Lattes maps and finite subdivision rules

TL;DR

This paper demonstrates that most Lattes maps can be realized as subdivision maps of finite subdivision rules with a single tile type, providing specific examples of exceptions with more complex tile configurations.

Contribution

It proves that all but finitely many Lattes maps can be represented as subdivision maps with one tile type, and presents an example of a Lattes map that cannot be realized with simpler subdivision rules.

Findings

Most Lattes maps are realizable with one tile type.

An explicit example of a Lattes map not realizable with simple subdivision rules.

Finitely many Lattes maps are exceptions to the main realization result.

Abstract

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.