Generic ergodic optimisation for invertible dynamical systems

遍历优化是动力系统的一个子领域，它考虑的问题是选择动力系统的初始条件，以便最大化沿着轨迹的一系列观测的长期平均输出。1999年，Guocheng Yuan和Brian R. Hunt提出，对于流形上的展开或双曲动力系统，对于C^1可观测值，通过将初始条件取为适当的周期点，该平均值是否一般成立（在Baire范畴意义上）。Gonzalo Contreras在2016年的一篇文章《基态一般是周期性轨道》（发表在Inventiones Mathematicae上）中积极地回答了这个问题。本项目的总体目标是利用Thierry Bousch介绍的双动力系统遍遍优化方法，将Contreras的结果扩展到可逆的双曲动力系统，在这个更广泛的背景下回答Yuan和Hunt的问题。最初，该项目将专注于可逆的符号动力系统，如果时间允许，将结果扩展到可逆的动力系统，如Anosov微分同态。如果时间允许，也可以将这些结果推广到连续时间的动力系统。

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.