For a broad class of families of 2-dimensional dissipative diffeomorphisms, sufficient conditions for the existence of attractors carrying Sinai-Ruelle-Bowen measures will be investigated. The study of ergodic properties of these measures and of geometric structure of the respective attractors is proposed. The project is based on joint research with S. Newhouse. One of the main tools is the new technique of distortion estimates for two-dimensional maps with unbounded derivatives. The basic example is a family of piecewise smooth transformations defined on a finite number of rectangles which are all mapped hyperbolically except for one, which is mapped parabolically. No underlying one-dimensional maps are assumed. The technique is based on a system of initial conditions, which can be numerically checked, such as expansion, contraction, initial distortions and dependence of these quantities on the parameters and can be applied to a variety of models. This project involves the investigation of the phenomenon of random oscillations in deterministic systems. This type of oscillation arises in a variety of mathematical models in physics, chemistry, and biology. The behavior of such systems depends on exterior parameters. For some parameters, systems exhibit stationary or periodic solutions. However for other parameters, the same systems behave randomly. More precisely, random solutions may appear with higher probability than periodic ones. The goal of this work is to give strict mathematical conditions which imply random behavior. Sufficient conditions can be checked numerically, which provides straightforward applications to the real systems.
