We consider surface automorphisms of infinite order.
If the topological entropy of a surface automorphism is zero, then there is an invariant elliptic fibration.
We construct surface automorphisms with invariant elliptic fibration.
For some surface automorphism, all the eigenvalues, which are roots of unity, correspond to periodic cycles of curves.
The degree of a rational function that defines such invariant fibration can be 3, 4, 5, 6.