Proceedings of the American Mathematical Society, 1999
A one-dimensional shift of finite type can be described as the collection of bi-infinite “walks&q... more 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.
We classify n-point extensions of ergodic Z-actions up to relative orbit equivalence and establis... more We classify n-point extensions of ergodic Z-actions up to relative orbit equivalence and establish criteria under which one n-point extension of an ergodic Z-action can be sped up to be relatively isomorphic to an n-point extension of another ergodic Z-action. Both results are characterized in terms of an algebraic object associated to each n-point extension which is a conjugacy class of subgroups of the symmetric group on n elements.
Proceedings of the American Mathematical Society, 1999
A one-dimensional shift of finite type can be described as the collection of bi-infinite “walks&q... more 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.
We classify n-point extensions of ergodic Z-actions up to relative orbit equivalence and establis... more We classify n-point extensions of ergodic Z-actions up to relative orbit equivalence and establish criteria under which one n-point extension of an ergodic Z-action can be sped up to be relatively isomorphic to an n-point extension of another ergodic Z-action. Both results are characterized in terms of an algebraic object associated to each n-point extension which is a conjugacy class of subgroups of the symmetric group on n elements.
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