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The projective fundamental group of a ℤ2-shift

Published online by Cambridge University Press:  14 October 2010

William Geller  and
James Propp
William Geller
Affiliation:
Institute of Mathematics, University of Warwick, Coventry CV4 7AL, UK
James Propp
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA (propp@math.mit.edu.)
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Abstract

We define a new invariant for symbolic ℤ2-actions, the projective fundamental group. This invariant is the limit of an inverse system of groups, each of which is the fundamental group of a space associated with the ℤ2-action. The limit group measures a kind of long-distance order that is manifested along loops in the plane, and roughly speaking bears the same relation to the mixing properties of the ℤ2-action that π1; of a topological space bears to π0. The projective fundamental group is invariant under topological conjugacy. We calculate this invariant for several important examples of ℤ2-actions, and use it to prove non-existence of certain constant-to-one factor maps between two-dimensional subshifts. Subshifts that have the same entropy and periodic point data can have different projective fundamental groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 6 , December 1995 , pp. 1091 - 1118
Copyright
Copyright © Cambridge University Press 1995

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