Thurston's algorithm and rational maps from quadratic polynomial matings

Figure 1.  The conformal isomorphism $ \phi $ which determines external rays for $ z\mapsto z^2+i $. Shown on the right are external rays landing at points on the critical orbit of this polynomial.

Figure 2.  Steps in the formation of the formal mating.

Figure 3.  The Medusa and pseudo-equator algorithms are based upon Thurston's algorithm, highlighted in the commutative diagram above.

Figure 4.  A rudimentary finite subdivision rule on $ \hat{\mathbb{C}} $.

Figure 5.  The Julia set and Hubbard trees for $ f_{1/4} $.

Figure 6.  The preimage of a Hubbard tree under its associated polynomial.

Figure 7.  On the left, $ T_{1/4} $. On the right, the subdivision complex $ S_\mathcal{R} $ for the essential self-mating of $ f_{1/4} $.

Figure 8.  On the left, the expected pullback of $ S_\mathcal{R} $ by the essential mating as based on local behavior of Hubbard trees. The essential mating is locally homeomorphic everywhere except on the critical set, so we complete the pullback as shown on the right.

Figure 9.  The finite subdivision rule associated with $ f_{1/4}\;╨_e\;f_{1/4} $, along with marked pseudo-equator curves. $ C_0 $ is marked in blue on the left and its pullback $ C_1 $ is marked in blue on the right.

Figure 10.  Pullbacks of the equator by a rational map that is Thurston-equivalent to the topological self-mating of $ f_{1/4} $. These pullbacks approximate the Julia set of the rational map, $ \hat{\mathbb{C}} $. (Image generated in Mathematica.)

Figure 11.  Top: The Julia sets of $ f_{1/4} $ and $ f_{1/8} $, with external angles marked at postcritical points for reference. Middle: The Hubbard trees associated with these polynomials. Bottom: the finite subdivision rule associated with the essential mating $ f_{1/4}\;╨_e\;f_{1/8} $.

Figure 12.  The critical orbit portrait and finite subdivision rule associated with $ f_{1/4}\;╨_e\;f_{1/8} $, along with marked pseudo-equator curves. $ C_0 $ is marked in blue above and its pullback $ C_1 $ is marked in blue below. We have relabeled the marked points to emphasize angle markings given by the parameterizations of $ C_0 $ and $ C_1 $.

Figure 13.  Pullbacks of the equator by a sequence of rational maps which approximate the geometric mating of $ f_{1/4} $ and $ f_{1/8} $. (Image generated in Mathematica.)

Figure 14.  The problem with using the canonical branch of the square root for pullbacks of $ C_n $: orientation is important, but harder to keep record of when our pullback curve is cut into several pieces.

Figure 15.  The "pseudo-equator" is pinched by $ \sim_e $ into a non-Jordan curve.