Counterexamples to a rigidity conjecture

We discuss several counterexamples to a rigidity conjecture of K. Khanin, which states that under some quantitative condition on non-existence of periodic orbits, $C^0$ conjugacy implies $C^1$ (even $C^\infty$) conjugacy. We construct examples of non-rigid diffeomorphisms on the $2$-torus, which satisfy the assumptions of Khanin's (but not of Krikorian's) conjecture. We also construct examples of flows which are topologically conjugate, but not $C^1$ conjugate, in contradiction to a natural generalization of the conjecture to flows. These latter examples are based on results on solutions of the cohomological equation and suggest that the structure of the space of invariant distributions has to be taken into account in rigidity questions.