Entropy of a symbolic extension of a dynamical system

Residual entropy of a topological system is defined as the infimum increase of entropy necessary to build a symbolic extension of this system. If no symbolic extension exists then residual entropy is set at infinity. In this paper we provide a direct formula for the residual entropy of a system on a totally disconnected compact space in terms of basic notions of conditional entropies viewed as functions of invariant measures. This formula allows us to evaluate residual entropy in many examples as well as to construct new examples with arbitrarily preset topological and residual entropies. The appendix contains a condition equivalent to asymptotic h-expansiveness.