# Hypercyclic homogeneous polynomials on $H(\mathbb C)$

## TL;DR

This paper demonstrates the existence of hypercyclic, frequently hypercyclic, and chaotic homogeneous polynomials on the space of entire functions, providing the first such example on an F-space and contrasting with non-hypercyclic cases.

## Contribution

It constructs the first example of a hypercyclic homogeneous polynomial on an F-space, specifically on $H(\mathbb C)$, and analyzes its dynamical properties.

## Key findings

- The polynomial defined as the product of a translation operator and evaluation at 0 is mixing.
- This polynomial is also frequently hypercyclic and chaotic.
- Some related natural polynomials do not exhibit hypercyclicity.

## Abstract

It is known that homogeneous polynomials on Banach spaces cannot be hypercyclic, but there are examples of hypercyclic homogeneous polynomials on some non-normable Fr\'echet spaces. We show the existence of hypercyclic polynomials on $H(\mathbb C)$, by exhibiting a concrete polynomial which is also the first example of a frequently hypercyclic homogeneous polynomial on any $F$-space. We prove that the homogeneous polynomial on $ H(\mathbb C)$ defined as the product of a translation operator and the evaluation at 0 is mixing, frequently hypercyclic and chaotic. We prove, in contrast, that some natural related polynomials fail to be hypercyclic.