On the Global Curve Attractor for polynomial gluing

2026/05/31 by Panjing Wu
#math.DS

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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.

Citations

• Thurston's pullback map, invariant covers, and the global dynamics on curves
• Jordan mating is always possible for polynomials
• On the pullback relation on curves induced by a Thurston map
• Recognizing topological polynomials by lifting trees
• Tischler graphs of critically fixed rational maps and their applications
• Pullback invariants of Thurston maps

Related

• Curve attractors for marked rational maps
• Jordan mating is always possible for polynomials
• Gluing polynomials along the circle
• Elliptic Curves as Attractors in \mathbb P² Part 1: Dynamics
• Coexistence of non-periodic attractors