Zeckendorf representations and mixing properties of sequences

Abstract

We use generalised Zeckendorf representations of natural numbers to investigate mixing properties of symbolic dynamical systems. The systems we consider consist of bi-infinite sequences associated with so-called random substitutions. We focus on random substitutions associated with the Fibonacci, tribonacci and metallic mean numbers and take advantage of their respective numeration schemes.

Mixing properties and entropy bounds of a family of Pisot random substitutions

“…(2). We now define the random analogues of the deterministic substitutions we have in the previous section, which were first mentioned in the previous work of the second and third authors [30]. Definition 4.…”

“…We focus here on a family {ψ n,p } of random substitutions first mentioned in [30] which we call the random noble Pisa substitutions. This is a two-parameter generalisation of both the random noble means family [5] and the Pisa family [4].…”

“…In Theorem 7, we showed that the random noble Pisa substitution ψ n,p induces a dynamical system (X n,p , S) that is not topologically mixing, i.e., there exist legal words t, w ∈ L(X n,p ) such that, for all K ∈ N, there is at least one k K such that tvw / ∈ L(X n,p ) for all v ∈ L k (X n,p ). We now show that (X n,p , S) satisfies a property that is weaker than mixing called semi-mixing, which was first introduced and studied for certain types of random noble Pisa substitutions in [40,30]. Even though it is a weaker property, it is still preserved under topological conjugacy, and thus is a well-defined invariant; see [40,Prop.…”

“…In [30], the generalised Zeckendorf representation of natural numbers was used to show that some well-known random noble Pisa substitutions (namely, the random Fibonacci substitution ψ 2,1 , random tribonacci substitution ψ 3,1 , and random metallic means substitutions ψ 2,p ) are semimixing. We now show that every random noble Pisa substitution ψ n,p gives rise to a numeration system and utilise this numeration system to prove that all random noble Pisa substitutions are semi-mixing.…”

“…We will need the following generalisation of the Zeckendorf and Ostrowski numerations, which is satisfied by the noble Pisa family; compare [15,30].…”