Value distribution of derivatives in polynomial dynamics

Abstract

AbstractFor every$m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in$\mathbb {C}\setminus \{0\}$under the$m$th order derivatives of the iterates of a polynomials$f\in \mathbb {C}[z]$of degree$d>1$towards the harmonic measure of the filled-in Julia set offwith pole at$\infty $. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number fieldkfor a sequence of effective divisors on$\mathbb {P}^1(\overline {k})$having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of$\mathbb {C}^2$has a given eigenvalue.