Introduction: an overview

Summary

Rigidity in dynamics

In a very general sense, modern theory of smooth dynamical systems deals with smooth actions of “sufficiently large but not too large” groups or semigroups (usually locally compact but not compact) on a “sufficiently small” phase space (usually compact, or, sometimes, finite volume manifolds). Important branches of dynamics specifically consider actions preserving a geometric structure with an infinite-dimensional group of automorphisms, two principal examples being a volume and a symplectic structure. The natural equivalence relation for actions is differentiable (corr. volume preserving or symplectic) conjugacy.

One version of the general notion of rigidity in this context would refer to a certain class A of actions being described by a finite set of parameters, usually smooth moduli. Examples of such classes are all actions in the neighborhood of a given one, or all actions of a continuous group with the same orbits, or all G-extensions of a given action α to a given principal G-bundle. In some situations this is too strong and, rather than classifying all actions from A, one may require that actions equivalent to a given one have a finite codimension in a properly defined sense, e.g., appear in typical or generic finite-parametric families of actions.