Entropy and Translation Length in the Ray Graph

Let $\mathrm{\Gamma }$ be the mapping class group of the plane minus a Cantor set, acting on the ray graph $\mathcal{R}$, and for $\gamma \in \mathrm{\Gamma }$ let $\tau (\gamma )$ denote the translation length of $\gamma$ on $\mathcal{R}$. We prove that $\tau (\gamma )\le h(f)\mathrm{/}\log (2)$ for every $C^{\mathrm{\infty }}$ diffeomorphism $f$ of $S^2$ representing $\gamma$, where $h$ denotes topological entropy. The proof falls into two parts: a combinatorial argument bounding distance between two rays in $\mathcal{R}$ by the logarithm of the geometric intersection number; and a geometric argument that promotes geometric control (coarse length of iterates of a fixed ray) to combinatorial control (geometric intersection number). We conjecture that the $C^{\mathrm{\infty }}$ hypothesis on $f$ can be removed.