Probabilistic and algebraic aspects of smooth dynamical systems

Abstract

In general terms, the objective of research in the area of Dynamical systems and Ergodic Theory is to describe the long term behavior of systems with an evolution law. There are deep interactions among Dynamical systems, other areas in mathematics like probability theory, geometry and number theory. There are also connections with many sub-areas in physics. In the Smooth ergodic theory, typically one tries to compare the evolution of an observable by action of a group with the long term behavior of independent random variables. In the last 60s and 70s many fundamental results in smooth ergodic theory were obtained and the vast majority were obtained in the context of uniformly hyperbolic dynamics which was mainly introduced by S. Smale (without forgetting Andronov and Pontryagin and Moscow school). Bowen, Ruelle and Sinai realized a connection between statistical mechanic and uniformly hyperbolic dynamics and constructed physical measures for such systems. After progresses in uniformly hyperbolic dynamics, weak forms of hyperbolicity like dominated splitting, partial hyperbolicity and non-uniform hyperbolicity were introduced mainly by Mañé, Pugh, Pesin and Shub. In the one-dimensional dynamical systems, inspirations from statistical mechanics (Feigenbaum, Coullet and Tresser) yields the origins of a rich teory of renormalization operator (defined dynamically) and universality in dynamics. Sullivan and Lyubich developped in a fundamental way the theory and it is interesting to mention that the Lyubich works, the (Smale) Horseshoe appears in the infinite dimensional model dynamics. The ideas from ergodic theory have been used in a very effective way to solve problems in number theory and Lie algebras too. We may highlight ergodic methods in the study of geodesic flow and horocyclic flow in the manifolds with negative curvature. The solution of Margulis for the oppenheim conjecture is also based on the methods from smooth ergodic theory and dynamical systems. This project is divided into following sub-projects which have both probabilistic and algebraic aspects. Possible solutions to the questions will have a positive impact in the development of the theory: Ergodic properties of dynamical systems preserving a "natural" probability measure: Equilibrium states for partially hyperbolic diffeomorphisms (in particular measures of maximal entropy, Margulis measures)Stable ergodicity for conservative diffeomorphisms Equilibrium states for action of groups Piecewise expanding transformations with low regularity Comparing classical partially hyperbolic dynamical systems and random walks, Invariance Principle (introduced by Avila-Viana generalizing Furstenberg, Ledrappier (and others) results into non-linear dynamical setting)Renormalization Theory and universality in one-dimensional dynamics Deformation theory in one-dimensional dynamics Dynamics defined by action of groups: Anosov (sub)systems and their representationsin SL(n,R) (AU)