+1,
    \quad k\in{{\mathbb{N}}},$$ where $\operatorname{\overline{ind}}(\gamma)\geq 0$ is the average index of $\gamma$ defined by $$\operatorname{\overline{ind}}(\gamma) := \lim_{k\to\infty} \frac{\operatorname{ind}(\gamma^k)}{k}.$$ Let $p\in M$ and $\Omega_p M\subset \Lambda M$ be the set of loops based at $p$, that is $H^1$-paths $\gamma:[0,1]\to M$ such that $\gamma(0)=\gamma(1)=p$. Given a closed geodesic $\gamma\in\Lambda M$, we denote by $\operatorname{ind}_\Omega(\gamma)\in{{\mathbb{N}}}$ the Morse index $$\operatorname{ind}_\Omega (\gamma) := \operatorname{ind}\left(E|_{\Omega_{\gamma(0)}M},\gamma\right).$$ By inclusion, $\operatorname{ind}_\Omega (\gamma)\leq \operatorname{ind}(\gamma)$. In fact, we have the concavity inequality [@BTZ Eq. (1.5)]: $$\label{eq:conc}
    \operatorname{ind}(\gamma) - \dim(M) +1 \leq
    \operatorname{ind}_\Omega(\gamma)\leq
    \operatorname{ind}(\gamma).$$

A Jacobi field of the geodesic path $\gamma$ is a smooth map $J:{{\mathbb{R}}}\to\gamma^*TM$, satisfying $$J(t)\in T_{\gamma(t)}M, \quad \forall t\in{{\mathbb{R}}}\quad \text{and} \quad
    \nabla_{\dot{\gamma}}^2 J = R(\dot{\gamma},J)\dot{\gamma},$$ where $R$ denotes the Riemann tensor. Let $\mu(t)$ be the number of linearly independent Jacobi fields of $\gamma$ such that $J(0)=J(t)=0$; the Morse index theorem states that $$\label{eq:MorseIndexThm}
    \operatorname{ind}_\Omega (\gamma) = \sum_{0<t<1} \mu(t).$$

The local homology of a critical circle $S^1\cdot\gamma$ is by definition $${\textup{C}}_*(S^1\cdot\gamma) := H_* (\{ E < E(\gamma) \} \cup S^1\cdot \gamma,
    \{ E < E(\gamma) \}),$$ where the set $\{ E<E(\gamma)\}$ is $\{\delta\in\Lambda M\ |\ E(\delta)<E(\gamma)\}$, and $H_*$ denotes the singular homology with integral coefficients. A closed geodesic is said to be homologically visible if ${\textup{C}}_*(S^1\cdot\gamma)\neq 0$ and is 