New developments in differentiable rigidity of orbit structure for actions
of higher rank abelian groups
There is a striking difference between the orbit structure of classical smooth dynamical systems (diffeomorphisms and flows on compact
differentiable manifolds) and their higher-rank counterparts. In the former case the differentiable orbit structure is never stable under small
perturbations. In contrast to that, for the actions of higher rank abelian groups rigidity of the differentiable orbit structure does appears.
During the 1990's it was understood that the circumstances which lead to structural stability for the classical case, i.e. global hyperbolic
behavior, typically produce differentiable rigidity in the higher rank case.