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

Wagoner, J. B.

doi:10.1090/S0002-9947-1990-1027363-9

Eventual finite order generation for the kernel of the dimension group representation
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Trans. Amer. Math. Soc. 317 (1990), 331-350 Request permission

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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