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

We consider a continuous dynamical system f : X → X on a compact metric space X equipped with an m-dimensional continuous potential Φ = (φ1, · · · , φm) : X → R . We study the set of ground states GS(α) of the potential α · Φ as a function of the direction vector α ∈ S. 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 Fα(Φ) of the rotation set associated with α. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in Fα(Φ). We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any m ∈ N examples with an exposed boundary point (i.e. Fα(Φ) being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face Fα(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.