Some Properties of Quasiperiodic Subshifts
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.
Palacio, J. D., Walo, Ma. L. B., & Miro, E. D. P. (2019). Some properties of quasiperiodic subshifts. AIP Conference Proceedings, 2192(1), 040009. https://doi.org/10.1063/1.5139135