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Orbits of homogeneous polynomials on Banach spaces

Profile image of Santiago García MuroSantiago García Muro

2020, Ergodic Theory and Dynamical Systems

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 • OrbP (x) is dense for every Γ ⊂ 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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