CAREER: Unifying approaches to non-uniform hyperbolicity

职业：统一非均匀双曲性的方法

• 批准号: 1554794
• 负责人: Vaughn Climenhaga
• 金额: $ 50万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1554794&HistoricalAwards=false
• 关键词: CAREER; Unifying; approaches; non; uniform

There are many phenomena in the real world that we perceive as having random, unpredictable behavior, despite the fact that they follow underlying rules that are completely deterministic and well understood. For example, when we roll a pair of dice, they fly and bounce according to physical laws that are non-random, but the outcome of the roll is nevertheless random. A more sophisticated example is weather; although we understand atmospheric dynamics and can make reliable short-term predictions, the exact behavior of the weather several weeks from now remains largely random and unpredictable. The theory of hyperbolic dynamical systems studies the mechanism that drives this evolution from predictability to randomness. When this mechanism operates consistently no matter what state the system is in, the system is said to be "uniformly hyperbolic". Such systems are well-understood, but this uniformity condition is so restrictive that it rarely applies to physically realistic examples. A more realistic condition is "non-uniform hyperbolicity", where the increase in randomness only happens some of the time, but nevertheless appears eventually for typical initial conditions. Several different approaches have been used to study non-uniformly hyperbolic systems, each with its own advantages and disadvantages. The goal of this project is to develop a unified theory of non-uniform hyperbolicity by clarifying and strengthening the connections between these existing approaches; by introducing new tools; and by studying new classes of examples.Historically, the two most successful approaches to non-uniform hyperbolicity are Pesin theory (introduced in 1976) and Young towers (introduced in 1998). Recent work by the PI and his co-authors has introduced "effective hyperbolicity" and "non-uniform specification". All four approaches study non-uniform hyperbolicity via invariant measures, but the connections between them are not yet clear, and different approaches have different applications. For example, Pesin theory gives powerful results once an appropriate invariant measure has been selected, but the other three approaches are more useful for finding distinguished invariant measures such as SRB measures and equilibrium states. Similarly, Young towers yield stronger statistical properties than the other approaches, but construction of a tower may be quite difficult. Recently Sarig gave a construction of countable Markov partitions using Pesin theory, which allows a tower to be built for any hyperbolic measure, but gives no information on the rate of decay of the tail of the tower, which is necessary for statistical properties such as the central limit theorem. Preliminary results by the PI and his co-authors give an alternate construction that yields estimates on the decay rate; the PI will develop this approach to obtain strong statistical properties for a broad class of hyperbolic measures, including the SRB measures constructed using effective hyperbolicity and the equilibrium states constructed using non-uniform specification. As a first concrete example, geodesic flow in non-positive curvature will be considered; these techniques are expected to give rapid decay of correlations for the unique measure of maximal entropy and other equilibrium states, and to give polynomial decay of correlations for the regular component of Liouville measure (subject to geometric conditions on the manifold). More generally, the connection between the four approaches will significantly strengthen the tools of effective hyperbolicity and non-uniform specification; it will also make the the powerful theory of Young towers easier to apply, and give a precise sense in which all four approaches are equivalent. A longer-term goal is to extend the class of rigorously understood non-uniformly hyperbolic examples; one important planned application is to Teichmuller flow, which has deep geometrical significance.

在现实世界中，我们认为有许多现象是具有随机，不可预测的行为，尽管事实上它们遵循了完全确定性且知名度充分的基本规则。 例如，当我们滚动一对骰子时，它们会根据非随机的物理定律飞行并反弹，但是卷的结果仍然是随机的。 一个更复杂的例子是天气。尽管我们了解大气动态并可以做出可靠的短期预测，但从现在起几周后天气的确切行为仍然很大程度上是随机的且无法预测的。 双曲动力学系统的理论研究了从可预测性到随机性发展的机制。 当该机制始终如一地运行，无论系统所在状态如何，该系统被称为“均匀的双曲线”。 这样的系统被很好地理解了，但是这种统一性条件是如此限制，以至于它很少适用于身体上现实的例子。 一个更现实的条件是“非均匀的双曲线”，其中随机性的增加仅在某些时候发生，但是最终在典型的初始条件下出现。 已经使用了几种不同的方法来研究非均匀的双曲系统，每个系统都有其自身的优势和缺点。 该项目的目的是通过澄清和加强这些现有方法之间的联系来发展统一的非均匀双曲线理论。通过引入新工具；通过研究新的示例。从历史上讲，佩辛理论（1976年引入）和年轻的塔（于1998年引入），两种最成功的非均匀双曲线方法是佩辛理论。 PI和他的合着者最近的工作引入了“有效的双曲线”和“不均匀的规格”。 所有四种方法都通过不变措施研究了非均匀的双曲线，但是它们之间的连接尚不清楚，不同的方法具有不同的应用。 例如，一旦选择了适当的不变度度量，佩辛理论就会产生强大的结果，但是其他三种方法对于找到诸如SRB测量和均衡状态之类的杰出不变措施更有用。 同样，年轻塔比其他方法产生的统计特性更强，但是塔楼的建造可能很困难。 最近，萨利格（Sarig）使用佩辛理论（Pesin Theory）构造了可数的马尔可夫分区，该理论允许为任何双曲线措施构建塔楼，但没有提供有关塔尾巴衰减速率的信息，这对于诸如中心极限定理等统计属性是必需的。 PI和他的合着者的初步结果提供了一种替代结构，可以估计衰减率； PI将开发这种方法，以获得一系列多种双曲线测量的强统计特性，包括使用有效双曲线构建的SRB度量和使用不均匀规范构建的平衡状态。 作为第一个具体的例子，将考虑非阳性曲率中的大地测量流。预计这些技术将对最大熵和其他平衡状态的独特度量的相关性快速衰减，并给出了liouville度量的常规组成部分的相关性多项式衰减（符合歧管上的几何条件）。 更一般而言，这四种方法之间的联系将显着增强有效的双曲线和不均匀规范的工具。这也将使年轻塔的强大理论更容易应用，并具有确切的意义，使所有四种方法都是等效的。 一个长期的目标是扩展严格理解的非均匀双曲线实例的类别；一项重要的计划应用是Teichmuller流，具有深刻的几何意义。