RELATIVE ENTROPY, ASYMPTOTIC PAIRS AND CHAOS

In this paper, we prove that positive conditional entropy implies the existence of asymptotic pairs and scrambled sets on fibers. Moreover, we introduce the notion of conditional topological entropy for a subset using Bowen’s definition of separated and spanning sets, and prove that the conditional topological entropy of a system relative to a factor is the supremum of conditional topological entropy of its scrambled sets on fibers.