Sinai-Ruelle-Bowen Measures for Non-Hyperbolic Attractors

非双曲吸引子的 Sinai-Ruelle-Bowen 测度

• 批准号: 9803635
• 负责人: Michael Jakobson
• 金额: $ 4.68万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-15 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803635&HistoricalAwards=false
• 关键词: Sinai; Ruelle; Bowen; Measures; Non

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.

对于一系列二维耗散差异性的家庭，将研究有足够的有携带西奈 - 腐烂措施的吸引者的条件。 提出了这些措施的千古特性和各自吸引子的几何结构的研究。 该项目基于与S. Newhouse的联合研究。 主要工具之一是具有无限衍生物的二维图的新技术估计技术。 基本的示例是在有限数量的矩形上定义的分段平滑转换家族，除了一个矩形尺寸夸张，一个矩形均匀绘制了一个，该矩形是一个拼图绘制的。 不假定潜在的一维图。 该技术基于可以检查数值检查的初始条件系统，例如扩展，收缩，初始失真和这些数量对参数的依赖性，并且可以应用于各种模型。 该项目涉及对确定性系统中随机振荡现象的研究。 这种振荡在物理，化学和生物学的多种数学模型中产生。 此类系统的行为取决于外部参数。 对于某些参数，系统具有固定或周期性解决方案。但是，对于其他参数，相同的系统行为随机行为。 更确切地说，随机溶液的概率可能比周期性溶液更高。 这项工作的目的是给出严格的数学条件，这意味着随机行为。 可以通过数值检查足够的条件，从而为真实系统提供直接的应用程序。