Time averages of polynomials

TL;DR

This paper investigates when polynomials and polynomial automorphisms have local or global time averages, linking these properties to dynamical features like Siegel-discs and elementary mappings, with implications for higher-dimensional automorphism groups.

Contribution

It introduces the concept of time averages for polynomials and automorphisms, proves a conjecture for generic polynomials, and characterizes automorphisms with global time averages.

Findings

A polynomial has a local time average near a point if and only if the point is eventually in a Siegel-disc.

A polynomial automorphism of C^2 has a global time average if and only if it is conjugate to an elementary map.

The conjecture holds generically for polynomials with maximal critical value count.

Abstract

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.