the choice of $\tilde v$ there is an element $h\in
\Gamma$ which preserves $\eta$ and whose restriction to $\eta$ is the translation $\eta(t)\to \eta(t+\tau)$ with translation length $\tau$. Hence we have $\Psi(f)(h^m)=m\int_0^\tau f(\Phi^t v)dt$ and $(\Psi(f)-\rho)(h^m)= mc$ for all $m\in \mathbb{Z}$. In particular, the function $\Psi(f)-\rho$ is unbounded and consequently $\Psi(f)$ is not equivalent to $\rho$. Since $\rho$ was arbitrary this means that the projection of $\Psi(f)$ into $\tilde {{\mathcal{Q}}}$ does not vanish.

Fix a number $\epsilon_0>0$ which is smaller than half of the injectivity radius of $M$ and for $f\in {{\mathcal{A}}}$ define $\Vert
f\Vert_{{\mathcal{A}}}$ to be the infimum of the numbers $k>0$ with the property that $\vert \int_0^T f(\Phi^t v)dt-\int_0^T f(\Phi^t w)dt \vert
\leq k$ whenever $v,w\in L$ and $T>0$ are such that $d(\Phi^t v,\Phi^t w)\leq \epsilon_0$ for every $t\in [0,T]$. We have.

[**Lemma 2.4:**]{} [*$\Vert \,\Vert_{{\mathcal{A}}}$ is a norm on ${{\mathcal{A}}}$.*]{}

[*Proof:*]{} We observed above that $\Vert f\Vert_{{\mathcal{A}}}<\infty$ for every Hölder continuous function $f\in {{\mathcal{A}}}$. Moreover, we clearly have $\Vert a f\Vert_{{\mathcal{A}}}= \vert a\vert \Vert f\Vert_{{\mathcal{A}}}$ for all $f\in {{\mathcal{A}}}$ and all $a\in \mathbb{R}$ and $\Vert
f+g\Vert_{{\mathcal{A}}}\leq \Vert f\Vert_{{\mathcal{A}}} +\Vert g\Vert_{{\mathcal{A}}} $ by a simple application of the triangle inequality. Thus we are left with showing that $\Vert 