2015/03/21 by Joshua Frisch, Omer Tamuz · 19 citations
Computer Science · Mathematics · #Alphabet #Amenable group #Cellular Automata and Applications #Combinatorics #Computer science #Entropy (arrow of time) #Mathematical Dynamics and Fractals #Mathematics #Physics #Pure mathematics #Quantum mechanics #Space (punctuation) #Subshift of finite type #Symbolic dynamics #Transitive relation #math.DS #math.GR #semigroups and automata theory
paper · pdf · doi:10.1017/etds.2015.84
published in Ergodic Theory and Dynamical Systems 37(4), 1187-1210 (Cambridge University Press)
arxiv created 2015/03/21 · openalex publication_date 2016/01/28 · arxiv updated 2018/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let G be a finitely generated amenable group. We study the space of shifts on G over a given finite alphabet A . We show that the zero entropy shifts are generic in this space, and that, more generally, the shifts of entropy c are generic in the space of shifts with entropy at least c . The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that, for every entropy value c∈ [0,log |A|] , there is a weakly mixing subshift of AG with entropy c . We also show that the set of strongly irreducible shifts does not form a G_\unicode[STIX]x1D6FF in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.