The decomposition theorem for two-dimensional shifts of finite type

Johnson, Aimee; Madden, Kathleen

doi:10.1090/S0002-9939-99-04678-X

The decomposition theorem for two-dimensional shifts of finite type
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by Aimee S. A. Johnson and Kathleen M. Madden PDF

Proc. Amer. Math. Soc. 127 (1999), 1533-1543 Request permission

Abstract:

A one-dimensional shift of finite type can be described as the collection of bi-infinite “walks" along an edge graph. The Decomposition Theorem states that every conjugacy between two shifts of finite type can be broken down into a finite sequence of splittings and amalgamations of their edge graphs. When dealing with two-dimensional shifts of finite type, the appropriate edge graph description is not as clear; we turn to Nasu’s notion of a “textile system" for such a description and show that all two-dimensional shifts of finite type can be so described. We then define textile splittings and amalgamations and prove that every conjugacy between two-dimensional shifts of finite type can be broken down into a finite sequence of textile splittings, textile amalgamations, and a third operation called an inversion.

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