  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.
