2016/04/21 by Tamara Kucherenko, Christian Wolf, Kucherenko, Tamara +1
Mathematics · Physics and Astronomy · #37B10 #37D35 #37E45 #37E45 (Primary) #37L40 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1604.06512
openalex publication_date 2016/04/21 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We consider a continuous dynamical system f:X\→ X on a compact metric\nspace X equipped with an m-dimensional continuous potential\n\Φ=(\φ1,\⋯,\φm):X\→ bRm. We study the set of ground states \nGS(\α) of the potential \α\⋅ \Φ as a function of the direction\nvector \α\∈ Sm-1. %We also study the corresponding rotation vectors\n rv(GS(\α)). We show that the structure of the ground state sets is\nnaturally related to the geometry of the generalized rotation set of \Φ. In\nparticular, for each \α the set of rotation vectors of GS(\α)\nforms a non-empty, compact and connected subset of a face F_\α(\Φ) of\nthe rotation set associated with \α. Moreover, every ground state\nmaximizes entropy among all invariant measures with rotation vectors in\nF_\α(\Φ). We further establish the occurrence of several quite\nunexpected phenomena. Namely, we construct for any m\∈ bN examples with an\nexposed boundary point (i.e. F_\α(\Φ) being a singleton) without a\nunique ground state. Further, we establish the possibility of a line segment\nface F_\α(\Φ) with a unique but non-ergodic ground state. Finally, we\nestablish the possibility that the set of rotation vectors of GS(\α) is a\nnon-trivial line segment.\n