Singularities of mappings on ICIS and applications to Whitney equisingularity

We study germs of analytic maps $f:(X,S)\rightarrow(\mathbb{C}^p,0)$, when $X$ is an ICIS of dimension $n<p$. We define an image Milnor number, generalizing Mond's definition, $\mu_I(X,f)$ and give results known for the smooth case such as the conservation of this quantity by deformations. We also use this to characterise the Whitney equisingularity of families of corank one map germs $f_t\colon(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instabilities in terms of the constancy of the $\mu_I^*$-sequences of $f_t$ and the projections $\pi\colon D^2(f_t)\to\mathbb{C}^n$, where $D^2(f_t)$ is the ICIS given by double point space of $f_t$ in $\mathbb{C}^n\times\mathbb{C}^n$. The $\mu_I^*$-sequence of a map germ consist of the image Milnor number of the map germ and all its successive transverse slices.