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We show that under certain boundedness condition, a \({{C}^{r}}\) conservative irrational pseudo-rotations on \({\mathbb{T}^{2}}\) with a generic rotation vector is \({{C}^{r-1}}\)-rigid. We also obtain C0-rigidity for Hölder pseudo-rotations with similar properties. These provide a partial generalisation of the main results in [Bra15, AFLXZ15]. We then use these results to study conservative irrational pseudo-rotations on \({\mathbb{T}^{2}}\) with a generic rotation vector that is semi-conjugate to a translation via a semi-conjugacy homotopic to the identity. We show that the conservative centralizer of any such diffeomorphism is isomorphic to an uncountable subgroup of \({\mathbb{R}^{2}/\mathbb{Z}^{2}}\). In connection with a question of Alec Norton and Dennis Sullivan, we describe the topologically linearizable maps within this class using the topology of the conservative centralizer group. In the minimal case, we obtain a precise characterization of topological linearizability for all totally irrational vectors. We also construct a \({{C}^\infty}\) conservative and minimal totally irrational pseudo-rotation that is semi-conjugate to a translation, but is topologically nonlinearizable. This gives a negative answer to the question of Norton and Sullivan in the \({{C}^\infty}\) category.

The rigidity of pseudo-rotations on the two-torus and a question of Norton-Sullivan