Dimension groups for self-similar maps and matrix representations of the core of the associated C*-algebras

TL;DR

This paper introduces a dimension group for self-similar maps via the K_0-group of associated C*-algebras, explicitly describing the core as an inductive limit and analyzing the structure for the tent map.

Contribution

It provides an explicit matrix representation of the core and computes the dimension group for the tent map, revealing new insights into the structure of these C*-algebras.

Findings

Dimension group for the tent map is isomorphic to Z^∞ with the unilateral shift.

Canonical endomorphisms on K_0-groups are generally not automorphisms.

Singularity structures can be counted within the dimension groups.

Abstract

We introduce a dimension group for a self-similar map as the ${\rm K}_0$-group of the core of the $C^*$-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit description of the core as the inductive limit using their matrix representations over the coefficient algebra, which can be described explicitly by the singularity structure of branched points. We compute that the dimension group for the tent map is isomorphic to the countably generated free abelian group ${\mathbb Z}^{\infty}\cong {\mathbb Z}[t]$ together with the unilatral shift, i.e. the multiplication map by $t$ as an abstract group. Thus the canonical endomorphisms on the ${\rm K}_0$-groups are not automorphisms in geneal. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.