Minimal subshifts of arbitrary mean topological dimension

Abstract

Let $G$ be a countable infinite amenable group and $P$ be a polyhedron. We give a construction of minimal subshifts of $P^G$ with arbitrarily mean topological dimension less than $\dim P$.Comment: 15 page

Mean dimension and a non-embeddable example for amenable group actions

“…We list some properties of tilings, which are going to be used in our main proof. Some of these properties may be found in [Dou17]. Here we reproduce their proofs for completeness.…”

“…Let {δ n } ∞ n=1 be a strictly decreasing sequence of positive numbers converging to zero. Take an increasing sequence {P n } ∞ n=1 of finite subsets of ( 9 ) The term "syndetic" here corresponds to "irreducible" in [Dou17].…”

Mean dimension and a non-embeddable example for amenable group actions

“…We list some propositions of tilings as follows, which are going to be used in our main proof. Some of these propositions may be found in [Dou17]. Here we reproduce their proofs for completeness.…”

“…The outer limit exists because Widim ǫ (X, d Fn ) is monotone with respect to ǫ. 10 The term "syndetic" here corresponds to the term "irreducible" in [Dou17].…”

Dynamical alternating groups, stability, property Gamma, and inner amenability

“…More generally, if Γ is of the form Γ 0 × Z where Γ 0 is infinite then Γ has property ID. To verify this we can use tilings which are products (in the obvious sense) of a suitable tiling of Γ 0 , as given by Theorem 3.5 of [15], and a monotiling of Z by translates of an interval of the form {−n, −n + 1, . .…”