Lattès maps and finite subdivision rules

Abstract

Abstract. This paper is concerned with realizing Lattès maps as subdivision maps of finite subdivision rules. The main result is that the Lattès 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 Lattès 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.This paper is concerned with realizing rational maps by subdivision maps of finite subdivision rules. If R is an orientation-preserving finite subdivision rule such that the subdivision complex S R is a 2-sphere, then the subdivision map σ R is a postcritically finite branched map. Furthermore, R has bounded valence if and only if σ R has no periodic critical points. In [1] and [4], Bonk-Meyer and CannonFloyd-Parry prove that if f is a postcritically finite rational map without periodic critical points, then every sufficiently large iterate of f is the subdivision map of a finite subdivision rule R with two tile types such that the 1-skeleton of S R is a circle. Since finite subdivision rules are essentially combinatorial objects, this gives good combinatorial models for these iterates. It is especially convenient to realize a postcritically finite map by the subdivision map of a finite subdivision rule with either a single tile type or with two tile types and 1-skeleton of the subdivision complex a circle.While passing to an iterate is not usually a serious obstacle, it would be preferable if one didn't need to do this. Suppose f is a postcritically finite rational map without periodic critical points. We consider the following questions.(1) Is f the subdivision map of a finite subdivision rule? (2) Is f the subdivision map of a finite subdivision rule with two tile types and 1-skeleton of the subdivision complex a circle? (3) Is f the subdivision map of a finite subdivision rule with one tile type? In this paper we consider these questions for Lattès maps. The main result of this paper is to answer question 3 in the affirmative for the maps in all but finitely many analytic conjugacy classes of Lattès maps. In addition we exhibit a Lattès map of degree 2 for which the answer to questions 2 and 3 is negative, although the answer to question 1 is positive. This paper has four sections. The first section develops the setting of Lattès maps. These results can be used to easily enumerate all Lattès maps of very small degree to test (not so easy) the above three questions for them.

Nearly Euclidean Thurston maps

“…Thus the number of essential, nonperipheral components of f −1 (δ) is c 3 − c 2 . This proves statement (2). Statements (3) and (4) can be proven similarly.…”

“…A fundamental domain F 1 for the action of Γ 1 on R 2 is hatched in Figure 2. We give F 1 a cell structure so that the boundary of F 1 is its 1-skeleton and its vertices are at (0, 0), (2, −1), (4, −2), (4,3), (2,4) and (0, 5). We regard the hatched region in Figure 2 as a subdivision of F 1 .…”

“…. , (0, 5), (1,5) and h maps the image of (2,4) to the image of (2,5) and the image of (2, −1) to the image of (2, 0). Let f = h • g. The action of f is indicated in Figure 4.…”

“…Let A be a finite Abelian group. Let H be a subset of A which is the disjoint union of four inverse pairs {±h 1 }, {±h 2 …”

“…Our motivation in this work is to better understand Thurston obstructions and the issue of conformality of finite subdivision rules (for which, see [2]). We were led to a class of Thurston maps which are as simple as possible but yet nontrivial in this regard.…”