Closed Orbits in Homology Class for Hyperbolic Flows

Let $M$ be a $X^\infty$ compact manifold, $\phi_t :\Lambda (\subset M)\to Lambda$ is a hyperbolic flow. For a closed orbit $\gamma$ of $\phi_t$, the homology 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 that how the error terms depend on the homology classes, we also discuss the asymptotic behaviour of certain orbital average, that is if $\chi$ is a non-trivial unitary character of $H_1(M,{\bf Z})$ we discuss the error terms of $\frac{1}{\pi (T)} \Sigma\limits_{\ell )\leq T} \chi ([\gamma ])$ as $T\to\infty$.