Then $\tilde \beta$ is exact and hence the integral of $\tilde \beta$ over every piecewise smooth closed curve in $\tilde M$ vanishes.

By Livshicz’ theorem [@HK] and the choice of $\beta$, there is a Hölder continuous flip anti-invariant function $\psi:L\to \mathbb{R}$ such that $\int_0^Tf(\Phi^t v) dt= \psi(\Phi^T v)+\int_0^T\beta(\Phi^t
v)dt-\psi(v)$ for every $v\in L$ and all $T>0$. As in the proof of Lemma 2.1 we extend $\psi$ to a locally Hölder continuous flip anti-invariant function on all of $T^1M$ which we denote by the same symbol. Let $\tilde \psi$ be the lift of $\psi$ to $T^1\tilde M$. Fix a point $p\in {\rm Conv}(\Lambda)$ and for $g\in \Gamma$ let $\gamma_g$ be the geodesic arc connecting $p=\gamma_p(0)$ to $g(p)=\gamma_p(T)$. Define $\alpha(g)=\tilde\psi(\gamma_g^\prime(T))+\int_0^T
\tilde\beta(\gamma_g^\prime(t))dt
-\tilde \psi(\gamma_g^\prime(0))$. By Lemma 2.1 and Lemma 2.2, $\alpha$ is a quasi-morphism for $\Gamma$ which defines the cohomology class $\Theta(f)$. On the other hand, $\alpha$ differs from the quasi-morphism defined by $\beta$ by a bounded function. Since the integral of $\tilde\beta$ over every piecewise smooth closed curve in $\tilde M$ vanishes, the cohomology class $\Theta(f)$ of the quasi-morphism $\alpha$ vanishes.

On the other hand, let $f\in {{\mathcal{A}}}$ and assume that there is no morphism $\rho:\Gamma\to \mathbb{R}$ such that $\int_{\gamma^\prime}f=\rho(\gamma)$ for every 