On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups

Motivated by a classic theorem of Birman and Series about the set of complete simple geodesics on a hyperbolic surface, we study the Hausdorff dimension of the set of endpoints in $\partial F_r$ of some abstract algebraic laminations associated with free group automorphisms.
For an exponentially growing outer automorphism $\phi\in Out(F_r)$ we show that the set of endpoints $\mathcal E_{L}\subseteq \partial F_r$ of any of the \emph{attracting laminations} $L$ of $\phi$ has Hausdorff and packing dimension $0$ for any visual metric on the boundary $\partial F_r$. Similarly that $L\subseteq \partial^2 F_r$ (where $\partial^2 F_r$ is equipped with the product metric of a visual metric) has Hausdorff dimension $0$ and packing dimension $0$. If $\phi\in Out(F_r)$ is an atoroidal and fully irreducible, we deduce the same conclusion for the set of endpoints of the ending lamination $\Lambda_\phi$ of $\phi$ that gets collapsed by the Cannon-Thurston map $\partial F_r\to \partial G_\phi$ for the associated free-by-cyclic group $G_\phi=F_r\rtimes_\phi\mathbb Z$. By contrast, the set of endpoints of any of these laminations has upper box dimension $>0$ for any visual metric on $\partial F_r$.