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Eventual finite order generation for the kernel of the dimension group representation

Author: J. B. Wagoner
Journal: Trans. Amer. Math. Soc. 317 (1990), 331-350
MSC: Primary 54H20; Secondary 57S99, 60J10
MathSciNet review: 1027363
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Abstract: The finite order generation problem (FOG) in symbolic dynamics asks whether every element in the kernel of the dimension group representation of a subshift of finite type $ ({X_A},{\sigma _A})$ is a product of elements of finite order in the group $ {\operatorname{Aut}}({\sigma _A})$ of homeomorphisms of $ {X_A}$ commuting with $ {\sigma _A}$. We study the space of strong shift equivalences over the nonnegative integers, and the first application is to prove Eventual FOG which says that every inert symmetry of $ {\sigma _A}$ is a product of finite order homeomorphisms of $ {X _A}$ commuting with sufficiently high powers of $ {\sigma _A}$. Then we discuss the relation of FOG to Williams' lifting problem (LIFT) for symmetries of fixed points. In particular, either FOG or LIFT is false. Finally, we also discuss $ p$-adic convergence and other implications of Eventual FOG for gyration numbers.

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Keywords: Kernel of the dimension group representation, Eventual FOG, the space of strong shift equivalences, LIFT, $ p$-adic asymptotic gyration numbers
Article copyright: © Copyright 1990 American Mathematical Society

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