Rigidity of Lyapunov exponents for polynomials

Let $f,g\in \bar{\mathbb{Q}}[z]$ be polynomials of degree $d\ge 2$ 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 $\bar{g}$ 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.