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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 
$\unicode[STIX]{x1D6FF}$-dense (the orbit meets every ball of radius 
$\unicode[STIX]{x1D6FF}$), weakly dense and such that 
$\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every 
$\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.
