Milnor-L\^e type fibrations for subanalytic maps

We prove a Milnor-L\^e type fibration theorem for a subanalytic map $f: X \to Y$ between subanalytic sets $X \subset \mathbb{R}^m$ and $Y \subset \mathbb{R}^n$. Moreover, if $f$ extends to an analytic map $\mathbb{R}^m \to \mathbb{R}^n$, we define the singular set and the discriminant set of $f$, in a stratified sense. Then we give Milnor-L\^e type fibration theorems for $f$ outside its discriminant, as well as over it. We give examples of each situation approached in this paper.