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 $\mathrm{\partial }{F}_r$ 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 ${\mathcal{E}}_L\subseteq \mathrm{\partial }{F}_r$ of any of the \emph{attracting laminations} $L$ of $\phi$ has Hausdorff dimension $0$ for any tree $T\in c{v}_r$ and any visual metric on the boundary $\mathrm{\partial }T=\mathrm{\partial }{F}_r$. 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 ${\mathrm{\Lambda }}_{\phi }$ of $\phi$ that gets collapsed by the Cannon-Thurston map $\mathrm{\partial }{F}_r\to \mathrm{\partial }{G}_{\phi }$ for the associated free-by-cyclic group $G_{\phi }=F_r{⋊}_{\phi }\mathbb{Z}$.