Entropy and Translation Length in the Ray Graph

Let $Γ$ be the mapping class group of the plane minus a Cantor set, acting on the ray graph $\mathcal{R}$, and for $γ\in Γ$ let $τ(γ)$ denote the translation length of $γ$ on $\mathcal{R}$. We prove that $τ(γ) \le h(f)/\log(2)$ for every $C^\infty$ diffeomorphism $f$ of $S^2$ representing $γ$, where