Quantitative Reducibility of Cᵏ Quasi-Periodic Cocycles

This paper establishes an extreme Cᵏ reducibility theorem of quasi-periodic SL(2, R) cocycles in the local perturbative region, revealing both the essence of Eliasson [Commun.Math.Phys.1992] and Hou-You [Invent.Math.2022] in respectively the non-resonant and resonant cases. By paralleling further the reducibility process with the almost reducibility, we are able to acquire the least initial regularity as well as the least loss of regularity for the whole KAM iterations. This, in return, makes various spectral applications of quasi-periodic Schr\"odinger operators wide open.