Matsumoto -groups associated to certain shift spaces

  • Toke Meier Carlsen

    University of Copenhagen, Denmark
  • Søren Eilers

    University of Copenhagen, Denmark
Matsumoto $K$-groups associated to certain shift spaces cover
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Abstract

In [24] Matsumoto associated to each shift space (also called a subshift) an Abelian group which is now known as Matsumoto's -group. It is defined as the cokernel of a certain map and resembles the first cohomology group of the dynamical system which has been studied in for example [2], [28], [13], [16] and [11] (where it is called the dimension group).

In this paper, we will for shift spaces having a certain property , show that the first cohomology group is a factor group of Matsumoto's -group. We will also for shift spaces having an additional property , describe Matsumoto's -group in terms of the first cohomology group and some extra information determined by the left special elements of the shift space.

We determine for a broad range of different classes of shift spaces if they have property and property and use this to show that Matsumoto's -group and the first cohomology group are isomorphic for example for finite shift spaces and for Sturmian shift spaces.

Furthermore, the ground is laid for a description of the Matsumoto -group as an emphordered group in a forthcoming paper.

Cite this article

Toke Meier Carlsen, Søren Eilers, Matsumoto -groups associated to certain shift spaces. Doc. Math. 9 (2004), pp. 639–671

DOI 10.4171/DM/182