Entropy points and applications

First notions of entropy point and uniform entropy point are introduced using Bowen's definition of topological entropy. Some basic properties of the notions are discussed. As an application it is shown that for any topological dynamical system there is a countable closed subset whose Bowen entropy is equal to the entropy of the original system.

Then notions of C-entropy point are introduced along the line of entropy tuple both in topological and measure-theoretical settings. It is shown that each C-entropy point is an entropy point, and the set of C-entropy points is the union of sets of C-entropy points for all invariant measures.