Rigidity of Lyapunov exponents for polynomials

TL;DR

This paper establishes a rigidity result for Lyapunov exponents of polynomials with disconnected Julia sets, showing they are equal only under specific intertwined conditions, and applies this to the injectivity of the multiplier spectrum map.

Contribution

It proves a rigidity theorem linking Lyapunov exponents to polynomial intertwining and extends the result to critical heights, with applications to moduli space mappings.

Findings

Lyapunov exponents are equal iff polynomials are intertwined or conjugate.

Provides a new proof of the generic injectivity of the multiplier spectrum map.

Extends rigidity results to critical heights.

Abstract

Let $f,g\in\overline{\mathbb{Q}}[z]$ be polynomials of degree $d\geq2$ with disconnected Julia sets. We prove that they have the same Lyapunov exponent $\mathcal{L}_f=\mathcal{L}_g$ if and only if either $f$ and $g$ are intertwined, or $f$ and $\overline{g}$ are intertwined. The analogous result for critical heights is also obtained. As an application, we provide a new proof of the theorem stating that the multiplier spectrum morphism on the moduli space of polynomials is generically injective.