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arXiv · 1703.04773

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

Rodrigo Cardeccia·Santiago Muro
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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.

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Rodrigo Cardeccia, Santiago Muro. 2017-03-14. Hypercyclic homogeneous polynomials on $H(\mathbb C)$. https://arxiv.org/abs/1703.04773

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