Entropy preservation under a finite-to-one factor
= Entropy preservation under a finite-to-one factor
For standard probability systems, a <factor of a measure-preserving system> with at most $q$ points in every fibre preserves <Kolmogorov-Sinai entropy>. Conditional on the complete factor point, every finite orbit name has at most $q$ possibilities, so its conditional entropy is at most $\log q$. Dividing by the orbit length gives zero relative entropy.