Let $M$ a compact manifold with surfaces $S_1,...,S_p$ as boundaries. Let us suppose that $M$ admits a complete hyperbolic structure. Then, from the ending lamination theorem, given either laminations or conformal structures (the end invariants) on $S_1...S_n$ one is able to find a unique complete hyperbolic structure on $M$ with ends $E_1,...E_N$ admitting exactly these end invariants.
My question is, are these invariants localized in the ends ?
More precisely, given $f$ a pseudo-Anosov diffeomorphism of the surface $S_1$ and its associated lamination when $f$ is iterated in positive time. Then, one can consider the hyperbolic structure given by the ending lamination theorem associated to this lamination for the surface $S_1$ and with whatever end invariants you want for the rest of the surfaces ($S_2...S_N$).
Does the resulting hyperbolic structure satisfies that the end associated to $S_1$ is asymptotically isometric to a $\mathbb{Z}$-cover of a hyperbolic compact 3-fold which fibers over the circle with fiber $S_1$ and monodromy $f$ ?
I know from McMullen's book (theorem 3.12)
http://www.math.harvard.edu/~ctm/papers/pup.html
that this is true for one degenerated end quasi-Fuchsian manifolds or, more obviously, for $\mathbb{Z}$-cover of compact manifold I wondered whether this result may be generalised.
A very first question is what happens if we consider two pseudo-Anosov diffeomorphisms $(\Phi_1, \Phi_2)$ for which the laminations given by iterating these in positive time are in transversal position. So that the sequence of quasi-Fuchsian manifolds given in Bers coordinates by $(\Phi_1^n, \Phi_2^n)$ converges by Thurston's theorem.
Can I say that the end $E_1$ (resp $E_2$) of the limit representation are asymptotically isometric to the $Z$-covering of the compact manifold which fibres over the circle of monodromy $\Phi_1$ (resp $\Phi_2)$ ?
Thank you for your reading.
