Cannon-Thurston fibers for iwip automorphisms of $F_N$

For any atoroidal iwip $\varphi \in Out(F_N)$ the mapping torus group $G_{\varphi }=F_N{⋊}_{\varphi }<t>e$ is hyperbolic, and the embedding $\iota: F_N \overset{\lhd}{\longrightarrow} G_\phi$inducesacontinuous,$F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat \iota: \partial F_N \to \partial G_\phi$. We prove that for any $\varphi$ as above, the map $\widehat{\iota }$ is finite-to-one and that the preimage of every point of $\mathrm{\partial }{G}_{\varphi }$ has cardinality $\le 2N$. We also prove that every point $S\in \mathrm{\partial }{G}_{\varphi }$ with $\ge 3$ preimages in $\mathrm{\partial }{F}_N$ has the form $(w{t}^m{)}^{\mathrm{\infty }}$ where $w\in F_N,m\mathrm{\ne }0$, and that there are at most $4N-5$ distinct $F_N$-orbits of such {\em singular} points in $\mathrm{\partial }{G}_{\varphi }$ (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 $S\in \partial G_\phi$(andthusuncountablymany$F_N$-orbitsofsuch$S$) with exactly $k$ preimages in $\mathrm{\partial }{F}_N$.