Infinite chains of perfect fits for expanding Thurston maps

Abstract

The topological mating of two postcritically finite polynomials with dendritic Julia sets is encoded by a pair of circle laminations $\Lambda^{\pm}$, whose collapse produces a sphere-filling curve. When a leaf of $\Lambda^{+}$ and a leaf of $\Lambda^{-}$ share an endpoint they form a perfect fit. An unpublished proposition of Epstein, recorded by Petersen and Meyer, shows that for matings of honest degree-$d$ polynomials an infinite-diameter ray equivalence class, i.e. an infinite chain of perfect fits is impossible. We show that this finiteness is a genuinely holomorphic phenomenon. Allowing the dynamics to carry a periodic critical orbit, we construct combinatorially expanding Thurston maps-realized by no expanding rational map-that admit invariant sphere-filling curves yet whose laminations $\Lambda^{\pm}$ contain infinite chains of perfect fits, in fact infinitely many of them.