Colloquium

One question frequently asked in dynamics is: what can be said about perturbations of a given dynamical system? Is any classification of perturbations possible? For \(\bf R\)-actions, i.e. flows, a reasonable classification exists only for flows tangent to Diophantine vector fields on tori. This is part of the classical Kolmogorov-Arnold-Moser (KAM) result from 1960's. These flows have a very strong property of global hypoellipticity. In this talk I will discuss this property for general differential operators and the consequences of having a higher rank abelian action whose leafwise Laplacian is globally hypoelliptic. Conjecturally for such actions one should be able to classify perturbations. This has been confirmed for a class of actions on 2-step nilmanifolds.