The rigidity conjecture (Q1749009)

One of the aims of dynamics is to understand whether two dynamical systems are topologically the same. This is determined by the existence of a homeomorphism which conjugates the two systems. A related question is then to ask when two systems are geometrically the same, that is, when is the conjugacy differentiable? This geometrical equivalence question has been studied in the last forty years in the case of circle diffeomorphisms, unimodal maps, critical circle maps and so on. It turned out that, under mild topological restrictions (these restrictions are essential), the conjugacy between two systems is differentiable whenever it exists. We say that two maps are in the same topological class if they are conjugated on their attractors. We say that two maps are in the same rigidity class if they are \(C^{1+\alpha}\)-conjugated on their attractors for some \(\alpha > 0\). In this paper, the authors state a conjecture which describes how topological classes are organized into rigidity classes.