We introduce a dimension group for a self-similar map as the $\mathrm {K}_0$ -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 unilateral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the $\mathrm {K}_0$ -groups are not automorphisms in general. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.

我们为自相似映射引入了一个维度组，作为与自相似映射相关的C * -代数核心的 $\mathrm {K}_0$ - 组以及规范自同态。计算的关键步骤是使用它们在系数代数上的矩阵表示将核心明确描述为归纳限制，这可以通过分支点的奇点结构明确描述。我们计算帐篷映射的维度群与可数生成的自由阿贝尔群 ${\mathbb Z}^{\infty }\cong {\mathbb Z}[t]$ 同构，并带有单边位移，即乘法t映射 作为一个抽象的群体。因此， $\mathrm {K}_0$ -群上的典型自同态一般不是自同构。与拓扑马尔可夫位移的维度组相比，这是一个不同的点。我们可以计算维度组中的奇异结构。