good: indeed, for $n > 1$, the codimension *n* submanifold consisting of “birth values” $u_0$ occupies almost no room in the boundary of $\mathcal{V}$; near the other boundary points, turbulence will appear, making it practically impossible to understand the phenomenon via families with fewer parameters.</footnote>
family $Z_u$ (resp., $h_u$) of vector fields on (resp., transformations of) $\mathbb{C}^n$ having third order contact at $0 \in \mathbb{C}^n$ with a normal form

$$N_u(z) = (z_j(\lambda_j(u) + i\mu_j(u) - \sum_{\ell=1}^{n} (a_{j\ell}(u) + ib_{j\ell}(u))|z_\ell|^2))_{1\le j \le n},$$

(resp., the time one of its flow), where $\lambda_j, \mu_j, a_{j\ell}, b_{j\ell}$ are differentiable real functions with $\lambda_j(u_0) = 0$ (ellipticity).

Under those hypotheses<fnref n="16" />, the main result of [16] is the following *birth lemma*: assume that, for some tangent vector $v_0$ at $u_0$, the vector field

$$\tilde{\xi}_{u_0, v_0}(\zeta) = (\zeta_j(D(\lambda_j + i\mu_j)(u_0)v_0 - \sum_{\ell=1}^{n} (a_{j\ell}(u_0) + ib_{j\ell}(u_0))|\zeta_\ell|^2))_{1\le j \le n}$$

on $\mathbb{C}^n$ admits a normally hyperbolic compact invariant manifold $\tilde{\Sigma} \subset \mathbb{C}^n$. Then, there is an open subset $\mathcal{U}_{u_0, v_0}$ of parameter space $\mathcal{U}$ with the following properties:

i) Its closure contains $u_0$.

ii) Its tangent cone at $u_0$ is an open cone with vertex 0 containing $\mathbb{R}_+^* v_0$.

iii) Every $Z_u$ (resp. $h_u$) with $u \in \mathcal{U}_{u_0, v_0}$ has a compact normally hyperbolic invariant manifold $S_u$ diffeomorphic to $\tilde{\Sigma}$, whose index<fnref n="17" /> is that of $\tilde{\Sigma}$ for $\tilde{\xi}_{u_0, v_0}$, depending nicely on $u$ and tending to $\{0\}$ when $u \to u_0$.

iv) Precisely, there is an open cone $V \ni v_0$ of $T_{u_0}\mathcal{U}$ with vertex 0 such that each $\tilde{\xi}_{u_0, v}$ with $v \in V$ has a unique normally hyperbolic compact invariant manifold $\tilde{\Sigma}_v$ diffeomorphic to $\tilde{\Sigma}$ and $C^1$-close to it up to homothety<fnref n="18" />; every smooth $\gamma: (\mathbb{R}_+, 0) \to (\mathcal{U}, u_0)$ with $\dot{\gamma}(0) = v \in V$ satisfies $\gamma(\varepsilon) \in \mathcal{U}_{v_0, v_0}$ for $\varepsilon > 0$ small enough, and $\lim_{\varepsilon \to 0} \varepsilon^{-1/2} S_{\gamma(\varepsilon)} = \tilde{\Sigma}_v$ in the at least $C^1$ sense<fnref n="19" />.

A key argument in the proof is that $\tilde{\xi}_{u_0, v}$ is the limit when $\varepsilon \to 0$ of the vector field obtained from $N_{\gamma(\varepsilon)}$ by subtracting to it the infinitesimal rotation $z \mapsto (i\mu_j(u_0)z_j)_{1\le j \le n}$ (which commutes with it) and making the variable change $z = \varepsilon^{1/2}\zeta$, and the time change $t = \varepsilon^{-1}\tau$

**Consequences.** The vector field $\tilde{\xi}_{u_0, v_0}$ being $\mathcal{U}(1)^n$-invariant, so is $\tilde{\Sigma}$ by local uniqueness; passing to the quotient, we see that the $\mathcal{O}(1)^n$-invariant submanifold $\Sigma = \tilde{\Sigma} \cap \mathbb{R}^n$ of $\mathbb{R}^n$ is a normally hyperbolic invariant manifold of the $\mathcal{O}(1)^n$-invariant vector field

$$\xi_{u_0, v_0}(r) = \sum_j r_j (D\lambda_j(u_0)v_0 - \sum_\ell a_{j\ell}(u_0)r_\ell^2) \frac{\partial}{\partial r_j}$$

on $\mathbb{R}^n$. Here is a weak converse: if $\xi_{u_0, v_0}$ has a normally hyperbolic $\mathcal{O}(1)^n$-invariant submanifold $\Sigma$ on which it vanishes identically, then the hypothesis of the *birth lemma* is satisfied by $\tilde{\Sigma} = \{z \in \mathbb{C}^n : (|z_1|, \dots, |z_n|) \in \Sigma\}$. Two cases of interest:

<footnote n="16">¹⁶Even when there are less than $n$ parameters or the family is not generic.</footnote>

<footnote n="17">¹⁷Dimension of the leaves of the stable foliation.</footnote>

<footnote n="18">¹⁸Because normal hyperbolicity is open and $\eta^{1/2} \tilde{\Sigma}_v$ is invariant by the flow of $\tilde{\xi}_{u_0, \eta v}$ for all positive $\eta$ when $\tilde{\Sigma}_v$ is invariant by the flow of $\tilde{\xi}_{u_0, v}$.</footnote>

<footnote n="19">¹⁹Hence the invariant manifold $S_u$ arises rather suddenly, as in classical Hopf bifurcations.</footnote>
– *Tori*. When $\Sigma = \{r : \forall j D\lambda_j(u_0)v_0 = \sum_l a_{jl}(u_0) r_l^2\}$ with $(a_{jl}(u_0))$ invertible, it consists of equilibrium points; if they are hyperbolic, we get the *n*-tori mentioned before.

– *Moment-angle manifolds.* If $\Sigma = \{r : F(r) = b\}$ with $F(r) = \sum_j \Lambda_j r_j^2, b, \Lambda_1, \dots, \Lambda_n \in \mathbb{R}^c$, $\text{conv}(\Lambda_1, \dots, \Lambda_n) \not= \emptyset$ and $b$ a regular value of $F$, then $\tilde{\Sigma}$ is called a *moment-angle manifold* and can have various topologies [7, 24]. When $\xi_{u_0, v_0}(r) = \sum_j r_j \Lambda_j \cdot (b-F(r)) \frac{\partial}{\partial r_j}$ (the dot stands for the scalar product), it equals $-\frac{1}{2} \nabla |F(r)-b|^2$ and therefore admits $\Sigma$ as a pointwise invariant normally hyperbolic attractor<sup>20</sup>; hence, the birth lemma applies.

In this example, $\tilde{\Sigma}$ is a $(2n-1)$-sphere if $c=1$, an $n$-torus if $c=n$. This $\xi_{u_0,v_0}$ is too particular to arise in generic $n$-parameter families<sup>21</sup> but, as normal hyperbolicity is open, the birth of normally hyperbolic attractors diffeomorphic to $\tilde{\Sigma}$ will be observed in generic $n$-parameter families nearby. This is the idea of the following corollary of the birth lemma:

Assume that the vector field $\xi(r) = \sum_j r_j(v_j v - \sum_l \gamma_{jl} r_l^2) \frac{\partial}{\partial r_j}$ on $\mathbb{R}^n$, $v_j, \gamma_{jl} \in \mathbb{R}$, has an $\mathbb{O}(1)^n$-invariant normally hyperbolic invariant manifold $\Sigma$ whose intersection with $\mathbb{R}_+^n$ is connected<sup>22</sup>. Then, if $M$ and $\mathcal{U}$ are separable manifolds with $\dim M \ge 2n$ and $\dim \mathcal{U} \ge n$, there exists a nonempty, $\mathbb{C}^3$-open set of smooth families $X : \mathcal{U} \times M \to TM$ of vector fields (resp., $f : \mathcal{U} \times M \to M$) for which the birth lemma ensures at some point $(u_0, x_0)$ the birth of normally hyperbolic invariant submanifolds of $X_u$ (resp., $f_u$) diffeomorphic to $\tilde{\Sigma}$. The same holds true if “normally hyperbolic” is replaced by “normally hyperbolic and attracting”.

For example, if $n=3$, $\Sigma$ can be a periodic orbit, yielding a 4-torus $\tilde{\Sigma}$ in $\mathbb{C}^3$.

**Spheres.** Of special interest is the case of the birth lemma where $\tilde{\Sigma}$ is an attracting embedded sphere of codimension 1 around the origin, a *bona fide* generalisation of the Hopf bifurcation in which every nonzero forward orbit of $Z_u$ (resp., $h_u$) in a fixed neighbourhood of the origin in $\mathbb{C}^n$ tends to $S_u$ for $u \in \mathcal{U}_{u_0, v_0}$ close enough to $u_0$.

The approach via the case $c=1$ of moment-angle manifolds [22] is interesting because the dynamics on $S_u$ can vary a lot, $\xi_{u_0,v_0}$ having no dynamics on $\Sigma$, but this provides quite a narrow set in parameter space.

A very wide set is furnished by the rough birth lemma [16] stating that, for positive $D\lambda_j(u_0)v_0$ and $a_{jl}(u_0)$, a family of attracting *Čech* homology $(2n-1)$-spheres (“Birkhoff attractors”) bifurcates as in the birth lemma. Conditions for these “spheres” to be normally hyperbolic differentiable hypersurfaces $S_u$ will be studied in a forthcoming paper with Santiago López de Medrano, together with the bifurcations that can occur inside $S_u$—the birth lemma can indeed apply for the same $(u_0, v_0)$ (with different $\mathcal{U}_{u_0, v_0}$) to many manifolds $\tilde{\Sigma}$, among which a big embedded $(2n-1)$-sphere containing all the others.

**References**

[1] V. I. Arnold, Sur une propriété topologique des applications globalement canoniques de la mécanique classique. C. R. Acad. Sc. Paris **261** (1965), Groupe 1, 3719–3722

<footnote n="20"><sup>20</sup>This remark is due to S. López de Medrano [18].</footnote>

<footnote n="21"><sup>21</sup>Normally hyperbolic invariant manifolds $\Sigma$ of large dimension are not always easy to find.</footnote>

<footnote n="22"><sup>22</sup>Technically, normal hyperbolicity is assumed absolute when $\Sigma$ meets some coordinate hyperplane $\{r_j = 0\}$ in which it is not contained.</footnote>
[2] —, *Mathematical methods of classical mechanics* (translated from the Russian original which appeared in 1974), *Appendix 9*. Springer Graduate Texts in Mathematics 60, 1989

[3] A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique. *Comment. Math. Helvetici* **53** (1978), 174–227

[4] —, R. de la Llave and C.E. Wayne, Cohomology equations and commutators of germs of contact diffeomorphisms. *Trans. Am. Math. Soc.* **312** (1989), No.2, 755–778

[5] —, Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem. *J. Geom. Anal.* **6** (1996), No. 4, 613–649

[6] A. Banyaga and P. Rukimbira, On Characteristics of Circle-Invariant Presymplectic Forms. *Proceedings of the American Mathematical Society*, Vol. **123**, No. 12 (Dec., 1995), 3901–3906

[7] F. Bosio and L. Meersseman, Real quadrics in $C^n$, complex manifolds and convex polytopes. *Acta Math.* **197** (1) (2006) 53–127

[8] M. Chaperon, Quelques questions de géométrie symplectique [d'après, entre autres, Poincaré, Arnold, Conley et Zehnder]. *Séminaire Bourbaki, Astérisque* **105–106** (1983), 231–249

[9] —, Quelques outils de la théorie des actions différentiables. *Astérisque* **107–108** (1983), 269–275

[10] —, Une idée du type “géodésiques brisées” pour les systèmes hamiltoniens. *C. R. Acad. Sc. Paris* **298** (1984) série A, 293–296

[11] —, $C^k$-conjugacy of holomorphic flows near a singularity. *Publ. Math. I.H.E.S.* **64** (1986), 143–183

[12] —, *Géométrie différentielle et singularités de systèmes dynamiques*. Astérisque **138–139** (1986), 434 pages

[13] —, Variétés stables et formes normales. *C. R. Acad. Sc. Paris* **317**, Série 1 (1993), 87–92

[14] —, Invariant manifolds revisited. *Proceedings of the Steklov Institute* **236** (2002), 415–433

[15] —, Stable manifolds and the Perron-Irwin method. *Ergodic Theory and Dynamical Systems* **24** (2004), 1359–1394

[16] —, Generalized Hopf bifurcations: a birth lemma. *Moscow Mathematical Journal* **11**, Vol. 3 (2011) dedicated to the memory of V.I. Arnold, 413–438

[17] — and F. Coudray, Invariant manifolds, conjugacies and blow-up. *Ergodic Theory and Dynamical Systems* **17** (1997), 783–791
[18] M. Chaperon and S. López de Medrano, Birth of attracting compact invariant sub-manifolds diffeomorphic to moment-angle manifolds in generic families of dynamics. *C. R. Acad. Sci. Paris, Sr. I* **346** (2008), 1099–1102

[19] A. Chenciner and G. Iooss, Bifurcations de tores invariants. *Arch. Rational Mech. Anal.* **69** (1979), no. 2, 109–198

[20] C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol'd. *Invent. math.* **73** (1983), 33–49

[21] H. Hofer, Lagrangian embeddings and critical point theory. *Ann. Inst. Henri Poincaré, Analyse non linéaire* **2** (1987), 337–356

[22] M. Kammerer-Colin de Verdière, Stable products of spheres in the non-linear coupling of oscillators or quasi-periodic motions. *C. R. Acad. Sc. Paris* **339** (2004), *Groupe 1*, 625–629

[23] F. Laudenbach and J.-C. Sikorav, Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibré cotangent. *Invent. math.* **82** (1985), 349–357

[24] S. López de Medrano, The space of Siegel leaves of a holomorphic vector field, in: *Dynamical Systems*, Lecture Notes in Mathematics, vol. 1345, Springer-Verlag, 1988, pp. 233–245

[25] R. Roussarie, *Modèles locaux de champs et de formes*. Astérisque **30** (1975), 181 pages

[26] D. Ruelle and F. Takens, On the nature of turbulence, *Com. Math. Phys.* **20** (1971), 167–192