Pseudo grading on cluster automorphism group with application to cluster algebras of rank $3$

We introduce a pseudo $\mathbb{N}$-grading on the cluster auotmorphism group $\mathrm{Aut}(\mathcal{A})$ with respect to an initial seed of $\mathcal{A}$, which consists of a family of subsets $\{G_i{\}}_{i\in \mathbb{N}}$ of $\mathrm{Aut}(\mathcal{A})$ such that $\mathrm{Aut}(\mathcal{A})={\bigcup }_{i\in \mathbb{N}}{G}_i$ and $G_k\cdot G_l\subset {\bigcup }_{i=0}^{k+l}{G}_i$. We prove that $\mathrm{Aut}(\mathcal{A})$ is generated by $G_0\cup G_1$, leading to an elementary approach for calculating cluster automorphism groups of certain cluster algebras. As an application, we completely determined the cluster automorphism groups of cluster algebras of rank $3$ with indecomposable exchange matrices.