On the Global Curve Attractor for polynomial gluing

Abstract

Pilgrim's Finite Global Attractor Conjecture has been verified for polynomials [2], but remains open for general rational maps. In this paper, we prove the conjecture for a family of rational maps obtained by gluing two PCF polynomials along the boundaries of their finite superattracting basins. Adapting the idea of [19], we show that a suitably defined intersection number with a finite family of separating arcs eventually decays under pullback, yielding a finite collection of homotopy classes that attracts all non-peripheral curves under iteration.