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Proceedings of the American Mathematical Society

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The decomposition theorem
for two-dimensional shifts of finite type


Authors: Aimee S. A. Johnson and Kathleen M. Madden
Journal: Proc. Amer. Math. Soc. 127 (1999), 1533-1543
MSC (1991): Primary 58F03
Published electronically: January 29, 1999
MathSciNet review: 1476140
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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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Additional Information

Aimee S. A. Johnson
Affiliation: Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania 19081
Email: aimee@swarthmore.edu

Kathleen M. Madden
Affiliation: Department of Mathematics and Computer Science, Drew University, Madison, New Jersey 07940
Email: kmadden@drew.edu

DOI: https://doi.org/10.1090/S0002-9939-99-04678-X
Keywords: Decomposition Theorem, two-dimensional shifts of finite type, textile systems
Received by editor(s): June 24, 1997
Received by editor(s) in revised form: September 2, 1997
Published electronically: January 29, 1999
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society