Closed Orbits in Homology Class for Hyperbolic Flows

Let $M$ be a $C^\infty$ compact manifold, $\phi_t :\Lambda (\subset M)\to \Lambda$ is a hyperbolic flow. For a closed orbit $\gamma$ of $\phi_t$, the homololgy class of $\gamma$ denoted by $[\gamma ](\in H_1(M,{\bf Z}))$, we give an asymptotic expansion including error terms for the number of closed orbits in homology classes $\pi (T,\alpha ) := \sharp (\gamma\leq T, [\gamma ] = \alpha )\}$. In particular, we discuss how the error terms depend on the homology classes, we also discuss the asymptotic behavior of certain orbital average, that is if $\chi$ is a non-trivial unitary character of $H_1(M,{\bf Z})$ we discuss the error tems of $\frac{1}{\pi (T)}\sum\limits_{\ell (\gamma )\leq T} \chi ([\gamma ])$ as $T\to\infty$.