M\"obius structures and timed causal spaces on the circle

We discuss a conjectural duality between hyperbolic spaces on one hand and spacetimes on the other hand, living on the opposite sides of the common absolute. This duality goes via M\"obius structures on the absolute, and it is easily recognized in the classical case of symmetric rank one spaces. In a general case, no trace of such duality is known. As a first step in this direction, we show how M\"obius structures on the circle from a large class including those which stem from hyperbolic spaces give rise to 2-dimensional spacetimes, which are axiomatic versions of de Sitter 2-space, and vice versa. The paper has two Appendices, one of which is written by V.Schroeder.