4. Extending the BurnsClimenhagaFisherThompson Technology

Problem 4.1.
[Vaughn Climenhaga] Let $S$ be a surface which has a flat cylinder of parallel periodic orbits. In [MR3856792], a criterion for the existence and uniqueness of equilibrium states includes a pressure gap, $P_{\operatorname{sing}}(\varphi) < P(\varphi)$. Can this be made explicit for the surface $S$? 
Problem 4.2.
[Dan Thompson] Define a decomposition and pressure gap for $\mbox{CAT}(0)$ geodesic flows, and extend [MR3856792] for these flows. 

Problem 4.4.
[Kiho Park and Keith Burns] Extend [MR3856792] to the case of no focal points. In particular, can one apply this to the case of the geometric potential? In particular, the Donnaytype sphere examples of surfaces with “spherical” caps. 
Cite this as: AimPL: Equilibrium states for dynamical systems arising from geometry, available at http://aimpl.org/equibdynsysgeom.