Value distribution of derivatives in polynomial dynamics

For every $minmathbb{N}$, we establish the equidistribution of the sequence of the averaged pull-backs 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 $fin mathbb{C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of $f$ with 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 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 Henon-type polynomial automorphism of $mathbb{C}^2$ has a given eigenvalue.