Entropy pair realization

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