  The almost periodic eigenvalue problem described by the Harper equation is
connected to other classes of quasiperiodic behaviour: the dissipative dynamics
on critical invariant tori and quasiperiodically driven maps. Firstly, the
strong coupling limit of the supercritical Harper equation and the strong
dissipation limit of the critical standard map play equivalent role in the
renormalization analysis of the self-similar fluctuations of localized
eigenfunctions and the universal slope of the projected map on the invariant
circle. Secondly, we use a simple transformation to relate the Harper equation
to a quasiperiodically forced one-dimensional map. In this case, the localized
eigenstates of the supercritical Harper equation correspond to strange but
nonchaotic attractors of the driven map. Furthermore, the existence of
localization in the eigenvalue problem is associated with the appearance of
homoclinic points in the corresponding map.

Chaotic Dynamics;Mesoscale and Nanoscale Physics;Chaotic Dynamics