weak$^*$-topology.

For $w\in T^1M$ let $\delta(w)\geq 0$ be the distance between $w$ and $L$. For $w\in T^1M -L$ define $$f_0(w)=\int_{B(w,2\delta(w))\cap L}f d\mu/\mu(B(w,2\delta(w)))$$ and let $f_0(w)=f(w)$ for $w\in L$. By assumption on the measures $\mu\vert B(w,r)$ and since $f$ is Hölder continuous, the function $f_0$ is locally Hölder continuous and its restriction to $L$ coincides with $f$. Thus we obtain a locally Hölder continuous flip anti-invariant extension $F$ of $f$ to $T^1M$ by assigning to $w\in T^1M-L$ the value $F(w)=\frac{1}{2}(f_0(w)-f_0(-w))$. For every compact subset $K$ of $T^1M$ the Hölder norm of the restriction of $F$ to $K$ only depends on $K$ and on the Hölder norm of $f$. If $F,H$ are the extensions of $f,h$ constructed in this way and if $a,b\in \mathbb{R}$ then $aF+bH$ is the extension of $af+bh$.

Let again $\Lambda$ be the limit set of $\Gamma$. The closure ${\rm Conv}(\Lambda)\subset \tilde M$ of the convex hull of $\Lambda$ in $\tilde M$ is invariant under the action of $\Gamma$. The *convex core* ${{\mathcal{C}}}(M)={\rm Conv}(\Lambda)/\Gamma$ of $M$ is compact. Let $\tilde F$ be the lift of $F$ to $T^1\tilde M$ and choose a point $p\in {\rm Conv}(\Lambda)$. For an element $g\in \Gamma$ define $\Psi(f)(g)$ to be the integral of $\tilde 