近可加序列的极大族与典型周期优化
Maximizing Families and Typical Periodic Optimization for Almost Additive Sequences
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中文总结 AI 辅助
本文在两个扰动空间中研究近可加势的典型周期优化，推广相关极大化理论并建立结构定理，构造了具有全局TPO的无限维紧系统，还发展了其利普希茨叶上的相对TPO理论。
中文摘要 AI 辅助
本文在两个扰动空间中研究近可加势的典型周期优化（TPO）。我们的主要研究环境是轨道-利普希茨近可加势的巴拿赫商空间$\boldsymbol{\textit{E}}_{\rm orb}(X,T)$，在此我们推广了W. Huang、O. Jenkinson、L. Xu和Y. Zhang建立的可极大化集与可数极大化族理论，并证明了一个全局结构定理。对于可数极大化族，若每个非边界成员具有$X$-可延拓TPO且边界区域内部为空，则全局TPO成立。作为应用，我们构造了一个具有全局TPO的紧系统，其中$\boldsymbol{\textit{E}}_{\rm orb}(X,T)$是无限维的，且开锁区中的极大化周期无界。对于固定的近可加势$\boldsymbol{\textit{\u03a6}}$，我们还在其利普希茨叶上发展了相对TPO理论；当$\boldsymbol{\textit{\u03a6}}=0$时，该框架退化为经典利普希茨TPO。我们证明了对应的叶状结构定理，并给出了全移位上的一个非可加秩-1矩阵例子。
英文摘要
In this paper, we study typical periodic optimization (TPO) for almost additive potentials in two perturbation spaces. Our main setting is the Banach quotient $\mathcal E_{\rm orb}(X,T)$ of orbit-Lipschitz almost additive potentials, where we extend the theory of maximizable sets and countable maximizable families developed by W. Huang, O. Jenkinson, L. Xu and Y. Zhang [Typical periodic optimization for dynamical systems: symbolic dynamics, Invent. Math. 245 (2026), 1--63], and establish a global structural theorem. For a countable maximizable family, global TPO holds if every non boundary member has $X$-extendable TPO and the boundary region has empty interior. As an application, we construct a compact system with global TPO for which $\mathcal E_{\rm orb}(X,T)$ is infinite-dimensional and the maximizing periods in open locking regions are unbounded. For a fixed almost additive potential $Φ$, we also develop relative TPO theory on its Lipschitz leaf. When $Φ=0$, this framework reduces to classical Lipschitz TPO. We prove the corresponding leafwise structural theorem and give a non additive rank-one matrix example on a full shift.
