  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.
