Maximizing Families and Typical Periodic Optimization for Almost Additive Sequences

Abstract

In this paper, we study typical periodic optimization (TPO) for almost additive potentials in two perturbation spaces. Our main setting is the Banach quotient ${\mathcal{E}}_{\mathrm{orb}}(X,T)$ 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 ${\mathcal{E}}_{\mathrm{orb}}(X,T)$ is infinite-dimensional and the maximizing periods in open locking regions are unbounded. For a fixed almost additive potential $\Phi$, we also develop relative TPO theory on its Lipschitz leaf. When $\Phi =0$, 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.