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Transactions of the American Mathematical Society

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



Periodic seeded arrays and automorphisms of the shift

Author: Ezra Brown
Journal: Trans. Amer. Math. Soc. 339 (1993), 141-161
MSC: Primary 58F03; Secondary 28D20, 54H20
MathSciNet review: 1145960
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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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  • [BK] M. Boyle and W. Krieger, Periodic points and automorphisms of the shift, Trans. Amer. Math. Soc. 302 (1987), 125-149. MR 887501 (88g:54065)
  • [BLR] M. Boyle, D. Lind, and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc. 306 (1988), 71-114. MR 927684 (89m:54051)
  • [Br] E. Brown, Commuting block maps in the shift dynamical system, submitted.
  • [CHR] E. M. Coven, G. A. Hedlund, and F. Rhodes, The commuting block maps problem, Trans. Amer. Math. Soc. 249 (1979), 113-138. MR 526313 (81k:54072)
  • [H] G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory 3 (1969), 320-375. MR 0259881 (41:4510)
  • [N] L. P. Neuwirth, private communication.
  • [R1] F. Rhodes, The principal part of a block map, J. Combin. Theory Ser. A 33 (1982), 48-64. MR 665655 (84d:58045)
  • [R2] -, The sums of powers theorem for commuting block maps, Trans. Amer. Math. Soc. 271 (1982), 225-236. MR 648088 (84i:54043)
  • [R3] -, Left cancellation of block maps, Bull. London Math. Soc. 16 (1984), 19-24. MR 719800 (85f:54091)
  • [R4] -, The role of the principal part in factorizing block maps, Math. Proc. Cambridge Philos. Soc. 96 (1984), 223-235. MR 757657 (86d:54062)
  • [R5] -, The enumeration of certain sets of block maps, J. Combin. Theory Ser. A 45 (1987), 263-276. MR 894821 (88h:05016)
  • [Ry] J. P. Ryan, The shift and commutativity, Math. Systems Theory 6 (1972), 82-85. MR 0305376 (46:4506)

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Keywords: Block maps, shift dynamical system, automorphism group, symbolic dynamics, $ 0{\text{-}}1$ arrays, periodic points
Article copyright: © Copyright 1993 American Mathematical Society

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