  Quasiperiodic systems extend the concept of Anderson transition to the
quasi-random and low-dimensional realm, exhibiting intricate behaviors even in
the one-dimension, while their investigation in higher dimensions remains less
explored. Here, we delve into a series of two-dimensional lattice models of
Hall systems with asymptotically incommensurate flux, and reveal the impact of
asymptotic incommensurability together with relaxation on transport phenomena.
Specifically, we demonstrate anomalous bulk transport with universal scaling
characteristics in the wave-packet dynamics and conductivity, and predict novel
interplay effects involving asymptotic incommensurability, temperature, and
relaxation, leading to unprecedented multiple anisotropic metal-insulator
transitions. The asymptotic quasiperiodicity also leads to the quantized
anisotropic edge tunneling transport. Our work enriches the universal quantum
transport phenomena, and add to the fundamental mechanisms underlying the
metal-insulator transitions driven by incommensurability in higher dimensions,
potentially opening a new avenue for exploring novel transport physics in
quasiperiodic systems.
