On dimension functions and topological Markov chains

A Homology Theory for Smale Spaces: A Summary

“…The aim of this article is to introduce a new type of homology theory for such spaces. This takes as its starting point the notion of the dimension group of a shift of finite type introduced by Krieger [7] and the fundamental result of Bowen [2] that every basic set is a factor of a shift of finite type.…”

Operator Algebras and C*-correspondences: A Survey

“…Unlike strong shift equivalence, the study of shift equivalence, from the viewpoint of C * -correspondences, has been met with limited success [38]. (Other operator theoretic viewpoints however have been quite successful [36].) The concept of strong Morita equivalence for C * -correspondences was first developed and studied by Abadie, Eilers and Exel [1] and Muhly and Solel [42], and plays the role of a generalized Conjugacy (see Example 3.2 below).…”

Shift equivalence and the Jordan form away from zero

“…One can give a less elementary but more geometric demonstration of (3.4) which bypasses (3.2) and (3.3). Here one applies the direct limit viewpoint of Krieger [9] and considers the group automorphism B n obtained by restriction of B n to {xe V n :x(B n ) k eZ" +3 n V m for some fc>0}.…”

“…Shift equivalence is an invariant of conjugacy which is conjectured to be complete and which is more computable-it is often practical to decide if two matrices are shift equivalent, and significant partial results suggest there is a general decision procedure [7]. In addition, from Krieger's work we find shift equivalence intimately related to dimension groups [9] and the construction of factor maps [10].…”