The Rigidity Conjecture for topological classes of dynamical systems

A smooth dynamical system has an attractor in the sense of Milnor. Two maps are in the same topological class if they are conjugated on their attractors, and they are in the same probabilistic rigidity class if the conjugacy is $C^{1+\alpha}$ almost everywhere with respect to the dynamically relevant measure, for some $\alpha>0$. A stratification of a topological class is a partition into finite-codimension submanifolds.

Rigidity Conjecture. The topological class is a finite-codimension manifold which is stratified by probabilistic rigidity classes.

The conjecture aims to describe how the topological and geometric properties of a system determine the geometry of its attractor. It is formulated for topological classes with bounded combinatorics; rigidity is known in several classical settings, while the general structure of the stratification remains open.