under the flip ${{\mathcal{F}}}$ if ${{\mathcal{F}}}(c)$ is cohomologous to $-c$. If the cohomology class of a Hölder continuous cocycle $c$ is anti-invariant under the flip then there is a Hölder continuous function $f$ which is anti-invariant under the flip, i.e. which satisfies $f(v)=-f(-v)$ for all $v\in L$, such that the cocycle $c_f$ defined by $f$ is cohomologous to $c$ (compare [@H97]). Denote by ${{\mathcal{A}}}$ the vector space of all Hölder continuous functions $f$ on $L$ which are anti-invariant under the flip ${{\mathcal{F}}}$.

Since $L$ is a compact invariant hyperbolic topologically transitive set for the geodesic flow on $T^1M$, for every Hölder continuous function $f$ on $L$ and every number $\epsilon_0 >0$ which is smaller than half of the injectivity radius of $M$ there is a number $k>0$ only depending on the Hölder norm of $f$ with the following property. Let $v,w\in L$ and let $T>0$ be such that $d(\Phi^tv,\Phi^t w)\leq \epsilon_0$ for all $t\in
[0,T]$; then $\vert \int_0^Tf(\Phi^t
v)dt-\int_0^Tf(\Phi^t w)dt\vert \leq k$.

A *quasi-morphism* for $\Gamma$ is a function $\phi:\Gamma\to \mathbb{R}$ such that $$\Vert \phi\Vert_0=
\sup_{g,h\in \Gamma}\vert \phi(g)+\phi(h)-\phi(gh)\vert<\infty.$$ The set ${{\mathcal{Q}}}$ of all quasi-morphisms for $\Gamma$ naturally has the structure of a vector space. The function $\Vert \,\Vert_0:{{\mathcal{Q}}}\to
[0,\infty)$ which 