Strict germs on normal surface singularities

We show that any holomorphic germ $f\text{\hspace{0.17em}⁣}:(X,x_0)\to (Y,y_0)$ of topological degree $1$ between normal surface singularities can be written as $f=\pi \circ \sigma$, where $\pi \text{\hspace{0.17em}⁣}:Y^{\mathrm{\prime }}\to (Y,y_0)$ is a modification and $\sigma \text{\hspace{0.17em}⁣}:(X,x_0)\to (Y^{\mathrm{\prime }},y_1)$ is a local isomorphism sending $x_0$ to a point $y_1\in {\pi }^{-1}(y_0)$. A result by Fantini, Favre and myself guarantees that when $f$ is a selfmap, then $(X,x_0)$ is a sandwiched singularity. We give here an alternative proof based on the construction of the associated Kato surfaces, and valuative dynamics.