Ergodic optimization in dynamical systems
Ergodic optimization in dynamical systems
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DOI：
10.1017/etds.2017.142
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发表时间：
2017-12
影响因子：
0.9
通讯作者：
O. Jenkinson
中科院分区：
文献类型：
--
作者：
O. Jenkinson
遍历优化是研究与最大化轨道、不变测度和最大遍历平均值有关的问题。如果一个动力系统的轨道上的实值函数f的时间平均值大于其他轨道上的时间平均值，那么它就被称为f最大化；如果一个不变的概率测度给f的空间平均值大于其他任何不变的概率测度，那么它就被称为f最大化。在本文中，我们考虑遍历优化的主要方面，从一个有影响的模型问题开始，并将遍历优化解释为热力学形式主义的零温度极限。我们描述了各种函数空间的最大化测度的典型性质，增加共边界以揭示这些测度的性质的关键工具，以及某些已知最大化测度为Sturmian的函数类。
Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called $f$ -maximizing if the time average of the real-valued function $f$ along the orbit is larger than along all other orbits, and an invariant probability measure is called $f$ -maximizing if it gives $f$ a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.