PROBABILISTIC METHODS IN NON-HYPERBOLIC DYNAMICS 

Below are the participating countries.

Click on a flag, and you get a link to particapting people in that country.

	Belgium		Finland		UK		Italy		Switzerland		Sweden
	Poland		Portugal		France		Germany		Netherlands

Probabilistic and statistical methods are becoming increasingly important for  understanding deterministic dynamical systems. Starting in the 50s and 60s it became apparent that deterministic dynamical systems have many of the features of random and stochastic systems. This connection has been rigorously established for so-called hyperbolic systems. A large class of systems however are not hyperbolic and this includes most dynamical systems which arise out of modeling natural and physical processes. The great challenge of the 80s and 90s and of at least the first decade of the next century is to extend and generalize the techniques of hyperbolic dynamics to study non-hyperbolic systems. There have been several recent breakthroughs in the last few years and many of the researchers directly involved are European.

There are two main ways in which probabilistic and statistical methods come into this area of research. The first, more classical, is through the study of statistical properties of a given class of systems, like ergodicity and decay of correlations. The second is through the methods used to prove the existence of specific kinds of non-hyperbolic systems. Indeed some of the most interesting recent developments have involved the discovery of systems which occur very rarely from the classical point of view of topology but nevertheless with positive probability in a natural sense. Main examples of this phenomena occur in the well-known logistic family of one-dimensional maps, the Hénon family of plane diffeomorphisms, geometric models of the Lorenz family of differential equations and in rational holomorphic maps (see full references below).

Europe plays an important role in this research but at the moment many of research projects and activites are fragmented. This grant will unify the European efforts on this exciting project.

See a copy of the original grant application 1997

(see also the PRODYN page at the European Science Foundation web site).