Polynomials with core entropy zero

Figure 2.1.  The Julia set of $ - 1.3513 z^3 -2.73903 z^2 $

Figure 2.2.  The directed graph associated to the Hubbard tree in Figure 2.1

Figure 3.1.  The Julia set of $ f(z) = z^3-\frac{3}{2}z+\frac{1}{\sqrt{2}} $

Figure 3.2.  A 1-dimensional slice of the parameter space. A sequence of hyperbolic components converges to the 'tip', which corresponds to the post-critically finite polynomial $ f(z) = z^3-\frac{3}{2}z+\frac{1}{\sqrt{2}} $

Figure 4.1.  Local picture of tuning a post-critically finite polynomial $ g $ with a post-critically finite family of polynomials $ \mathscr{F} $ whose first return map is the Rabbit polynomial, where $ \phi_v^{-1}(\partial \mathscr{T}_v) = \{e^{2\pi i \cdot 1/5}, e^{2\pi i \cdot 2/5}, e^{2\pi i \cdot 3/5}, e^{2\pi i \cdot 4/5}\} $

Figure 4.2.  The quartic polynomial $ f $ for the right Hubbard tree is obtained by replacing each of the cubic fixed points of the quartic polynomial $ g $ for the left Hubbard tree by the bitransitive cubic Basilica

Figure 4.3.  (A) Basilica (above, $ z^2-1 $) and the doubled Basilica(below, $ z^2-1.3107 $); (B) Kokopelli tuned with the doubled Basilica ($ z^2-0.15652 + 1.03225 i $) and the zoomed-in image near 0; (C) Kokopelli tuned with Basilica ($ z^2-0.16267+1.037123 $); (D) Kokopelli ($ z^2-0.15652 + 1.03224 $). Cores of (B) and (C) are both homeomorphic to the Julia set of the Basilicas. (C) is the simplicial quotient of (B), and (D) is the simplicial quotient of (C)

Figure 5.1.  An illustration of the conjugacy $ \eta_f $ for a degree $ 3 $ boundary-hyperbolic Blaschke product. The Julia set is a Cantor set, constructed by removing the backward orbits of the interval $ I $

Figure 5.2.  The split modification and dual laminations

Figure 6.1.  The Hubbard tree of the airplane polynomial $ f $. Note that $ f(E_1) = E_1 \cup E_2 $ and $ f(E_2) = E_1 $. Let $ a_i $ be the corresponding vertices in $ \mathscr{G}_f $. We have intersecting cycles $ a_1 \to a_1 $ and $ a_1 \to a_2 \to a_1 $. The blue and red subintervals are mapped homeomorphically to $ E_1 $ by $ f^2 $ and $ f $

Figure 6.2.  Domains $ U_{I_1}, U_{I_2}\subset U_I $ mapped onto $ U_I $

Figure 6.3.  A sequence of pointed iterated simplicial tunings of the trivial pointed Hubbard tree, where the red vertex corresponds to the marked point

Figure 6.4.  The directed graph associated to the bottom Hubbard tree in Figure 6.3. The vertices are labeled so that $ a_i $ corresponds to the $ i $-th edge of the Hubbard tree from the left. It can be verified easily the maximal depth is $ 2 $, so $ \mathscr{C}_t(f) = 3 $ by Theorem 2.4

Figure A.1.  The left figure indicates the choice of $ \delta_0, \delta_1, g, h $. The right figure indicates the lifts of $ g $ and $ h $

Figure A.2.  The Moore diagram of the Basilica biset with the choice of basis $ \{\delta_0, \delta_1\} $ in Figure Figure A.1. For simplicity, we omit $ \delta $ in the labels of edges in the diagram, i.e., $ (i, j) $ means $ (\delta_i, \delta_j) $

Figure A.3.  Leaves of the Basilica lamination described in Table 1