Combining rational maps and controlling obstructions

Abstract

We apply Thurston's characterization of postcritically finite rational maps as branched coverings of the sphere to give new classes of combination theorems for postcritically finite rational maps. Our constructions increase the degree of the map but always yield branched coverings which are equivalent to rational maps, independent of the combinatorics of the original map. The main tool is a general theorem based on the intersection number of arcs and curves which controls the region in the sphere in which an obstruction may reside.

From rubber bands to rational maps: a research report

“…Pilgrim and Tan Lei showed that, if f is a rational map, these blow-ups frequently are as well [36]. We will restrict attention to cases where the initial map f is the identity, in which case the theorem becomes the following.…”

Exponential growth of some iterated monodromy groups

“…We mention in passing that we may think of as being constructed via a two‐tile subdivision rule in the sense of Cannon–Floyd–Parry (see [, Chapter 12; ]). Alternatively, is constructed from a Lattès map by ‘adding a flap’ or ‘blowing up an arc’ in the sense of .…”

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“…An analysis of quadratic Example #7 (see Figure 6) shows that this map has a cycle of four periodic basins meeting exactly in a common fixed point, with two of these basins meeting in an additional point which is a preimage of this fixed point. This example can be constructed by the operation of "blowing up an arc" [16]. This phenomena violates the classification of such intersections in [1], and thus leaves open the questions of just how the closures of such basins can intersect.…”

“…Such maps are in fact determined by their mapping scheme, and all may be constructed by the operation of blowing up an arc in a Möbius transformation. This technique of blowing up an arc given in [16] provides a way to produce new rational maps from ones of lower degree. For example, starting with z 2 − 1 one may produce six such cubic maps.…”

“…Later, we will also need the following result: Though this result is explicitly contained in [2], we now give a constructive proof based on the idea of "blowing up an arc"; see [16]. Given a branched covering F 0 and an arc α mapped homeomorphically under F 0 , blowing up α consists of slitting the sphere along α, inserting a disc, and defining a new map F defined as F 0 off of this disc, and which maps the disc homeomorphically to the complement of F 0 (α).…”