Properties of equilibrium states for geodesic flows over manifolds without focal points

Counting closed geodesics on rank one manifolds without focal points

“…For the the geodesic flow on manifolds of nonpositive curvature, Call and Thompson [8] showed that the unique equilibrium state for certain potential has the Kolmogorov property, based on the earlier work of [28]. Recently, this result is extended to manifolds without focal points in [10], via methods developed in [8]. In particular, we have Proposition 7.4.…”

“…It is proved in [30] that m is unique MME, which is called Knieper measure. Furthermore, m is proved to be mixing in [29] and Kolmogorov in [10]. In the appendix, we prove that m is Bernoulli.…”

“…In nonpositive curvature case, the Knieper measure is proved to be mixing in [2] and Bernoulli in [8]. For manifolds without focal points, Knieper measure is known to be mixing by [29] and have the Kolmogorov property by [10]. The method of Call and Thompson [8], actually proves the Kolmogorov property of the unique equilibrium states for certain potentials [8,10].…”

“…Gelfert and Ruggiero [22] also proved the uniqueness of MME for surfaces without focal points by different methods. Extending the approach in [6], Chen, Kao and Park proved the uniqueness of equilibrium states for certain potentials in [9,10]. Manifolds without conjugate points in general are far from being well understood.…”

“…For manifolds without focal points, Knieper measure is known to be mixing by [29] and have the Kolmogorov property by [10]. The method of Call and Thompson [8], actually proves the Kolmogorov property of the unique equilibrium states for certain potentials [8,10]. For Knieper measure, to lift the Kolmogorov property to the Bernoulli property, we adapt the classical argument in [33,43] for Anosov flows.…”