Ground states and zero-temperature measures at the boundary of rotation sets

Abstract

We consider a continuous dynamical system$f:X\rightarrow X$on a compact metric space$X$equipped with an$m$-dimensional continuous potential$\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{m}):X\rightarrow \mathbb{R}^{m}$. We study the set of ground states$GS(\unicode[STIX]{x1D6FC})$of the potential$\unicode[STIX]{x1D6FC}\cdot \unicode[STIX]{x1D6F7}$as a function of the direction vector$\unicode[STIX]{x1D6FC}\in S^{m-1}$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of$\unicode[STIX]{x1D6F7}$. In particular, for each$\unicode[STIX]{x1D6FC}$the set of rotation vectors of$GS(\unicode[STIX]{x1D6FC})$forms a non-empty, compact and connected subset of a face$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$of the rotation set associated with$\unicode[STIX]{x1D6FC}$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any$m\in \mathbb{N}$examples with an exposed boundary point (that is,$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of$GS(\unicode[STIX]{x1D6FC})$is a non-trivial line segment.