Universality conjecture for the unique non-trivial renormalization fixed point

Consider the renormalization operator acting on the class of twist maps under discussion, and call a property universal when it holds on an open set of functions. A fixed point is non-trivial if it is not the integrable or otherwise trivial fixed point of this operator. Unique fixed-point universality conjecture. The existence of one and only one non-trivial fixed point of the renormalization operator is a universal property. This conjecture is described as longstanding and is said in the source to follow rigorously from an extension of the standard renormalization-group picture; the exact scope of that result and the relevant function space should be checked.