Dynamical Systems ( IF 0.5 ) Pub Date : 2021-06-17 , DOI: 10.1080/14689367.2021.1933914 K. Burns 1 , J. Buzzi 2 , T. Fisher 3 , N. Sawyer 4
We study the one parameter family of potential functions associated with the unstable Jacobian potential (or geometric potential) for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For q<1, it is known that there is a unique equilibrium state associated with , and it has full support. For q>1 it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value q = 1 and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure or measures supported on the singular set. In particular, when q = 1, there is a unique ergodic equilibrium state that gives positive measure to the regular set.
中文翻译:
具有非正曲率的一级曲面的测地线流的相变
我们研究势函数的单参数族 与不稳定的雅可比势(或几何势)相关联 用于非正曲率的紧凑秩 1 表面的测地线流动。对于q <1,已知有一个独特的平衡状态与,并得到全力支持。对于q >1,已知不变测度是均衡状态当且仅当它在奇异集上得到支持。我们研究了临界值q = 1 并表明遍历平衡状态要么是对 Liouville 测度的正则集的限制,要么是在奇异集上支持的测度。特别是,当 q = 1 时,存在一个独特的遍历平衡状态,它对正则集给出了正测度。