Ground States and Zero-Temperature Measures at the Boundary of Rotation Sets

Abstract

We consider a continuous dynamical system $f:X\to X$ on a compact metric space $X$ equipped with an $m$-dimensional continuous potential $\Phi=(\phi_1,\cdots,\phi_m):X\to \bR^m$. We study the set of ground states $ GS(\alpha)$ of the potential $\alpha\cdot \Phi$ as a function of the direction vector $\alpha\in S^{m-1}$. %We also study the corresponding rotation vectors $\rv(GS(\alpha))$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of $\Phi$. In particular, for each $\alpha$ the set of rotation vectors of $ GS(\alpha)$ forms a non-empty, compact and connected subset of a face $F_\alpha(\Phi)$ of the rotation set associated with $\alpha$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in $F_\alpha(\Phi)$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any $m\in\bN$ examples with an exposed boundary point (i.e. $F_\alpha(\Phi)$ being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face $F_\alpha(\Phi)$ with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of $GS(\alpha)$ is a non-trivial line segment.