Let $f(x,y) \in \mathcal{O}^\ast_{\mathbb{C}^2,0}$, a germ of holomorphic function at the origin of $\mathbb{C}^2$ with $f(0,0)=1$. Let $$\varphi(x,y)=(ax+by,cx+dy)$$ be a linear germ of biholomorphism ( $ad-bc\neq0$) of finite order, $\varphi^n(x,y)=(x,y)$. Suppose that $$\prod_{i=0}^{n-1}f\circ\varphi^i =1.$$ Is it true that $f=1$? I think this shuold not be a hard question but I could not figure out how to prove or find a counterexample.