In this paper we formulate a unified theory of multiple sine functions by using a view point of absolute zeta functions and absolute automorphic forms.

Let $f\colon X\to X$ be a non-isomorphic étale endomorphism of a smooth projective variety $X$. Suppose that there exists a $K_{X}$-negative extremal ray $R'\subset \overline{\mathrm{NE}}(X)$ of fiber type. Then we give a sufficient condition for a $K_{X}$-negative extremal ray $R\subset \overline{\mathrm{NE}}(X)$ of divisorial type to terminate under a suitable power $f^{k}$ of $k > 0$.