Time averages of polynomials

We define and study when a polynomial mapping has a local or global time average. We conjecture that a polynomial f in the complex plane has a time average near a point z if and only if z is eventually mapped into a Siegel-disc of f. We prove that the 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 iterated monodromy groups. We also show that a polynomial automorphism of C^2 has a global time average if and only if the map is conjugate to an elementary mapping. The definition of a time average is motivated by an attempt to understand the polynomial automorphism groups in dimensions 3 and higher.