Some properties of quasiperiodic subshifts

Abstract

Let 𝒜 be a finite alphabet. A word w over 𝒜 is said to be quasiperiodic if it has a finite proper subword q such that every position of w falls under some occurrence of q. In this case, q is said to be a quasiperiod of w. A quasiperiodic word with an infinite number of quasiperiods is called multi-scale quasiperiodic. We show that, unlike in the case of right infinite words, biinfinite multi-scale quasiperiodicity does not imply uniform recurrence. The relation between biinfinite quasiperiodicity and other notions of symmetry of words were explored. It was also shown that the q-quasiperiodic subshift Xq, which is the set of all q-quasiperiodic biinfinite words, is a (2|q| – 2)-memory shift of finite type. By identifying all periodic points in Xq, a necessary and sufficient condition for the q-quasiperiodic subshift Xq to be mixing was established. Lastly, disjoint unions of these qi-quasiperiodic subshifts, where the qi’s are non-empty finite words over 𝒜, were looked into. A sufficient condition for such a union be a shift of finite type was given.