  We show that every family of isolated hypersurface singularity with constant
Milnor number has constant multiplicity. To achieve this, we endow the A'Campo
model of "radius zero" monodromy with a symplectic structure. This new approach
allows to generalize a spectral sequence of McLean converging to fixed point
Floer homology of iterates of the monodromy to a more general setting which is
well suited to study $\mu$-constant families.
