Generic ergodic optimisation for invertible dynamical systems

遍历优化是动力系统的一个子领域，它考虑选择动力系统的初始条件的问题，以便最大化沿轨迹沿着的观测序列的长期平均输出。1999年，Guocheng Yuan和Brian R.亨特问，对于流形上的扩张动力系统或双曲动力系统，对于C^1可观测量，通过将初始条件取为适当的周期点，这个平均值是否被最大化，这在一般意义上是正确的（在贝尔范畴的意义上）。贡萨洛·孔特雷拉斯（Gonzalo Contreras）在2016年的文章《基态一般是一个周期轨道》（Ground States are generally a Periodic Orbit）（发表在《数学发明》（Inventiones Mathematicae））中对这个问题做出了积极的回答。本项目的总体目标是将Contreras的结果扩展到可逆的双曲动力系统，使用Thierry Bousch介绍的双动力系统的遍历优化方法，在更广泛的背景下回答袁和亨特的问题。最初，该项目将集中在可逆的符号动力系统，以期扩展的结果，可逆的动力系统，如Anosov同构，如果时间允许。如果时间允许的话，也可以探讨这些结果在连续时间的动力系统的扩展。

Ergodic optimisation is a subfield of dynamical systems which considers the problem of choosing the initial condition of a dynamical system in order to maximise the long-term average output of a sequence of observations along the trajectory. In 1999, Guocheng Yuan and Brian R. Hunt asked whether for expanding or hyperbolic dynamical systems on manifolds, it is generically true (in the sense of Baire category) that for a C^1 observable this average is maximised by taking the initial condition to be an appropriate periodic point. This question was answered positively by Gonzalo Contreras in the expanding case in the 2016 article "Ground States are generically a Periodic Orbit" (published in Inventiones Mathematicae). The overall objective of the present project is to extend Contreras' result to invertible, hyperbolic dynamical systems using the methodology of ergodic optimisation for amphidynamical systems introduced by Thierry Bousch, answering the question of Yuan and Hunt in this broader context. Initially the project will focus on invertible symbolic dynamical systems, with a view to extending the results to invertible dynamical systems such as Anosov diffeomorphisms if time permits. Extensions of these results to dynamical systems in continuous time may also be explored if time permits.