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M. Pilgrim: LNM 1827, pp. 49–57, 2003. c Springer-Verlag Berlin Heidelberg 2003 50 3 Combinations ∓ • the map π A ◦ ρ ◦ (π S )−1 induces the involution b± 0 → b0 on connected components. The new sphere S 2 is the quotient space (S − B) ρ A0 ≈ S 2 . The projection maps π S , π A then deﬁne a continuous projection π : S 2 → T . 2 Critical gluing data At this point it is possible, after making some choices of extensions over U and C, to deﬁne a new continuous map F : S 2 → S 2 on the quotient space S0 ρ A0 ≈ S 2 .

Let M be a compact, oriented, irreducible (every embedded two-sphere bounds a three-ball) three-manifold. 6 Analogy with three-manifolds 25 sided surface S ⊂ M (which is neither a sphere, projective plane, or disc isotopic into ∂M ) is incompressible if the inclusion S → M induces an injection on fundamental groups. S is said to be peripheral if it is isotopic into ∂M . If M contains a nonperipheral incompressible surface S then M is called Haken; M is toroidal if it contains a nonperipheral incompressible torus.

The output is a rational function R. e. a map which is proceeds by ﬁrst constructing a K-quasiregular map R, locally a rational map followed by a K-quasiconformal homeomorphism. Since quasiconformal maps are quite ﬂexible, this step is often not especially diﬃ˜ is to check that the iterates cult. A common method for producing R from R ˜ are uniformly K -quasiregular (often, a very delicate and technical step) of R for some constant K , and then apply a theorem of Sullivan [Sul] (sometimes referred to as the “Shishikura principle”) which asserts that under these as˜ is quasiconformally conjugate to a rational map R.