Conditional entropy and fiber entropy for amenable group actions

In this paper, we introduce the notions of topological conditional entropy and fiber entropy for a given factor map between two amenable group actions, and prove three variational principles for conditional entropy and fiber entropy. Moreover, as an application of our variational principle we prove that the countable-to-one extension and the distal extension have zero conditional topological entropy. In particular, the topological entropy of a distal system is zero.