Let $f:S^2\to S^2$ be an orientation-preserving branched covering such that the forward orbit of any critical point is finite. Thurston studied the dynamics of $f$ using an induced holomorphic self-map $T(f)$ of the Teichmuller space of complex structures on $S^2$ marked by the postcritical set. Koch found that this holomorphic dynamical system on Teichmuller space descends to an algebraic dynamical system on the moduli space $\mathcal{M}_{0,n}$.

I will introduce these related topological, holomorphic and algebraic dynamical systems, and discuss how the last can be studied on compactifications of $\mathcal{M}_{0,n}$.