Explicit formulae and topological descriptions of action-minimizing sets for 2-locally potentials of the XY model
Yuika Kajihara, Shoya Motonaga, Mao Shinoda
TL;DR
This work analyzes ergodic optimization for the XY model with an uncountable alphabet under 2-locally constant potentials and a twist condition. By leveraging the Mañé potential, Peierl's barrier, and calibrated subactions, it derives explicit descriptions of the Mather and Aubry sets, and characterizes the Mañé set in this setting. For potentials phi of the form phi(x)=h(x0,x1) with h in H, the Mather and Aubry sets coincide with the a_infty fixed points for a in m_h, the quotient Aubry set is totally disconnected, while the Mañé set is far larger, containing preimages of Mather periodic points and higher-dimensional cubes. The paper also gives detailed equivalence criteria on the Aubry set and proves smallness results for the quotient Aubry set, including a unit-interval isometry in a key case, thereby clarifying the fine structure of action-minimizing dynamics in 2-locally constant XY models.
Abstract
We consider ergodic optimization of a symbolic dynamics with uncountable alphabet $[0,1]$ for 2-locally constant potentials with the twist condition, and give explicit formulae of the associated Mather set and the Aubry set. Moreover, we investigate the total disconnectedness of the (quotient) Aubry set, in which case the differentiability of the potential function makes a remarkable difference. Although these results imply that the (quotient) Aubry set is small enough, we give a complete characterization of an analogical object of the Aubry set, called the Mañé set, and show that it is much larger than the Aubry set so that it contains cubes of any finite dimension.