## Binary sequences which contain no $BBb$

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- by Earl D. Fife
- Trans. Amer. Math. Soc.
**261**(1980), 115-136 - DOI: https://doi.org/10.1090/S0002-9947-1980-0576867-5
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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. ${M^ + } = \{ {x_0}{x_1}{x_2} \ldots : \ldots {x_{ - 1}}{x_0}{x_1} \ldots \in M\}$. 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 ${P^ + }$.

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## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**261**(1980), 115-136 - MSC: Primary 05B30; Secondary 54H20
- DOI: https://doi.org/10.1090/S0002-9947-1980-0576867-5
- MathSciNet review: 576867