Let (X, 0) be a complex analytic surface germ embedded in \(({\mathbb {C}}^n,0)\) with an isolated singularity and \(\Phi =(g,f):(X,0) \longrightarrow ({\mathbb {C}}^2,0)\) be a finite morphism. We define a family of analytic invariants of the morphism \(\Phi \) , called inner rates of \(\Phi \) . By means of the inner rates we study the polar curve associated with the morphism \(\Phi \) when fixing the topological data of the curve \((gf)^{-1}(0)\) and the surface germ (X, 0), allowing to address a problem called polar exploration. We also use the inner rates to study the geometry of the Milnor fibers of a non constant holomorphic function \(f:(X,0) \longrightarrow ({\mathbb {C}},0)\) . The main result is a formula which involves the inner rates and the polar curve alongside topological invariants of the surface germ (X, 0) and of the curve \((gf)^{-1}(0)\) .
