In this talk, I will discuss a dynamical analogue of the N'eron-Ogg-Shafarevich criterion for the good reduction of abelian varieties. Along the way, we quantify (in an effective way) the asymptotic growth of ramification in dynamical extensions in terms of simpler geometric features of the dynamical system. To do so, we reformulate the question in anabelian terms to leverage Tamagawa’s anabelian version of the N'eron-Ogg-Shafarevich criterion. Although the proof can be reformulated in elementary terms, our anabelian approach better clarifies the interaction between geometry, arithmetic, and dynamics of the situation, and raises a host of natural questions which are less visible in the elementary context.

Some of the results (but only the elementary versions) are available in this preprint: https://arxiv.org/abs/2208.00359. The much more recent anabelian improvements will be available soon (although not necessarily before this talk)