f\Vert_{{\mathcal{A}}} =0$ only if $f\equiv 0$. For this assume that $0\not\equiv f\in {{\mathcal{A}}}$. Since $f$ is anti-invariant under the flip by assumption, $f$ is not the constant function. Hence by continuity there are points $v,w\in W$ with $d(v,w)<\epsilon_0/2$ and numbers $\delta >0$, $T\in
(0,\epsilon_0)/2$ with $f(\Phi^tv)\geq f(\Phi^tw)+\delta$ for all $t\in [0,T]$. Then $\Vert f\Vert_{{\mathcal{A}}} \geq \delta T$ by the definition of $\Vert\,\Vert_{{\mathcal{A}}}$.

Call two Hölder functions $f,g\in {{\mathcal{A}}}$ *weakly cohomologous* if $f-g$ is cohomologous to a closed one-form on $M$, viewed as a function on $T^1M$. The class of $f$ under the thus defined equivalence relation will be called the *weak cohomology class* of $f$. The set ${{\mathcal{H}}}$ of weak cohomology classes of Hölder functions is a vector space. For $\psi\in {{\mathcal{H}}}$ let $\Vert \psi\Vert $ be the infimum of the norms $\Vert f\Vert
_{{\mathcal{A}}}$ where $f$ runs through all functions in ${{\mathcal{A}}}$ which define the weak cohomology class $\psi$. Then $\Vert \,\Vert$ is a pseudo-norm on ${{\mathcal{H}}}$.

The *Gromov norm* $\Vert\,\Vert$ of an element $\alpha\in H_b^2(\Gamma, \mathbb{R})$ is the infimum of the supremums-norms over all bounded 2-cocycles for $\Gamma$ representing $\alpha$ [@G83] (here a bounded 2-cocycle is a bounded $\Gamma$-invariant function on $\Gamma^3$ contained in the kernel of the 