Value distribution of derivatives in polynomial dynamics - OKUYAMA, YÛSUKE; VIGNY, GABRIEL (Ergodic theory and dynamical systems, 2021)
For 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 of f with pole at $∞ $ . 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 field k for 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.