Sinai-Ruelle-Bowen Measures for Non-Hyperbolic Attractors

对于一类广泛的二维耗散微分同胚族，研究了吸引子具有Sinai-Ruelle-Bowen测度的充分条件。研究了这些测度的遍历性质和相应吸引子的几何结构。该项目是基于与S·纽豪斯的联合研究。其中一个主要工具是对具有无界导数的二维映射的失真估计的新技术。基本的例子是定义在有限数量的矩形上的一族分段光滑变换，除了一个以抛物线映射的矩形外，所有这些矩形都是双曲线映射的。不假定底层的一维地图。该技术基于一个初始条件系统，这些初始条件可以进行数值检查，如膨胀、收缩、初始变形和这些量对参数的依赖，并可应用于各种模型。这个项目涉及对确定性系统中随机振荡现象的研究。这种类型的振荡出现在物理、化学和生物学的各种数学模型中。这类系统的行为取决于外部参数。对于某些参数，系统表现出定常或周期解。然而，对于其他参数，相同的系统行为是随机的。更准确地说，随机解可能比周期解出现的概率更高。这项工作的目标是给出严格的数学条件，暗示随机行为。充分条件可以用数值方法检验，这为实际系统提供了直接的应用。

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.