Thermodynamics and statistics of non-uniformly hyperbolic dynamical systems

非均匀双曲动力系统的热力学和统计

• 批准号: 1362838
• 负责人: Vaughn Climenhaga
• 金额: $ 12万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362838&HistoricalAwards=false
• 关键词: Thermodynamics; statistics; non; uniformly; hyperbolic

It is natural to ask how much our knowledge of the present tells us about the future. If we study some system that evolves in time according to known rules, what predictions can we make based on observing the present state of the system? It is by now well-understood that even relatively simple systems can display chaotic behavior, in which our ability to make exact predictions decays quite quickly as we look further ahead. In this case one may hope to treat the system as a random process and make statistical predictions using tools from probability theory. This approach has been successfully carried out for uniformly hyperbolic systems, the most strongly chaotic. However, most physically realistic examples fall outside of this class, including important models from meteorology (the Lorenz system), population dynamics (the logistic map), and others. This motivates the study of non-uniformly hyperbolic systems, where the present state of knowledge is much less complete. There has been progress towards understanding certain classes of non-uniformly hyperbolic systems, but many open problems remain, both for systems that have been studied and for broader classes of systems. This research project will give new results for several important classes of non-uniformly hyperbolic systems, and is an important step in developing a more complete understanding of physically relevant systems displaying chaotic behavior.Key elements of the uniformly hyperbolic theory include existence and uniqueness results for equilibrium states and SRB measures, together with statistical properties for these measures, such as the central limit theorem governing long-term fluctuations of observations around an expected average, and large deviations principles describing the probability of outcomes far from that average. Some of these results, but not all, are known for classes of one-dimensional maps and their perturbations (strongly dissipative maps), partially hyperbolic systems, and geodesic flows. The PI will extend these results to weakly dissipative maps and more general geodesic flows, and will strengthen existing results in all three categories. The key innovation making these extensions possible is the introduction by the PI and his co-authors of new tools for non-uniform hyperbolicity: a notion of "effective hyperbolicity" for the construction of SRB measures, and a notion of "thermodynamic specification" for uniqueness and statistical properties of equilibrium states. These tools have already yielded a number of new results and have clear applicability to broader classes of systems.

很自然地问我们对当前的知识告诉我们未来的知识。 如果我们研究一些根据已知规则及时演变的系统，那么根据观察系统状态，我们可以做出什么预测？ 到目前为止，即使是相对简单的系统也可以显示混乱的行为，在这种行为中，我们的确切预测能力很快就会衰落，因为我们进一步向前看。 在这种情况下，人们可能希望将系统视为一个随机过程，并使用概率理论中的工具进行统计预测。 这种方法已经成功地针对均匀的双曲系统，最混乱的。 但是，大多数身体上现实的例子都超出了这一类，包括来自气象学（Lorenz系统），人口动态（Logistic Map）等的重要模型。 这激发了对当前知识状态的非均匀双曲线系统的研究。 在理解某些类别的非均匀双曲线系统方面取得了进展，但是对于已经研究的系统和更广泛的系统，仍然存在许多开放问题。 该研究项目将为几类非统一的双曲系统提供新的结果，并且是对显示混乱行为的物理相关系统的更完整理解的重要一步。统一的双波利理论的关键要素包括存在和唯一性结果，包括均衡状态和SRB的统计范围，例如，这些量子的统计范围以及这些量子的统计范围，例如，这些量子的统计范围，例如，这些量子的统计范围以及范围限制了范围的范围，范围限制了范围的范围，范围限制了范围的范围，范围限制了范围的范围。偏离原则描述了结果的概率远非该平均值。 其中一些结果（但不是全部）以一维图的类别及其扰动（强烈耗散图），部分双曲线系统和地球流量而闻名。 PI将将这些结果扩展到弱耗散图和更一般的大地测量流，并将加强在这三个类别中的现有结果。 使这些扩展可能成为可能的关键创新是PI及其共同作者的新工具的非均匀双曲线的引入：用于构建SRB测量的“有效双光纤性”的概念，以及“热力学规范”的概念，以实现平衡状态的唯一性和统计特性。 这些工具已经产生了许多新的结果，并且对更广泛的系统具有明显的适用性。