Rigidity in Complex Dynamics: Multiplier Spectrum and Dynamical André-Oort Conjecture
Junyi Xie
TL;DR
The paper advances rigidity in one-dimensional complex dynamics by proving the Dynamical André-Oort conjecture for curves and establishing generic injectivity of the multiplier spectrum on moduli spaces ${\mathcal M}_d$, while developing a parallel length-spectrum theory. It blends algebraic geometry, Arakelov geometry, and complex dynamics to relate PCF points, bifurcation measures, and invariant correspondences through a four-step program: equidistribution of PCF parameters, parameter exclusion via dynamical relations, construction of phase/parameter-space renormalizations, and derivation of local entropy symmetries that yield contradictions unless rigidity holds. The multiplier spectrum is shown to determine the conjugacy class up to finitely many choices (excluding Lattès), and, crucially, the authors prove generic injectivity of the spectrum morphism $\tau_d$, solving a long-standing question of Poonen. The work suggests a robust framework for higher-dimensional analogues and opens several directions, including generic injectivity for small periods, length-spectrum rigidity, and extensions to endomorphisms on ${\mathbb P}^N$, thereby connecting dynamic invariants with arithmetic and geometric properties of moduli spaces.
Abstract
In this note, we present recent progress on rigidity problems in one-dimensional complex dynamics, including the proof of Dynamical André-Oort conjecture for curves and generic injectivity of multiplier spectrum. The proofs combine ideas from algebraic geometry, Arakelov geometry and complex dynamics.