The generic wickedness conjecture for chain recurrent maps

Let $M$ be a compact manifold. A continuous map on $M$ is chain recurrent if every point is chain recurrent, and a homeomorphism is considered with the analogous property. Let $\mathcal{C}$ be either (i) the set of chain recurrent continuous maps on $M$, with $M$ of any dimension, or (ii) the set of chain recurrent homeomorphisms on $M$, with $M$ of dimension at least $2$. Then a generic element of $\mathcal{C}$ is wicked.

Generic wickedness conjecture. A generic element of $\mathcal{C}$ is wicked.

The statement proposes that, among chain recurrent systems, wicked dynamics is generic in the indicated spaces. The paper motivates it by contrasting the abundance of trapping regions, associated with wholesome systems, with their absence in chain recurrent systems, where wickedness is expected to prevail.