On the Global Curve Attractor for polynomial gluing

TL;DR

This paper proves the Finite Global Attractor Conjecture for a specific family of rational maps formed by gluing two PCF polynomials, demonstrating convergence properties of non-peripheral curves under iteration.

Contribution

It extends the conjecture's verification to a new class of rational maps created by polynomial gluing, using intersection number decay techniques.

Findings

Intersection number with separating arcs decays under pullback

A finite collection of homotopy classes attracts all non-peripheral curves

Convergence behavior established for the specific family of rational maps

Abstract

Pilgrim's Finite Global Attractor Conjecture has been verified for polynomials [1], 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 [17], 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.