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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Periodic seeded arrays and automorphisms of the shift
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by Ezra Brown PDF
Trans. Amer. Math. Soc. 339 (1993), 141-161 Request permission

Abstract:

The automorphism group $\operatorname {Aut}({\Sigma _2})$ of the full $2$-shift is conjectured to be generated by the shift and involutions. We approach this problem by studying a certain family of automorphisms whose order was unknown, but which we show to be finite and for which we find factorizations as products of involutions. The result of this investigation is the explicit construction of a subgroup $\mathcal {H}$ of $\operatorname {Aut}({\Sigma _2})$ ; $\mathcal {H}$ is generated by certain involutions ${g_n}$, and turns out to have a number of curious properties. For example, ${g_n}$ and ${g_k}$ commute unless $n$ and $k$ are consecutive integers, the order of ${g_{n + k}} \circ \cdots \circ {g_k}$ is independent of $k$, and $\mathcal {H}$ contains elements of all orders. The investigation is aided by the development of results about certain new types of arrays of $0$’s and $1$’s called periodic seeded arrays, as well as the use of Boyle and Krieger’s work on return numbers and periodic points.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 339 (1993), 141-161
  • MSC: Primary 58F03; Secondary 28D20, 54H20
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1145960-X
  • MathSciNet review: 1145960