2019/04/02 by Salo, Ville
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1904.01285
We show that the CPE class α of Barbieri and García-Ramos contains a one-dimensional subshift for all countable ordinals α, i.e. 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 ⊃ 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.