Orbits of homogeneous polynomials on Banach spaces

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 delta-dense (the orbit meets every ball of radius delta), weakly dense and such that Gamma . OrbP(x) is dense for every Gamma subset of C that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinitedimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.