  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.
