This generalizes a result of Abhyankar. The simplest such surface corresponds to Pinchuk's construction in the real case. In fact, we prove one more equivalent formulation of the Jacobian conjecture using etale exotic surfaces. We consider polynomial vector fields on etale exotic surfaces and explore their properties in relation to the Jacobian conjecture. Conversely, each such tiling arises in this way.
Similar results hold for harmonic functions with more general boundary conditions.
Certain prescriptions of transition probabilities on edges in M give rise to tilings with prescribed shapes. This allows us to give necessary conditions for the existence of a tiling of an arbitrary polygon with squares, equilateral triangles, and so on. Using this method, we classify all polygons with at most one non-convex vertex which can be tiled with squares. A similar classification holds for tiling with equilateral triangles. We determine the Euclidean tori which can be square-tiled.