I am reading the famous and concise Thouless-Kohmoto-Nightingale-den Nijs (TKNN) paper Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405–408 (1982), where I have several subtle questions about the details.
At the beginning, it says with a weak periodic potential and flux $\varphi = p/q, p,q\in \mathbb Z$ per unit cell, "each Landu level is split into $p$ subbands..." But on the last page, it says with strong periodic potential, $p$ and $q$ are interchanged.
Are there $q$ subbands in the latter case?
Why $\varphi \mapsto 1/\varphi$ in the strong potential case?
In the figure below,
what do the arrows and dots all mean?
How do they explain the quantized conductance?!
Any ideas? Many thanks! To be continued...