Ergodic Theory of Weakly Hyperbolic Dynamical Systems

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Dmitry Dolgopyat.Ergodic theory of weakly hyperbolic dynamical systems.Smooth ergodic theory deals with long time asymptotic of autonomous systems which can be described by finite number of parameters. An amazing fact is that even though the evolution of such systems is completely deterministic sensitive dependence on initial conditions (that is exponential divergence of nearby trajectories)often leads the random behavior in the sense that there is asymptoticmeasure (equilibrium state) such that most orbits become distributed in the phase space according to this measure. During past decades powerful methods for proving the existence of the asymptotic measure have been developed. This theory already has found spectacular applications in many areas of pure mathematics most notably number theory and Lie theory. It is less clear if it can be appliedto real life problems because in practise the assumption that the system can be described by a finite number of parameters is merely an idealization which is harmless in short time range but may be crucial in the long run. The aim of this project is to combine recent advances in the fields of non-uniformly hyperbolic and partially systems in order to develop effective and reliable toolsfor solving the following problems. (1) Determining the rate of convergence to equilibrium;(2) Describing the change of equilibrium under various perturbations of the system.At first the theory developed in this project would be of perturbative nature describing small perturbations of well understood systems in particular of transversely hyperbolic systems with compact symmetry group but eventually it should be applicable to a large class of dynamical systems. This research will be of interest in any field of science where ordinary differential equations are used.