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

We consider a continuous dynamical system $f:X\to X$ on a compact metric space $X$ equipped with an $m$-dimensional continuous potential $Φ=(ϕ_1,\cdots,ϕ_m):X\to \bR^m$. We study the set of ground states $GS(α)$ of the potential $α\cdot Φ$ as a function of the direction vector $α\in S^{m-1}$. %We also study the corresponding rotation vectors $\rv(GS(α))$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of $Φ$. In particular, for each $α$ the set of rotation vectors of $GS(α)$ forms a non-empty, compact and connected subset of a face $F_α(Φ)$ of the rotation set associated with $α$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in $F_α(Φ)$. 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. $F_α(Φ)$ being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face $F_α(Φ)$ with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of $GS(α)$ is a non-trivial line segment.

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