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We prove that systems satisfying the specification property are saturated inthe sense that the topological entropy of the set of generic points of anyinvariant measure is equal to the measure-theoretic entropy of the measure. Westudy Banach valued Birkhoff ergodic averages and obtain a variationalprinciple for its topological entropy spectrum. As application, we examine aparticular example concerning with the set of real numbers for which thefrequencies of occurrences in their dyadic expansions of infinitely many wordsare prescribed. This relies on our explicit determination of a maximal entropymeasure.

Generic points in systems of specification and Banach valued Birkhoff ergodic average