Previous results about universality in phase space have been local in nature and only concern scaling about a single point. The authors proves that a much stronger global result holds for three important examples: period-doubling cascades, golden critical circle maps and certain diffeomorphisms of the circle. The author proves that the conjugacy between the appropriate phase space structures of two systems in the same stable manifold of the appropriate renormalisation transformation is or can be extended to a C1+ alphadiffeomorphism. This means that the structures are globally geometrically equivalent and have the same global quantitative scaling properties. These results are corollaries of a general theory for Markov families and the general techniques and results should be applicable to a much wider class of problems.
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