Orbits of homogeneous polynomials on Banach spaces

TL;DR

This paper explores the dynamic behavior of homogeneous polynomials on Banach spaces, providing examples with various density properties of orbits and extending the analysis to separable Fréchet spaces.

Contribution

It introduces new examples of homogeneous polynomials with complex orbit density properties and generalizes the construction to broader classes of infinite-dimensional spaces.

Findings

Existence of homogeneous polynomials with orbits that are d-dense and weakly dense.

Orbits whose scalar multiples are dense in the space.

Extension of the construction to separable Fréchet spaces.

Abstract

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show, a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $d$-dense (the orbit meets every ball of radius $d$), weakly dense and such that $\Gamma \cdot Orb_P(x)$ is dense for every $\Gamma\subset \mathbb C$ that is either unbounded or that has 0 as an accumulation point. Moreover we generalize the construction to arbitrary infinite dimensional separable Fr\'echet spaces. To prove this we study Julia sets of homogeneous polynomials on Banach spaces.