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^{itau 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 .
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