Stable Marked Points in Holomorphic Dynamics

A celebrated theorem of McMullen states that if an algebraic family of rational maps is J-stable, then it has to be a family of Lattès maps or an isotrivial family. DeMarco generalized partially this result proving that, if a pair $(f,a)$ constituted of a marked point and an algebraic family of rational maps is stable, i.e. the family of iterates of $a$ is a normal family of functions of the parameter, then it is either stably preperiodic, or isotrivial.

In this talk, I will explain how to adapt this result to families of endomorphisms of higher dimensional projective spaces together with a marked point. If time allows, I will give an arithmetic interpretation of this result. This is a joint work with Gabriel Vigny.

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