Conditional variational principles of conditional entropies for amenable group actions *

Let G be an infinite discrete countable amenable group acting continuously on two compact metrizable spaces X, Y. Assume that phi : (Y, G) -> (X, G) is a factor map. Using finite open covers, the conditional topological entropy of phi is defined. The conditional measure-theoretic entropy of phi equals the conditional measure-theoretic entropy of Y to X. With the aid of tiling system of G, the conditional variational principle of phi is studied when (X, G) is an asymptotically h-expansive system. If X = Y and phi is the identity map, the conditional topological entropy of system (X, G) is defined. In the Cartesian square (X x X, G), we define the conditional measure-theoretic entropy of (X, G) to be the defect of the upper semi-continuity of the conditional measure-theoretic entropy of X x X to the first axis. Then the conditional variational principle of (X, G) is obtained.