I'm trying to complete the proof of the Theorem 6.1 in the notes https://www.math.u-psud.fr/~nb/articles/coursX.pdf, which ensures, under certain conditions, that the support of the microlocal defect measure is a union of curves of the type $s \mapsto \left(x(s), \frac{\xi(s)}{|\xi(s)|}\right)$.

I am not able to prove two statements in the proof, being them:

1) d'où, pour $s$ assez petit de sorte que $a \circ \phi_s$ est définie sur tout $\Omega \times S^{d-1}$.

1) $\textbf{whence, for $s$ small enough so that $a\circ \phi_s$ is defined on all $\Omega \times S^{d-1}$.}$

2) Il en résulte que le support de $\mu$ est invariant par la projection de flot $\phi_s$ sur la sphère unité, donc est une union de projetées de courbes bicaractéristiques de $q$, ou encore de $p$, puisque $q=|\xi|^{1-m}p$.

2) $\textbf{ It follows the support of $\mu $ is invariant by the projection of the flux $ \phi_s $ on}$ $\textbf{the unit sphere, thus it is a union union of projected curves}$ $\textbf{bicharacteristics of $ q $}$, or still, of $ p $, since $ q = | \xi |^{1-m} p $.

The second statement can be rewritten as: Let $(x_0,\xi_0) \in \operatorname{supp} \mu$ and consider $s \in I \mapsto (x(s),\xi(s))$ a bicharacteristic starting in $(x_0,\xi_0)$, then $\left( x(s),\frac{\xi(s)}{|\xi(s)|}\right) \in \operatorname{supp} \mu$ for all $s \in I.$ To prove this statement, I am trying to use a characterization of the support of a Radon measure, which can be found here https://math.stackexchange.com/questions/1348050/support-of-radon-measures, combined with the fact that $$\int_{\Omega \times S^{d-1} } a \circ \phi_s d \mu=\int_{\Omega \times S^{d-1} } a d \mu,$$ since $$\frac{d}{ds} \int_{\Omega \times S^{d-1} } a \circ \phi_s d \mu =0.$$

Can anybody help me?


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