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

TL;DR

This paper establishes a connection between specification systems and entropy of generic points, introduces a variational principle for Banach valued Birkhoff averages, and applies these results to analyze frequency patterns in dyadic expansions.

Contribution

It proves saturation for systems with the specification property and develops a variational principle for the entropy spectrum of Banach valued Birkhoff averages, with explicit applications.

Findings

Systems with the specification property are saturated.

A variational principle for the entropy spectrum of Banach valued Birkhoff averages is established.

Explicit maximal entropy measures are determined for specific frequency pattern sets.

Abstract

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.