“Enlacements asymptotiques” revisited

Abstract

We give an alternative proof of a theorem of Gambaudo and Ghys (Topology 36(6):1355–1379, 1997) and Fathi (Transformations et homéomorphismes préservant la mesure. Systèmes dynamiques minimaux. Thèse Orsay, 1980) on the interpretation of the Calabi homomorphism for the standard symplectic disc as an average rotation number. This proof uses only basic complex analysis.

Résumé

Nous donnons une preuve alternative d’un théorème de Gambaudo and Ghys (Topology 36(6):1355–1379, 1997) et Fathi (Transformations et homéomorphismes préservant la mesure. Systèmes dynamiques minimaux. Thèse Orsay, 1980) sur l’interpretation de l’homomorphisme de Calabi pour le disque symplectique standard comme un nombre de rotation moyen. Cette preuve utilise seulement l’analyse compléxe de base.

1 The theorem of Gambaudo–Ghys and Fathi

Let $\mathcal {G}= Ham_c({\mathbb {D}},\omega )$ be the group of compactly supported Hamiltonian diffeomorphisms of the standard disc ${\mathbb {D}}= \{z \in {\mathbb {C}}:|z| < 1\}$ endowed with the standard symplectic form $\omega = \frac{i}{2}dz \wedge d\overline{z}.$ The Calabi homomorphism [2] from $\mathcal {G}$ to ${\mathbb {R}}$ is defined as

$$\begin{aligned} {{\mathrm{{\mathrm {Cal}}}}}(\phi ) = \int _0^1 \left( \int _{{\mathbb {D}}} H_t \,\omega \right) \, dt , \end{aligned}$$

where $H_t$ is the compactly supported Hamiltonian of a Hamiltonian isotopy $\{\phi _t\}_{t\in [0,1]}$ with $\phi _1 = \phi .$ In other words this isotopy is generated by a time-dependent vector field $X_t,$ that satisfies the relation

$$\begin{aligned} \iota _{X_t} \omega = - d H_t. \end{aligned}$$

The mean rotation number is defined in terms convenient for the proof as follows. Consider the differential form

$$\begin{aligned} \alpha =\frac{1}{2\pi }\frac{d(z_1 -z_2)}{z_1 - z_2} \end{aligned}$$

(used by Arnol’d [1] in his study of the cohomology of the pure braid groups) on the configuration space $X_2 = X_2({\mathbb {D}}) = \{(z_1,z_2)| z_j \in {\mathbb {D}}, \; z_1 \ne z_2\} = {\mathbb {D}}\times {\mathbb {D}}{\setminus } \Delta ,$ where $\Delta \subset {\mathbb {D}}\times {\mathbb {D}}$ is the diagonal. Denote by

$$\begin{aligned} \theta = Im(\alpha ) \end{aligned}$$

its imaginary part. Note that the two forms $\alpha$ and $\theta$ are closed. For each pair of points $(z_1,z_2) \in {\mathbb {D}}\times {\mathbb {D}}$ such that $z_1 \ne z_2,$ that is for each point $x = (z_1,z_2) \in X_2,$ consider the curve $\{\phi _t \cdot x\}$ in $X_2$ defined by

$$\begin{aligned} \phi _t \cdot x = (\phi _t(z_1), \phi _t(z_2)) \end{aligned}$$

for each $t \in [0,1].$ The average rotation number is

$$\begin{aligned} \Phi (\phi ) = \int _{X_2} dm ^2(x) \int _{\{\phi _t \cdot x\}} \theta , \end{aligned}$$

where $dm ^2(x) = dm (z_1) dm (z_2)$ is the Lebesgue measure on ${\mathbb {D}}\times {\mathbb {D}}$ restricted to $X_2$ (here $dm (z)$ denotes the Lebesgue measure on ${\mathbb {D}}$). By preservation of volume, it is clear that $\Phi$ is a homomorphism $\mathcal {G}\rightarrow {\mathbb {R}}.$

The theorem of Gambaudo and Ghys [5] and Fathi [4] is the following equality.

Theorem 1

$$\begin{aligned} \Phi = -2{{\mathrm{{\mathrm {Cal}}}}}, \end{aligned}$$

as homomorphisms $\mathcal {G}\rightarrow {\mathbb {R}}.$

Gambaudo and Ghys have presented several proofs of this result, and in [4] a different proof of Fathi is found. More proofs of this result are known today (cf. [3]). Here we present an alternative short proof, which is in fact a complex-variable version of the proof of Fathi.

2 The alternative proof

Put $\xi _t = dz(X_t),$ for the natural complex coordinate z on ${\mathbb {D}}.$ Hence $\xi _t$ is a smooth compactly supported complex-valued function on ${\mathbb {D}}.$ The computations

$$\begin{aligned} \iota _{X_t} \left( \frac{i}{2} dz\wedge d\overline{z}\right) = \frac{i}{2} \xi _t d\overline{z} - \frac{i}{2} \overline{\xi }_t dz \end{aligned}$$

and

$$\begin{aligned} -dH_t = - \frac{\partial H_t}{\partial \overline{z}} d\overline{z}- \frac{\partial H_t}{\partial z} dz \end{aligned}$$

give us

$$\begin{aligned} \xi _t = 2i \frac{\partial H_t}{ \partial \overline{z}}. \end{aligned}$$

(1)

Now

$$\begin{aligned} \Phi (\phi ) = Im\left( \int _{X_2} dm ^2(x) \int _{\{\phi _t \cdot x\}} \alpha \right) , \end{aligned}$$

and hence it is sufficient to compute

$$\begin{aligned}&\int _{X_2} dm ^2(x) \int _{\{\phi _t \cdot x\}} \alpha = \frac{1}{2\pi } \int _{X_2} dm ^2(x) \int _{\{\phi _t \cdot x\}} \frac{d(z_1 - z_2)}{z_1 - z_2}=\\&\quad = \frac{1}{2\pi } \int _{X_2} dm (z_1) dm (z_2)\int _0^1 dt \; \frac{\xi _t(\phi _t(z_1)) - \xi _t(\phi _t(z_2))}{\phi _t(z_1) - \phi _t(z_2)}=\\&\quad \text {as the function is absolutely integrable (see Lemma 1), by Fubini,}\\&\quad =\frac{1}{2\pi } \int _0^1 dt \; \int _{X_2} dm (z_1) dm (z_2) \; \frac{\xi _t(\phi _t(z_1)) - \xi _t(\phi _t(z_2))}{\phi _t(z_1) - \phi _t(z_2)} \\&\quad = \frac{1}{2\pi } \int _0^1 dt \; \int _{X_2} dm (z_1) dm (z_2) \; \frac{\xi _t(z_1) - \xi _t(z_2)}{z_1 - z_2} =\\&\quad \text {as both terms of the sum are absolutely integrable (a consequence of the proof}\\&\qquad \text {of Lemma 1 as well),}\\&\quad = 2 \cdot \frac{1}{2\pi } \int _0^1 dt \; \int _{{\mathbb {D}}} dm (w) \int _{{\mathbb {D}}\setminus \{w\}} \frac{\xi _t(z)}{z - w} dm (z) =\\&\quad \text {by Eq. 1,}\\&\quad = - 2 i \int _0^1 dt \; \int _{{\mathbb {D}}} dm (w) \int _{{\mathbb {D}}\setminus \{w\}} \frac{1}{2 \pi i}\frac{\partial H_t}{\partial \overline{z}}\frac{dz \wedge d\overline{z}}{z - w} \\&\quad = - 2 i \int _0^1 dt \; \int _{{\mathbb {D}}} dm (w) H_t(w) = - 2 i {{\mathrm{{\mathrm {Cal}}}}}(\phi ). \end{aligned}$$

The penultimate equality is a consequence of the Cauchy formula for smooth functions [6, Theorem 1.2.1]. Indeed for any $C^1$ function $f:{\mathbb {D}}\rightarrow {\mathbb {C}},$ we have

$$\begin{aligned} f(w) = \frac{1}{2 \pi i} \int _{\partial {\mathbb {D}}} \frac{f(z)}{z-w} dz + \frac{1}{2\pi i} \int _{{\mathbb {D}}} \frac{\partial f}{\partial \overline{z}} \frac{dz \wedge d\overline{z}}{z-w}. \end{aligned}$$

It remains to note that as $H_t$ is zero near the boundary, the first term of the sum vanishes.

Now we show the absolute integrability that we use to change the order of integration.

Lemma 1

$$\begin{aligned} \int _{X_2}\int _0^1 dm (z_1) dm (z_2) dt \; \frac{|\xi _t(\phi _t(z_1)) - \xi _t(\phi _t(z_2))|}{|\phi _t(z_1) - \phi _t(z_2)|} < \infty \end{aligned}$$

By the Tonnelli theorem, the following chain of inequalities suffices:

$$\begin{aligned}&\int _0^1 dt \int _{X_2} dm (z_1) dm (z_2) \; \frac{|\xi _t(\phi _t(z_1)) - \xi _t(\phi _t(z_2))|}{|\phi _t(z_1) - \phi _t(z_2)|}\\&\quad = \int _0^1 dt \int _{X_2} dm (z_1) dm (z_2) \; \frac{|\xi _t(z_1) - \xi _t(z_2)|}{|z_1 - z_2|}\\&\quad \le 2 \int _0^1 dt \int _{{\mathbb {D}}} dm (z) |\xi _t(z)| \int _{{\mathbb {D}}\setminus \{z\}} \; \frac{1}{|z - w|} dm (w)\\&\quad \le 8\pi \int _0^1 dt \int _{{\mathbb {D}}} |\xi _t| dm < \infty , \end{aligned}$$

because

$$\begin{aligned} \int _{{\mathbb {D}}\setminus \{z\}} \; \frac{1}{|z - w|} dm (w) \le 4\pi , \end{aligned}$$

as one verifies by direct calculation.