De la Llave will study the long term behavior of deterministic systems appearing in practical problems. The methods proposed for this study are usually based in finding landmarks that organize the long term behavior. Since these landmarks satisfy complicated functional equations, part of this project is devoted to the analytical study of these non-local functional equations. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics" can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.
