Quadratic Thurston maps with few postcritical points

Abstract

We use the theory of self-similar groups to enumerate all combinatorial classes of non-exceptional quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on moduli space can be realized by quadratic rational maps with fewer than four postcritical points.Holomorphic dynamics in one complex variable is largely concerned with the study of rational maps as dynamical systems, along with their parameter spaces. Postcritically finite rational maps have attracted much attention due to their relative simplicity and their structural significance in parameter space. Furthermore, W. Thurston has given a powerful characterization and rigidity theorem for postcritically finite rational maps up to combinatorial equivalence (see Theorem 1.1). With this tool at our disposal, we are free to consider rational maps in the more flexible topological category.

Eliminating Thurston obstructions and controlling dynamics on curves

“…The conjecture is also known to be true for all critically fixed rational maps (that is, rational maps for which each critical point is fixed) and some nearly Euclidean Thurston maps (that is, Thurston maps with exactly four postcritical points and only simple critical points); see [FKK + 17,Hlu19,Lod13]. In [KL19], Kelsey and Lodge verified the conjecture for all quadratic non-Lattès maps with four postcritical points. However, for general postcritcally finite rational maps, the conjecture remains wide open.…”

Eliminating Thurston obstructions and controlling dynamics on curves

“…The conjecture is also known to be true for all critically fixed rational maps (that is, rational maps for which each critical point is fixed) and some nearly Euclidean Thurston maps (that is, Thurston maps with exactly four postcritical points and only simple critical points); see [Hlu19] and [Lod13, FKK + 17]. In [KL19] Gregory Kelsey and Russell Lodge verified the conjecture for all quadratic non-Lattès maps with four postcritical points. However, for general postcritcally-finite rational maps the conjecture remains wide open.…”

Invariant Spanning Trees for Quadratic Rational Maps

“…In [KL18], all non-Euclidean Thurston maps with at most 4 postcritical points are classified, and an algorithm is suggested for solving the twisting problem for such maps. However, invariant spanning trees for rational maps from [KL18] are not immediate from the provided description.…”

“…This fundamental problem has applications beyond complex dynamics, e.g., in group theory; it is a focus of recent developments, see e.g. [BN06, BD17, CG+15, KL18,Hlu17]. We approach the problem via analogs of Hubbard trees for quadratic rational maps: invariant spanning trees.…”

“…The second approach of [BN06] has been further developed in [KL18], where all non-Euclidean Thurston maps with 4 or fewer post-critical points are classified. The authors also provide an algorithm for identifying the twists of all such maps.…”