Constant weighted sums of iterates

Abstract

We define and study when a polynomial mapping has a constant weighted sum of iterates, either locally or globally. We conjecture that a polynomial f in the complex plane has a constant weighted sum of iterates near a point z if and only if z is eventually mapped into a Siegel-disc of f. We prove that this conjecture holds generically, namely for those polynomials whose iterates have the maximal number of critical values. Important steps in the proofs rely on understanding the monodromy groups of the iterates of f. We also show that a polynomial automorphism of ℂ2 has a global constant weighted sum of iterates if and only if the map is conjugate to an elementary mapping. The definition of a constant weighted sum of iterates is motivated by an attempt to understand the polynomial automorphism groups in dimensions 3 and higher.