Eventual Extensions of Finite Codes

Suppose S and T are shift equivalent mixing shifts of finite type, and / is a conjugacy from a subsystem of S to a subsystem of T. Then for any sufficiently large n, f extends to a conjugacy of Sn and Tn. A consequence of the proof is a fortified version of Wagoner's Stable FOG Theorem. Recall [Wi] that a nonnegative integral matrix A is the adjacency matrix of a directed graph whose arc set is the coordinate state space of a shift of finite type (SFT), which we denote (X^, 5,4). In this paper we prove the following theorem. Eventual Extension Theorem. Let (X,S) be a subshift contained in a mixing SFT (X^S^). If f be a continuous injective map f: X —► Xb such that f S = SßfThen the following are equivalent. (1) A and B are shift equivalent. (2) For any sufficiently large n, there exists a homeomorphism f: Xa —* Xb such that f = JonXand f(SA)n = (SB)nfCOROLLARY. Let (X,S) be a mixing SFT and suppose U is an automorphism of a subshift of (X,S). Then for all large n, U extends to an automorphism of (X,Sn). As an application of the ideas involved in the proof of the theorem (specificially of Lemma 1), we provide in the appendix a more direct proof of Wagoner's Stable FOG Theorem [Wa2]. This proof also yields technical improvements in the result which lead to a concrete presentation theorem for automorphisms of the shift which lie in the kernel of the dimension representation. We find the theorem and its corollary of interest for three reasons. The first is its relevance to a fundamental unsolved problem of symbolic dynamics, due essentially to R. F. Williams: when does an automorphism of a subsystem of a mixing SFT extend to an automorphism of the SFT? This question is basic to understanding the dynamics: one wants to know whether isomorphic subsystems, especially finite subsystems such as fixed points, can sit within the SFT in essentially different ways. Also, Williams has pointed out that this extension problem provides a test for his conjecture [Wi] that shift equivalence implies conjugacy: for example, it is possible that a transposition of fixed points may extend to an automorphism in one SFT but not in a shift equivalent SFT. The corollary provides some insight into this Received by the editors December 1, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H20; Secondary 58F15, 28D20. This research was partially supported by the National Science Foundation under grant DMS8601619. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page