It is known that the robust absence of volume hyperbolicity (a weaker notion of hyperbolicity) on a chain recurrence class implies the generic coexistence of inifintely many sinks or sources, in particular the wildness (generic coexistence of infinitely many distinct chain recurrence classes) of the system. In other words, volume hyperbolicity is a necessary conditions for the tameness (robust finitude of chain recurrence classes).

In this talk, I will show that volume hyperbolicity is not a sufficient condition for the tameness, by giving examples of diffeomorphisms on any three manifold which is volume hyperbolic and simultaneously wild. As a by-product, I give an example of open region of diffeomorphisms in which generic diffeomorphism has neither topological attractors nor repellers.

The construction comprises the study of the shape of the attracting region and the repelling region from the (differential) topological viewpoint and the C^1 perturbation technique called flexible periodic points.