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We give examples of composition operators $C\_\Phi$ on $H^2 (\D^2)$ showingthat the condition $\|\Phi \|\_\infty = 1$ is not sufficient for theirapproximation numbers $a\_n (C\_\Phi)$ to satisfy $\lim\_{n \to \infty} [a\_n(C\_\Phi) ]^{1/\sqrt{n}} = 1$, contrary to the $1$-dimensional case. We alsogive a situation where this implication holds. We make a link with theMonge-Amp\`ere capacity of the image of $\Phi$.

Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk