Circle packings, renormalizations and subdivision rules

Abstract

In this paper, we use iterations of skinning maps on Teichmüller spaces to study circle packings and develop a renormalization theory for circle packings whose nerves satisfy certain subdivision rules. We characterize when the skinning map has bounded image. Under the corresponding condition, we prove that the renormalization operator $\mathfrak{R}$ is uniformly contracting. This allows us to give complete answers for the existence and moduli problems for such circle packings. The exponential contraction of $\mathfrak{R}^n$ means that despite the non-rigidity of such circle packings, they are geometrically inflexible. As an application, we show that any geometrically finite Kleinian circle packing is combinatorially rigid.

Figures (23)

• Figure 1.1: An example of an acylindrical subdivision rule on the top, and a cylindrical subdivision rule on the bottom, where each rule is iterated thrice. In both examples, a quadrilateral is divided into two quadrilaterals, but with different identifications. Each face is identified with the original quadrilateral by an orientation-preserving homeomorphism so that the corners marked by a star match.
• Figure 1.2: The finite circle packings of level $0, 1, 2, 3, 4, 15$ of the subdivision graph in Figure \ref{['fig:subda']}. These circle packings are chosen so that one can fit in a circle touching all four sides of each complementary quadrilateral region. One notices that the circles at a given level stabilize quite rapidly as asserted by Theorem \ref{['thm:LB']}. The region bounded by the outermost 4 circles is the skinning interstice for $\mathcal{P}$.
• Figure 1.3: An illustration of the renormalization operator. Note that $(\mathcal{P}, x), (\mathcal{P}, z), (\mathcal{P}, p) \in \Sigma^1 \subseteq \Sigma$, but $(\mathcal{P}, x) \in \Sigma - \Sigma^1$. The point $z \in \partial \overline{\Pi(\mathcal{P}_F)} \cap \partial \overline{\Pi(\mathcal{P}_{F'})}$ for two non-external faces, so $(\mathcal{P}, z)$ lifts to two points in $\widetilde{\Sigma}^1$. On the other hand, $x, p \in \overline{\Pi(\mathcal{P}_F)}$ for some unique non-external face $F$, so $(\mathcal{P}, x)$ (or $(\mathcal{P}, p)$) lifts to a point in $\widetilde{\Sigma}^1$.
• Figure 2.1: A non-Jordan face $F$ and its ideal boundary $I(F)$. The map $\Psi_F$ is a homeomorphism between the interiors. Both vertices $a,a'$ on the right is mapped to $a$ on the left.
• Figure 2.2: Interstice for a non-Jordan face with the same combinatorics as Figure \ref{['fig:ideal_boundary']}. Note that each vertex gives a side of the interstice and each non-adjacent pair in $I(F)$ gives an invariant simple closed curve on the double $X_F$, obtained by the double of a path connecting the pair of sides. Since $\Psi_F(a)=\Psi_F(a')$, a lift of the corresponding invariant curve is a loop in the domain of discontinuity. On the other hand, since $\Psi_F(b)\neq\Psi_F(d)$, a lift of the corresponding invariant curve is an axis of a loxodromic element.

Theorems & Definitions (141)

• Definition 1.2
• Definition 1.3
• Theorem A
• Remark
• Theorem B
• Theorem C: Geometric inflexibility
• Theorem D
• Remark
• Theorem E
• Theorem F