Unmating of expanding Thurston maps with Julia set 𝕊²

Abstract

Every expanding Thurston map f f without periodic critical points is known to have an iterate f n f^n which is the topological mating of two polynomials. This has been examined by Kameyama and Meyer; the latter has offered an explicit construction for finding two polynomials in the unmating of the iterate. Initializing Meyer’s algorithm depends on an invariant Jordan curve through the postcritical set of f f –but we propose adjustments to this unmating algorithm for Kameyama’s more general case where there exists a curve which is fully f f -invariant up to homotopy and not necessarily simple. When f f is a critically pre-periodic expanding Thurston map, extending the algorithm to accommodate non-Jordan curves in this manner may allow us to unmate without iterates.