Entropy pair realization

Abstract

We show that the complete positive entropy (CPE) class alpha of Barbieri and Garcia-Ramos contains a one-dimensional subshift for all countable ordinals alpha, that is, the process of alternating topological and transitive closure on the entropy pairs relation of a subshift can end on an arbitrary ordinal. This is the composition of three constructions. We first realize every ordinal as the length of an abstract 'close-up' process on a countable compact space. Next, we realize any abstract process on a compact zero-dimensional metrizable space as the process started from a shift-invariant relation on a subshift, the crucial construction being the implementation of every compact metrizable zero-dimensional space as an open invariant quotient of a subshift. Finally, we realize any shift-invariant relation E on a subshift X as the entropy pair relation of a supershift Y superset of X, and under strong technical assumptions, we can make the CPE process on Y end on the same ordinal as the close-up process of E.