Inert Actions on Periodic Points

The action of inert automorphisms on finite sets of periodic points of mixing subshifts of finite type is characterized in terms of the sign-gyrationcompatibility condition. The main technique used is variable length coding combined with a “nonnegative algebraic K-theory” formulation of state splitting and merging. One application gives a counterexample to the Finite Order Generation Conjecture by producing examples of infinite order inert automorphisms of mixing subshifts of finite type which are not products of finite order automorphisms. Introduction, main results, and applications Subshifts of finite type (XA, σA) constructed from a nonnegative integral matrices A appear in a number of areas ranging from smooth dynamical systems to coding and information theory. See [LM] for a comprehensive introduction to these model dynamical systems and to the field of symbolic dynamics in general. The automorphism group Aut(σA) of (XA, σA) consists of those homeomorphisms of XA which commute with the basic shift homeomorphism σA : XA −→ XA. The first systematic study of Aut(σA) appeared in [He], and it has subsequently been studied in a number of papers. For example, see [BF], [BK1], [BLR], [KR1], [KRW1]. In general, Aut(σA) is a huge countable group. It contains copies of the direct sum of any countable collection of finite groups and is highly nonabelian. It contains a copy of the free abelian group on countably many generators. It contains copies of the fundamental groups of closed, 2-dimensional surfaces. It is residually finite, and therefore does not contain a divisible group. See [BLR]. The group Aut(σA) is interesting both in its own right and for its relation to other questions in symbolic dynamics. For example, present understanding of the fundamental classification problem for subshifts of finite type is closely related to Aut(σA). See [B], [KRW1], [KR2], [W]. One way to study Aut(σA) is through representations of it to simpler groups, and there are currently two such representations which play a key role; namely, the periodic point and periodic orbit representations and the dimension group representation. Received by the editors October 25, 1996. 1991 Mathematics Subject Classification. Primary 54H20, 57S99, 20F99. The first two authors were partially supported by NSF Grants DMS 8820201 and DMS 9405004. The last author was partially supported by NSF Grants DMS 8801333, DMS 9102959, and DMS 9322498. c ©1997 American Mathematical Society