The projective fundamental group of a ℤ 2 -shift
William Geller, James Gary Propp · Ergodic Theory and Dynamical Systems · 1995
Abstract We define a new invariant for symbolic ℤ 2 -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 ℤ 2 -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 ℤ 2 -action that π 1 ; of a topological space bears to π 0 . The projective fundamental group is invariant under topological conjugacy. We calculate this invariant for several important examples of ℤ 2 -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.