Iterated polynomials are dense

TL;DR

This paper proves that for any infinite field and positive integer, the map sending a polynomial to its r-th iterate is dense in various topologies, showing a form of genericity of iterates.

Contribution

It provides a constructive proof that polynomial iteration maps are dense in multiple topologies over infinite fields, extending understanding of polynomial dynamics.

Findings

The r-th iterate map is dense in various topologies.

Constructive methods demonstrate the dominance of iteration maps.

Results hold for any infinite field and positive integer r.

Abstract

For any infinite field k and any positive integer r, we show constructively that the map sending each polynomial P $\in$ k[x] to its r-th iterate is dominant in various inductive limit topologies on the space of all polynomials.