Cannon-Thurston fibers for iwip automorphisms of F_N

For any atoroidal iwip ℓ Out(F_N), the mapping torus group G_ℓ =F_N→ _ℓ t is hyperbolic, and, by a result of Mitra, the embedding iota: F_N hd G_ℓ induces a continuous, F_N-equivariant and surjective Cannon-Thurston map F_N to G_ℓ. We prove that for any as above, the map is finite-to-one and that the preimage of every point of G_ℓ has cardinality at most 2N. We also prove that every point S in G_ℓ with at least three preimages in F_N has the form (wt_m) where w F_N, m ne 0, and that there are at most 4N-5 distinct F_N-orbits of such singular points in G_ℓ (for the translation action of F_N on G_ℓ). By contrast, we show that for k=1,2, there are uncountably many points S G_ℓ (and thus uncountably many F_N-orbits of such S) with exactly k preimages in F_N.