  We propose a solvable class of 1D quasiperiodic tight-binding models
encompassing extended, localized, and critical phases, separated by nontrivial
mobility edges. Limiting cases include the Aubry-Andr\'e model and the models
of PRL 114, 146601 and PRL 104, 070601. The analytical treatment follows from
recognizing these models as a novel type of fixed-points of the renormalization
group procedure recently proposed in arXiv:2206.13549 for characterizing phases
of quasiperiodic structures. Beyond known limits, the proposed class of models
extends previously encountered localized-delocalized duality transformations to
points within multifractal critical phases. Besides an experimental
confirmation of multifractal duality, realizing the proposed class of models in
optical lattices allows stabilizing multifractal critical phases and
non-trivial mobility edges without the need for the unbounded potentials
required by previous proposals.
