Expansion properties for finite subdivision rules II

Lattes maps and finite subdivision rules

This paper is concerned with realizing Lattes maps as subdivision maps of finite subdivision rules. The main result is that the Lattes maps in all but finitely many analytic conjugacy classes can be realized as subdivision maps of finite subdivision rules with one tile type. An example is given of a Lattes map which is not the subdivision map of a finite subdivision rule with either i) two tile types and 1-skeleton of the subdivision complex a circle or ii) one tile type.

Constructing subdivision rules from rational maps

This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f^n is the subdivision map of a finite subdivision rule.

Expansion properties for finite subdivision rules I

Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate relationships between various notions of expansion---combinatorial, dynamical, algebraic, and coarse-geometric.

Levy and Thurston obstructions of finite subdivision rules

For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. In order to show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and…

Subdivision rule constructions on critically preperiodic quadratic matings

Mating is an operation that identifies the domains of a polynomial pair in order to obtain a new map on the resulting quotient space. The dynamics of the mating are then dependent on the two polynomials and the manner in which the quotient space was defined, which can be difficult to visualize. This research addresses using Hubbard trees and finite subdivision rules as tools to examine quadratic matings with preperiodic critical points. In many cases, discrete parameter information on such quadratic pairs can be translated into topological information on the dynamics of their mating. The centr…

圆填充、重整化与细分规则

Circle packings, renormalizations and subdivision rules

本文利用 Teichmüller 空间上剥皮映射的迭代，为满足特定细分规则的圆填充发展了重整化理论，证明重整化算子一致收缩，从而完整解决了此类圆填充的存在性与模问题，并证明了其几何不灵活性与组合刚性。