  In the recent paper \cite{LoD1}, we classified closed geodesics on Finsler
manifolds into rational and irrational two families, and gave a complete
understanding on the index growth properties of iterates of rational closed
geodesics. This study yields that a rational closed geodesic can not be the
only closed geodesic on every irreversible or reversible (including Riemannian)
Finsler sphere, and that there exist at least two distinct closed geodesics on
every compact simply connected irreversible or reversible (including
Riemannian) Finsler 3-dimensional manifold. In this paper, we study the index
growth properties of irrational closed geodesics on Finsler manifolds. This
study allows us to extend results in \cite{LoD1} on rational and in
\cite{DuL1}, \cite{Rad4} and \cite{Rad5} on completely non-degenerate closed
geodesics on spheres and $\CP^2$ to every compact simply connected Finsler
manifold. Then we prove the existence of at least two distinct closed geodesics
on every compact simply connected irreversible or reversible (including
Riemannian) Finsler 4-dimensional manifold.
