The project proposes several research problems in Ergodic Theory of Dynamical Systems. The theory is in a period of great activity, fueled by both the necessary internal coherence and its many interactions with other domains of mathematics or of mathematical Physics. The projects anticipates developments on three topics. One is parabolic dynamics. The precise problems of interest here are the characterization of infinite invariant measures for horocyclic flows on cover spaces, the new theorems of convergence in average along orbits which appear in connection with these measures, the speed of convergence in these theorems and the possible applications of all these results. The second topic is almost hyperbolic dynamics. This covers questions where the lack of hyperbolicity is precisely localized. Some of the previous questions are related to frame flows, or more generally compact group extensions of hyperbolic flows. Another example is the geodesic flow on rank one manifolds, where the nonhyperbolicity is concentrated on a singular invariant closed set. There is some ongoing progress about the Martin boundary and the geometry at infinity of the universal cover of a rank one manifold. The third topic concerns several ergodic theoretic questions related to Quantum Chaos.
In general, the evolution of a dynamical system is represented by the action of a one parameter group -the time- over a space of configurations. Ergodic theory is the study of global statistical properties of such actions. The two extreme cases have been flourishing successes of last century mathematics. On the one side, the elliptic case, when the motion is periodic or quasi-periodic gave rise to KAM theory. The algebraic simple model is the translations on a compact group. On the other extreme, the hyperbolic case is the one where the trajectories diverge exponentially. It is also well understood now, with extraordinarily precise statistical results. The standard models are the geodesic flow in negative curvature or the algebraic action of an integer matrix on the torus, when no eigenvalue has modulus one. Many interesting programs today try to develop new tools to understand the vast territory between the two extreme cases. As summarized above, the propositions of this project explore some possible paths. In spite of their apparent diversity, they have common threads. An unexpected one is the role of invariant distributions, which appear for different reasons in several of these a priori unrelated questions.
