A continuous real-valued *cocycle* for the action of $\Phi^t$ on $L$ is a continuous function $c:L\times \mathbb{R}\to \mathbb{R}$ with the property that $c(v,t+s)=c(v,t)+c(\Phi^t v,s)$ for all $v\in L$, all $s,t\in \mathbb{R}$. Every continuous function $f:L\to \mathbb{R}$ defines such a cocycle $c_f$ by $c_f(v,t)=\int_0^t f(\Phi^s v)ds$. Two cocycles $b,c$ are called *cohomologous* if there is a continuous function $\psi:L\to \mathbb{R}$ such that $\psi(\Phi^t
v)+c(v,t)-\psi(v)=b(v,t)$. If $b, c$ are cocycles which are Hölder continuous with respect to the distance $d$ on $L$, i.e. if for fixed $t>0$ the maps $v\to b(v,t),v\to c(v,t)$ are Hölder continuous, then by Livshicz’ theorem, $b,c$ are cohomologous if and only if for every periodic point $v$ of the geodesic flow with period $\tau >0$ we have $b(v,\tau)=c(v,\tau)$ [@HK]. Every Hölder continuous cocycle is cohomologous to the cocycle of a Hölder continuous function $f$ (see e.g. [@H99]), and two Hölder functions $f,g$ on $L$ are *cohomologous*, i.e. their cocycles $c_f,c_g$ are cohomologous, if and only if we have $\int_{\gamma^\prime} f=\int_{\gamma^\prime}g$ for every closed geodesic $\gamma$ on $M$ (where $\gamma^\prime$ is the unit tangent field of $\gamma$).

The flip ${{\mathcal{F}}}$ acts on the space of cocycles preserving cohomology classes. We call the cohomology class of a cocycle $c$ *anti-invariant* 