A planar homeomorphism \(h: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) induces a homeomorphism \(h_*:\mathbb{P} \rightarrow \mathbb{P}\) on the space of prime ends \(\mathbb{P}\) associated to a non-separating Peano continuum \(K \subseteq \mathbb{R}^2\) strongly invariant under \(h\). Although the dynamics of \(h|_K\) is simple and for some continua the dynamical properties of \(h_*\) are similar to those of the restricted map \(h : K \rightarrow K\), the authors construct a continuum \(K\) and a planar dissipative homeomorphism \(h\) such that \(K\) is the global attractor of \(h\) and \(h_*\) is a Denjoy map. Therefore the dynamics induced on \(\mathbb{P}\) can be more complex than the original dynamics on \(K\).