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# Binary Sequences which Contain no $BBb$

Earl D. Fife
Transactions of the American Mathematical Society
Vol. 261, No. 1 (Sep., 1980), pp. 115-136
DOI: 10.2307/1998321
Stable URL: http://www.jstor.org/stable/1998321
Page Count: 22
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## 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. $M^+ = \{x_0x_1x_2 \ldots:\ldots x_{-1}x_0x_1 \ldots \in M\}$. 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 $P^+$.

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