  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 $C^0$-rigidity for H\"older pseudo-rotations
with similar properties. These provide a partial generalisation of the main
results in [B. Bramham, Invent. Math. (2015), no. 2, 561-580; A. Avila, B.
Fayad, P. Le Calvez, D. Xu and Z. Zhang, arXiv: 1509.06906v1].
  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 centralizers of any such diffeomorphism is isomorphic to a
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 diffeomorphism 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.
