*cohomology class* of $c$. The *flip* ${{\mathcal{F}}}:v\to -v$ acts on $W$ and on the space of cocycles for the geodesic flow preserving cohomology classes. The cohomology class of a cocycle $c$ is called *flip anti-invariant* if ${{\mathcal{F}}}(c)$ is cohomologous to $-c$. We denote by ${{\mathcal{D}}{\mathcal{C}}}(M)$ the vector space of all flip anti-invariant cohomology classes of cocycles for the geodesic flow on $W$ which are Hölder continuous, i.e. such that for a fixed number $t>0$ the function $v\to c(v,t)$ is Hölder continuous.

Every smooth closed 1-form on $M$ defines via integration along orbit segments of $\Phi^t$ a Hölder continuous cocycle for $\Phi^t$ which is anti-invariant under the flip. Two cocycles defined by closed one-forms in this way are cohomologous if and only if the one-forms define the same de Rham cohomology class on $M$. Thus $H^1(\Gamma,\mathbb{R})=H^1(M,\mathbb{R})$ is naturally a subspace of ${{\mathcal{D}}{\mathcal{C}}}(M)$. We show in Section 2.

[**Theorem D:**]{} [*Let $\Gamma$ be the fundamental group of a convex cocompact manifold $M$ of bounded negative curvature. Then the quotient space ${{\mathcal{D}}{\mathcal{C}}}(M)/H^1(\Gamma,\mathbb{R})$ naturally embeds into ${\rm ker}(H_b^2(\Gamma,\mathbb{R})\to
H^2(\Gamma,\mathbb{R}))$.*]{}

Section 3 contains the main technical result of this paper. Embarking from the concrete construction in Section 2, we present an abstract dynamical criterion for 