Rigidity in Complex Dynamics: Multiplier Spectrum and Dynamical Andr\'e-Oort Conjecture

TL;DR

This paper discusses recent advances in complex dynamics, proving the Dynamical André-Oort conjecture for curves and establishing the generic injectivity of the multiplier spectrum using interdisciplinary methods.

Contribution

It provides the first proof of the Dynamical André-Oort conjecture for curves and demonstrates the generic injectivity of the multiplier spectrum in complex dynamics.

Findings

Proof of the Dynamical André-Oort conjecture for curves

Generic injectivity of the multiplier spectrum

Integration of algebraic, Arakelov, and complex dynamics methods

Abstract

In this note, we present recent progress on rigidity problems in one-dimensional complex dynamics, including the proof of Dynamical Andr\'e-Oort conjecture for curves and generic injectivity of multiplier spectrum. The proofs combine ideas from algebraic geometry, Arakelov geometry and complex dynamics.