
## 4. Extending the Burns-Climenhaga-Fisher-Thompson Technology

1. #### 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 4.3.

[Keith Burns] Can one simplify the assumptions and/or arguments of [MR3046278] to the setting of [MR3856792]?
• #### 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 Donnay-type sphere examples of surfaces with “spherical” caps.
• #### Problem 4.5.

[Keith Burns] Can one find a simplified proof of Burns-Gelfert in [MR3124716] using [MR3856792], possibly with stronger conclusions? Similarly for the results of Paulin-Policott-Schapira in [MR3444431].

Cite this as: AimPL: Equilibrium states for dynamical systems arising from geometry, available at http://aimpl.org/equibdynsysgeom.