Entropy pair realization
: Salo Ville
Publisher: CAMBRIDGE UNIV PRESS
: 2022
: Ergodic Theory and Dynamical Systems
: ERGODIC THEORY AND DYNAMICAL SYSTEMS
: ERGOD THEOR DYN SYST
: 18
: 0143-3857
: 1469-4417
DOI: https://doi.org/10.1017/etds.2021.175
: https://research.utu.fi/converis/portal/detail/Publication/175398719
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.
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