Maximizing measures in hyperbolic dynamics

在动力系统的研究中，长期的渐近行为往往是最好的理解方面的性质不变的措施。在特定的设置拉格朗日流量，马内和马瑟制定了一些重要的问题，这些措施的积分最大化一个特定的积分。作为一般理论的一部分，人们可以对各种各样的动力系统提出类似的问题。在双曲（或混沌）动力系统的特殊情况下，我们可以考虑一类称为吉布斯测度的自然测度，其研究起源于统计力学的数学理论，以及Ruelle和Sinai的工作。这个建议涉及如何最大化措施的典型功能，这些双曲系统可以近似这些更好的表现吉布斯措施。这对理解相当复杂但重要的最大化措施有重要意义，这些措施是用简单得多的吉布斯措施来实现的。

In the study of dynamical systems, the long term asymptotic behaviour is often best understood in terms of properties of invariant measures. In the particular setting of Lagrangian flows, Mane and Mather formulated a number of important questions about those measures whose integrals maximized a particular integral. As part of a general theory, one can ask similar questions about quite diverse dynamical systems. In the particular case of hyperbolic (or chaotic) dynamical systems, we can consider a natural class of measures called Gibbs measures, whose study originated in the mathematical theory of Statistical Mechanics, and the work of Ruelle and Sinai. This proposal relates to how maximizing measures for typical functions for these hyperbolic systems can be approximated by these better behaved Gibbs measures. This has important implications for understanding the quite complicated, but important, maximizing measures in terms of much simpler Gibbs measures.