CAREER: Unifying approaches to non-uniform hyperbolicity

在真实的世界中有许多现象，我们认为它们具有随机的、不可预测的行为，尽管它们遵循的基本规则是完全确定的，而且是众所周知的。 例如，当我们掷一对骰子时，它们根据非随机的物理定律飞行和反弹，但掷出的结果仍然是随机的。 一个更复杂的例子是天气;虽然我们了解大气动力学，可以做出可靠的短期预测，但几周后天气的确切行为在很大程度上仍然是随机和不可预测的。 双曲动力系统理论研究了驱动这种从可预测性到随机性的演化的机制。 当这个机制无论系统处于什么状态都一致地起作用时，系统被称为“一致双曲”。 这样的系统是很好理解的，但是这种一致性条件是如此的限制，以至于它很少适用于物理上现实的例子。 一个更现实的条件是“非均匀双曲性”，其中随机性的增加只发生在某些时候，但最终会出现在典型的初始条件下。 研究非一致双曲方程组有几种不同的方法，每种方法都有其优缺点。 该项目的目标是通过澄清和加强这些现有方法之间的联系，通过引入新的工具，并通过研究新的例子，发展一个统一的非均匀双曲理论。历史上，两个最成功的方法，非均匀双曲是Pesin理论（1976年推出）和杨塔（1998年推出）。 PI和他的合著者最近的工作引入了“有效双曲性”和“非均匀规范”。 这四种方法都是通过不变测度来研究非一致双曲性的，但它们之间的联系尚不清楚，不同的方法有不同的应用。 例如，一旦选择了合适的不变测度，Pesin理论给出了强大的结果，但其他三种方法对于寻找诸如SRB测度和平衡态等独特的不变测度更有用。 类似地，年轻塔产生比其他方法更强的统计特性，但是塔的构造可能相当困难。 最近Sarig给出了一个建设可数马尔可夫分区使用Pesin理论，它允许一个塔，以建立任何双曲措施，但没有提供任何信息的衰减率的尾部的塔，这是必要的统计性质，如中心极限定理。 PI和他的合著者的初步结果给出了另一种构造，该构造产生对衰减率的估计; PI将开发这种方法来获得广泛的双曲测度的强统计特性，包括使用有效双曲性构造的SRB测度和使用非均匀规范构造的平衡态。 作为第一个具体的例子，在非正曲率的测地线流将被考虑;这些技术预计会给最大熵和其他平衡状态的唯一措施的相关性的快速衰减，并给出多项式衰减的相关性的刘维测度（受几何条件的流形上）。 更一般地说，这四种方法之间的联系将大大加强有效双曲性和非均匀规范的工具;它也将使强大的杨塔理论更容易应用，并给出一个精确的意义，其中所有四种方法都是等价的。 一个长期的目标是扩展严格理解的非一致双曲的例子类;一个重要的计划应用是Teichmuller流，这具有深刻的几何意义。

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.