CAREER: Unifying approaches to non-uniform hyperbolicity

在现实世界中，我们认为有许多现象具有随机、不可预测的行为，尽管事实上它们遵循着完全确定且易于理解的基本规则。例如，当我们投掷一对骰子时，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，并能做出可靠的短期预测，但从现在起几周内天气的确切变化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驱动这种从可预测性到随机性进化的机制。无论系统处于何种状态，当这种机制始终如一地运行时，系统就被称为“均匀双曲”。这样的系统是很容易理解的，但是这种均匀性条件是如此的严格，以至于它很少适用于物理现实的例子。更现实的情况是“非均匀双曲”，在这种情况下，随机性的增加只在某些时候发生，但最终会出现在典型的初始条件下。几种不同的方法被用来研究非均匀双曲系统，每种方法都有自己的优点和缺点。这个项目的目标是通过澄清和加强这些现有方法之间的联系来发展一个统一的非一致双曲理论；通过引入新工具；通过学习新的例子。历史上，研究非均匀双曲最成功的两种方法是Pesin理论（1976年提出）和Young塔（1998年提出）。PI和他的合著者最近的工作引入了“有效双曲性”和“非统一规范”。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，一旦选择了适当的不变测度，Pesin理论就会给出强有力的结果，但其他三种方法对于寻找区分不变测度（如SRB测度和平衡状态）更有用。同样，Young塔比其他方法产生更强的统计特性，但塔的建造可能相当困难。最近，Sarig用Pesin理论给出了一个可数马尔可夫分区的构造，它允许为任何双曲度量建立一个塔，但没有给出关于塔尾部衰减率的信息，而这对于中心极限定理等统计性质是必要的。PI和他的合著者的初步结果给出了另一种结构，可以估算衰变率；PI将发展这种方法，以获得广泛的双曲测度的强统计性质，包括使用有效双曲构造的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，我们将考虑非正曲率的测地线流；这些技术有望为最大熵和其他平衡状态的独特度量提供相关性的快速衰减，并为刘维尔度量的规则分量（受流形上的几何条件限制）提供相关性的多项式衰减。更一般地说，这四种方法之间的联系将大大加强有效双曲线和非统一规范的工具；它还将使杨塔理论更容易应用，并给出一种精确的感觉，即所有四种方法都是等效的。长期目标是扩展严格理解的非一致双曲例子的类别；一个重要的有计划的应用是Teichmuller流，它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.