Orbits of homogeneous polynomials on Banach spaces

2018/06/29 by Cardeccia, Rodrigo, Muro, Santiago
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1806.11543

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 Γ⋅ Orb_P(x) is dense for every Γ⊂ \mathbb C that is either unbounded or that has 0 as an accumulation point. Moreover we generalize the construction to arbitrary infinite dimensional separable Fréchet spaces. To prove this we study Julia sets of homogeneous polynomials on Banach spaces.

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