The Aubry set for the XY model and typicality of periodic optimization for 2-locally constant potentials
Yuika Kajihara, Shoya Motonaga, Mao Shinoda
TL;DR
This work addresses the problem of characterizing action-minimizing structures (Aubry and Mather sets) for the XY model with an uncountable symbol space and analyzes the typical periodic optimization (TPO) behavior for 2-locally constant potentials. It blends weak KAM methods for symbolic dynamics with variational techniques from twist-map theory, introducing the Mañé potential $S_{\varphi}$ and Peierl's barrier $H_{\varphi}$ to define the Aubry set $\Omega_{\varphi}$ and related objects. The authors obtain explicit descriptions of optimal periodic measures, establish complete Aubry/Mather set characterizations under a twist condition, and prove generic TPO results showing periodic minimizers are typical in this setting. These results extend ergodic optimization to systems with uncountable alphabets and demonstrate robust periodic minimization phenomena under twist, providing a framework for further study of action-minimization in complex symbolic and twist-dynamical contexts.
Abstract
We consider the Aubry set for the XY model, symbolic dynamics $([0,1]^{\mathbb{N}_0},σ)$ with the uncountable symbol $[0,1]$, and study its action-optimizing properties. Moreover, for a potential function that depends on the first two coordinates we obtain an explicit expression of the set of optimal periodic measures and a detailed description of the Aubry set. We also show the typicality of periodic optimization for 2-locally constant potentials with the twist condition. Our approach combines the weak KAM method for symbolic dynamics and variational techniques for twist maps.