We introduce a dimension group for a self-similar map as the K0-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 ℤ∞ ≅= ℤ[t] together with the unilatral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the K0-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.
46L35, 46L80, 46L55
|Publication status||Published - Mar 23 2018|
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