  We consider the effects of quasiperiodic spatial modulation on the quantum
Hall plateau transition, by analyzing the Chalker-Coddington network model for
the integer quantum Hall transition with quasiperiodically modulated link
phases. In the conventional case (uncorrelated random phases), there is a
critical point separating topologically distinct integer quantum Hall
insulators. Surprisingly, the quasiperiodic version of the model supports an
extended critical phase for some angles of modulation. We characterize this
critical phase and the transitions between critical and insulating phases. For
quasiperiodic potentials with two incommensurate wavelengths, the transitions
we find are in a different universality class from the random transition. Upon
adding more wavelengths they undergo a crossover to the uncorrelated random
case. We expect our results to be relevant to the quantum Hall phases of
twisted bilayer graphene or other Moir\'e systems with large unit cells.
