Value distribution of derivatives in polynomial dynamics

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