We report on a new class of fast-roll inflationary models. In a part of its parameter space, inflationary perturbations exhibit quite unusual phenomena such as scalar and tensor modes freezing out at widely different times, as well as scalar modes reentering the horizon during inflation. One specific point in parameter space is characterized by extraordinary behavior of the scalar perturbations. Freeze-out of
scalar perturbations as well as particle production at horizon crossing are absent. Also the behavior of the perturbations around this quasi-de Sitter background is dual to a quantum field theory in flat space-time. Finally, the form of the primordial power spectrum is determined by the interaction between different modes of scalar perturbations.