Cannon-Thurston fibers for iwip automorphisms of $F_N$
  For any atoroidal iwip $\phi \in Out(F_N)$ the mapping torus group $G_\phi=F_N\rtimes_\phi <t>e$ is hyperbolic, and the embedding $\iota: F_N \overset{\lhd}{\longrightarrow} G_\phi$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat \iota: \partial F_N \to \partial G_\phi$.   We prove that for any $\phi$ as above, the map $\hat \iota$ is finite-to-one and that the preimage of every point of $\partial G_\phi$ has cardinality $\le 2N$.   We also prove that every point $S\in \partial G_\phi$ with $\ge 3$ preimages in $\partial F_N$ has the form $(wt^m)^\infty$ where $w\in F_N, m\ne 0$, and that there are at most $4N-5$ distinct $F_N$-orbits of such {\em singular} points in $\partial G_\phi$ (for the translation action of $F_N$ on $\partial G_\phi$).   By contrast, we show that for $k=1,2$ there are uncountably many points $S\in \partial G_\phi$ (and thus uncountably many $F_N$-orbits of such $S$) with exactly $k$ preimages in $\partial F_N$. 
Ilya Kapovich and Martin Lustig
math.GR math.GT