  We prove that systems satisfying the specification property are saturated in
the sense that the topological entropy of the set of generic points of any
invariant measure is equal to the measure-theoretic entropy of the measure. We
study Banach valued Birkhoff ergodic averages and obtain a variational
principle for its topological entropy spectrum. As application, we examine a
particular example concerning with the set of real numbers for which the
frequencies of occurrences in their dyadic expansions of infinitely many words
are prescribed. This relies on our explicit determination of a maximal entropy
measure.
