To Symbolic Dynamics Through The Thue-Morse Sequence

Abstract

The celebrated Thue-Morse sequence, or the Prouhet-Thue-Morse sequence (A010060 in the OEIS), has a number of interesting properties and is a rich source to many (counter)examples. We introduce two different square-free sequences on three letters with one of them is equivalent (up-to permutations of letters) to Thue's original square-free sequence on three letters. Then we use them to introduce an explicit method to construct infinitely many number of recurrent points in $\{0,1{\}}^{\mathbb{N}}$ whose orbit closures under the shift map is minimal, uncountable, and for any two distinct such points their orbit closures are disjoint.