Resumen de Smooth classification of Cartan actions of higher rank semisimple Lie groups and their lattices
Edward R. Goetze, Ralf J. Spatzier
Anosov diffeomorphisms and flows (actions of ${\bold Z}$ and ${\bold R}$) are important in dynamics because they combine dynamical complexity with topological rigidity (structural stability): up to continuous coordinate change (topological conjugacy), these systems are unchanged by $C^1$-perturbations. For smooth conjugacy the situation is completely different (almost no $C^1$-perturbation is $C^1$-conjugate to the original system), but in the last decade it has become apparent that actions of larger groups (with some hyperbolicity) tend to be smoothly rigid, i.e., perturbing such an action leads to an action that is smoothly conjugate. The present paper is aimed at a conjecture that all higher-rank actions with an Anosov element are affine. (If $H$ is a connected simply-connected Lie group and $\Lambda\subset H$ is a cocompact lattice, then a diffeomorphism of $H/\Lambda$ is said to be affine if it lifts to a diffeomorphism of $H$ that respects right-invariant vector fields.) The authors give several sets of sufficient conditions for an action to be smoothly conjugate to an affine one. In the case of a connected semisimple Lie group without compact factors and with real rank at least three it suffices to have a volume-preserving Anosov action on a compact manifold $M$ that is trellised with respect to a maximal ${\bold R}$-split Cartan subgroup $A$ (i.e., there is a sufficiently large $A$-invariant collection of invariant one-dimensional foliations) and is multiplicity-free (the superrigidity homomorphism corresponding to the action consists of irreducible subrepresentations that are multiplicity-free). For lattices in a connected semisimple Lie group without compact factors and with no simple factor of real rank less than two it suffices to have a volume-preserving Cartan action on a compact manifold (i.e., suitable intersections of stable and unstable manifolds of certain commuting elements are one-dimensional). This was conjectured by Hurder to suffice even without the Cartan assumption.The paper has a helpful introduction that explains how it builds on prior work, and good references.
