Poincaré duality for the tropical Dolbeault cohomology of non-archimedean Mumford curves

We calculate the tropical Dolbeault cohomology for the analytifications of P1 and Mumford curves over non-archimedean fields. We show that the cohomology satisfies Poincaré duality and behaves analogously to the cohomology of curves over the complex numbers. Further, we give a complete calculation of the dimension of the cohomology on a basis of the topology.