  We study invariant Fatou components for holomorphic endomorphisms in
$\mathbb{P}^2$. In the recurrent case these components were classified by
Sibony and the second author in 1995. In 2008 Ueda completed this
classification by proving that it is not possible for the limit set to be a
punctured disk. Recently Lyubich and the third author classified non-recurrent
invariant Fatou components, under the additional hypothesis that the limit set
is unique. Again all possibilities in this classification were known to occur,
except for the punctured disk. Here we show that the punctured disk can indeed
occur as the limit set of a non-recurrent Fatou component. We provide many
additional examples of holomorphic and polynomial endomorphisms of
$\mathbb{C}^2$ with non-recurrent Fatou components on which the orbits converge
to the regular part of arbitrary analytic sets.
