Abstract
We study various aspects of periodic points for random substitution subshifts. In order to do so, we introduce a new property for random substitutions called the disjoint images condition. We provide a procedure for determining the property for compatible random substitutions—random substitutions for which a well-defined abelianisation exists. We find some simple necessary criteria for primitive, compatible random substitutions to admit periodic points in their subshifts. In the case that the random substitution further has disjoint images and is of constant length, we provide a stronger criterion. A method is outlined for enumerating periodic points of any specified length in a random substitution subshift.
Highlights
Random substitutions are a generalisation of the classical notion of a substitution on a finite alphabet
The dynamical systems and tilings associated with random substitutions provide good models for quasicrystaline structures that have long range order induced by an underlying hierarchical supertile structure, whilst possessing positive entropy
In the case of random substitutions, the question of identifying the existence of periodic points is decidedly more difficult. This is due to the intricate interplay between the shift dynamics and the long-range hierarchical structure induced by the inflation action of the random substitution, as well as the highly non-minimal nature of the subshift
Summary
Let An denote the set of words of length n over A given by all concatenations of letters from A and for u ∈ An, write |u| = n for the length of the word u. For a compatible random substitution θ, let Mθ denote the associated substitution matrix where the entry mi j is given by mi j := |θ(a j )|ai (which is well-defined by the compatibility of θ). The set of length-n legal words for θ is denoted by Lnθ := Lθ ∩ An. The random substitution subshift of θ (RS-subshift) is given by. Definition 6 A random substitution θ on a finite alphabet A is called primitive if there exists a k ∈ N such that for all ai , a j ∈ A we have ai θk(a j ). Before moving on to tackling questions about periodic points, we need to introduce and study an important property of random substitutions
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