  We provide several new examples in dynamics on the $2$-sphere, with the
emphasis on better understanding the induced boundary dynamics of invariant
domains in parametrized families. First, motivated by a topological version of
the Poincar\'e-Bendixson Theorem obtained recently by Koropecki and Passeggi,
we show the existence of homeomorphisms of $\mathbb{S}^2$ with Lakes of Wada
rotational attractors, with an arbitrarily large number of complementary
domains, and with or without fixed points, that are arbitrarily close to the
identity. This answers a question of Le Roux. Second, from reduced Arnold's
family we construct a parametrised family of Birkhoff-like cofrontier
attractors, where at least for uncountably many choices of the parameters, two
distinct irrational prime ends rotation numbers are induced from the two
complementary domains. This example complements the resolution of Walker's
Conjecture by Koropecki, Le Calvez and Nassiri from 2015. Third, answering a
question of Boyland, we show that there exists a non-transitive Birkhoff-like
attracting cofrontier which is obtained from a BBM embedding of inverse limit
of circles, such that the interior prime ends rotation number belongs to the
interior of the rotation interval of the cofrontier dynamics. There exists
another BBM embedding of the same attractor so that the two induced prime ends
rotation numbers are exactly the two endpoints of the rotation interval.
