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


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      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 $S$?


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          Problem 4.2.

          [Dan Thompson] Define a decomposition and pressure gap for $\mbox{CAT}(0)$ geodesic flows, and extend [MR3856792] for these flows.


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              Problem 4.3.

              [Keith Burns] Can one simplify the assumptions and/or arguments of [MR3046278] to the setting of [MR3856792]?


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                  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.


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                      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.