Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation

A classical Kamae-Weiss theorem states that an increasing sequence (ni)i∈ℕ of positive lower density is normality preserving, i.e., has the property for any normal binary (bn)n∈ℕ, $${({n_i})_{i \in \mathbb{N}}}$$ normal, if and only a deterministic sequence. Given countable cancellative amenable semigroup G, Følner $${\cal F} = {({F_n})_{n \in}}$$ in we introduce notions preservation, determinism subexponential complexity subsets G with respect to F}$$ show sets -density these three are equivalent. The proof utilizes apparatus theory tilings groups notion tile-entropy We also prove under natural assumption on follows from preservation. Finally, provide numerous examples preserving various semigroups.