Binary sequences which contain no $BBb$

Abstract:

A (one-sided) sequence or (two-sided) bisequence is irreducible provided it contains no block of the form BBb, where b is the initial symbol of the block B. Gottschalk and Hedlund [Proc. Amer. Math. Soc. 15 (1964), 70-74] proved that the set of irreducible binary bisequences is the Morse minimal set M. Let ${M^ + }$ denote the one-sided Morse minimal set, i.e. ${M^ + } = \{ {x_0}{x_1}{x_2} \ldots : \ldots {x_{ - 1}}{x_0}{x_1} \ldots \in M\}$. Let ${P^ + }$ denote the set of all irreducible binary sequences. We establish a method for generating all $x \in {P^ + }$. We also determine ${P^ + } - {M^ + }$. Considering ${P^ + }$ as a one-sided symbolic flow, ${P^ + }$ is not the countable union of transitive flows, thus ${P^ + }$ is considerably larger than ${M^ + }$. However ${M^ + }$ is the $\omega$-limit set of each $x \in {P^ + }$, and in particular ${M^ + }$ is the nonwandering set of ${P^ + }$.