Thurston Equivalence to a Rational Map is Decidable

Abstract

We demonstrate that the question whether or not a given topological ramified covering map of the 2-sphere is Thurston equivalent to a rational map is algorithmically decidable.

Origami, Affine Maps, and Complex Dynamics

“…So HS contains either (0, 1) or (2, 1). One verifies that [ 1 2 0 1 ] ∈ SL(2, Z) induces an automorphism on Z 4 ⊕ Z 2 which fixes (0, 0), ±(1, 0), ±(1, 1) and interchanges (0, 1) and (2,1). Hence there is only one equivalence class of these Hurwitz structure sets and only one impure Hurwitz class of such maps.…”

“…The static portrait is the bipartite directed graph whose vertex set is the disjoint union of A := C(f ) ∪ P (f ) and B := P (f ), directed edges → q 2 , p 3 → q 3 , p 4 → q 4 ), and the branch data is ( [2], [2], [1,1], [1,1]). The static portrait, and hence branch data, are impure Hurwitz invariants.…”

“…On the other hand, we will soon see from Figure 10 that the images of ρ 1 , ρ 2 and ρ 3 in PGL(2, Z) generate a subgroup with index 6. (Its intersection with PSL(2, Z) equals the image of Γ (2).) It easily follows that G f0 is generated by ρ 1 , ρ 2 , ρ 3 together with the elements of G f0 which lift to translations in Aff(2, Z).…”

Constructive Geometrization of Thurston Maps and Decidability of Thurston Equivalence

“…In [4] it was shown that, outside of some exceptional cases, the question of Thurston equivalence to a rational mapping is algorithmically decidable. Namely, there exists an algorithm A 1 which, given a combinatorial description of f , outputs 1 if f is equivalent to a rational mapping and 0 otherwise.…”

Algorithmic aspects of branched coverings

“…It has a portrait B˚: A ý induced by the map on peripheral conjugacy classes, and a local degree deg a pBq. There is a minimal orbisphere quotient of G associated with B, which is the quotient G of G by the additional relations Γ ord B paq a " 1, for (11) ord B paq " l. c. m.td | n ě 0 and Γ a has a lift of degree d under B bn u,…”