Eventual Finite Order Generation for the Kernel of the Dimension Group Representation

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.