# Dynamical Galois groups and rational portraits in one-parameter families of arithmetic dynamical systems

# Dynamical Galois groups and rational portraits in one-parameter families of arithmetic dynamical systems

Let K=**Q**(t) be a one-variable function field and $\phi$ a finite endomorphism of the projective line $**P**^1_K$. For all but finitely many rational numbers $c$, a process of reduction yields a specialized map $\phi_c$, an endomorphism of $**P**^1_**Q**$. Thus, $\phi$ gives rise to a one-parameter family of discrete dynamical systems on $**P**^1_**Q**$. We are then led to ask how the dynamical properties of $\phi$ are related to those of its specializations $\phi_c$. Examples of dynamical invariants we may wish to compare are the structure of the dynatomic groups, i.e., the Galois groups of extensions generated by $n$-periodic points (for any $n\ge 1)$, and the structure of the directed graph of pre-periodic points, i.e., points having a finite orbit. The dynatomic groups of the maps $\phi_c$ are known to form a finite collection of groups (even as the parameter $c$ is allowed to vary), and the graphs of pre-periodic points of the maps $\phi_c$ are conjectured to form a finite collection of directed graphs. In this talk we will discuss algorithmic methods for determining how the dynamical properties of $\phi$ are reflected in the properties of its specializations; as an application, we choose several one-parameter families of quadratic maps that are natural to study from the dynamical perspective, and we prove that a local-global principle for periodic points holds true across these families.