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 $d$-dense (the orbit meets every ball of radius $d$), weakly dense and such that $\Gamma \cdot Orb_P(x)$ is dense for every $\Gamma\subset \mathbb C$ that is either unbounded or that has 0 as an accumulation point. Moreover we generalize the construction to arbitrary infinite dimensional separable Fr\'echet spaces. To prove this we study Julia sets of homogeneous polynomials on Banach spaces.