We will describe an approach to the study of actions of various groups by diffeomorphisms of a compact manifold based primarily on the ideas from hyperbolic dynamics and smooth ergodic theory which leads to the conclusion that certain kinds of actions necessarily preserve a geometric structure.
Here is a sample of specific examples:
1. Let $\rho$ be an action of $\Bbb Z^k,\,\, k\ge 2$ on the flat torus by affine maps such that the linear part of each non-zero element is semi-simple and does not have one as an eigenvalue. Assume also that the linear part of some element is hyperbolic. Then any $C^1$ small perturbation of $\rho$ preserves a flat affine structure.
2.Let $\Gamma$ be a cocompact lattice in $SL(n,\Bbb R),\,\, n\ge 3$, and $\rho$ be the standard projective action on $\Bbb RP^{(n-1)}$. Then any $C^1$ small perturbation of $\rho$ preserves a projective structure.
3. Any action of $\Bbb Z^2$ on the three-torus, which is effective on the first homology group, preserves an absolutely continuous measure.