  Let (X_R, 0) be a germ of real analytic subset in (R^N, 0) of pure dimension
n+1 with an isolated singularity at 0. Let (f_R,0) : (X_R, 0) --> (R,0) a real
analytic germ with an isolated singularity at 0, such that its complexification
f_C vanishes on the singular set S of X_C. We also assume that X_R-[0] is
orientable. To each $ A \in H^{0}(X_{\mathbb{R}} - \lbrace 0 \rbrace ,\mathbb
{C}) $ we associate a $n-$cycle $ \Gamma(A) $ ("explicitly " described) in the
complex Milnor fiber of $f_{\mathbb{C}}$ at 0 such that the non trivial terms
in the asymptotic expansions of the oscillating integrals $ \int_{A} e^{i\tau
f(x)} \phi(x) $ when $ \tau \to \pm \infty $ can be read from the spectral
decomposition of $\Gamma(A) $ relative to the monodromy of $f_{\mathbb{C}}$ at
0 .
