Iteration of mapping classes and limits of hyperbolic 3-manifolds

Let ϕ∈Mod(S) be an element of the mapping class group of a surface S. We classify algebraic and geometric limits of sequences {Q(ϕiX,Y)}i=1∞ of quasi-Fuchsian hyperbolic 3-manifolds ranging in a Bers slice. When ϕ has infinite order with finite-order restrictions, there is an essential subsurface Dϕ⊂S so that the geometric limits have homeomorphism type S×ℝ-Dϕ×{0}. Typically, ϕ has pseudo-Anosov restrictions, and Dϕ has components with negative Euler characteristic; these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit. We show there is an s≥1 depending on ϕ and bounded in terms of S so that {Q(ϕsiX,Y)}i=1∞ converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits.