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 ∂ Fᵣ of some abstract algebraic laminations associated with free group automorphisms. For an exponentially growing outer automorphism ϕ∈ Out(Fᵣ) we show that the set of endpoints EL⊆ ∂ Fᵣ of any of the attracting laminations L of $ϕ$ has Hausdorff and packing dimension $0$ for any visual metric on the boundary ∂ Fᵣ. Similarly that L⊆ ∂² Fᵣ (where ∂² Fᵣ is equipped with the product metric of a visual metric) has Hausdorff dimension $0$ and packing dimension $0$. If ϕ∈ Out(Fᵣ) is an atoroidal and fully irreducible, we deduce the same conclusion for the set of endpoints of the ending lamination Λ_ϕ of $ϕ$ that gets collapsed by the Cannon-Thurston map ∂ Fᵣ→ ∂ G_ϕ for the associated free-by-cyclic group G_ϕ=Fᵣ_ϕ Z. By contrast, the set of endpoints of any of these laminations has upper box dimension $>0$ for any visual metric on ∂ Fᵣ. Published in the journal of Groups, Complexity, Cryptology