Polynomial dynamical pairs

Abstract

This chapter discusses polynomial dynamical pairs. A polynomial dynamical pair is a family of polynomials together with a marked point. The chapter begins by reviewing basic notions of bifurcation and activity for holomorphic dynamical pairs, and proving the following important rigidity property when the bifurcation locus is included in a smooth real curve. It then turns to algebraic dynamical pairs, explaining how to attach a canonical line bundle to such a pair, and discussing the continuity of the Green function associated to a non-isotrivial pair. The chapter concludes by looking at dynamical pairs defined over a number field and proving that they induce a natural height arising from an adelic semi-positive metrization of a suitable divisor. This allows one to characterize isotrivial adelic pairs in terms of their height function.