Pressures for flows on arbitrary subsets

With the help of the theory of Carathéodory structure, the topological pressure for flows on non-compact sets is introduced, and the measure-theoretic pressure for flows is defined as well. It is shown that the topological pressure for flows on arbitrary sets is equal to the one of its time one map. Moreover, given an ergodic measure for a flow, there exists a certain set on which the topological pressure is equal to the measure-theoretic one. A variational principle for the topological pressure defined above is also established. At the end, the pressures for a frame flow are studied.