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Transactions of the American Mathematical Society

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Binary sequences which contain no $ BBb$


Author: Earl D. Fife
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
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0576867-5
Keywords: Morse minimal set, nonrepetitive sequences
Article copyright: © Copyright 1980 American Mathematical Society

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