Quantitative reducibility of Ck quasi-periodic cocycles

This paper establishes an extreme C-k reducibility theorem of quasi-periodic SL( 2, I[8) cocycles in the local perturbative region, revealing both the essence of Eliasson [Floquet solutions for the 1-dimensional quasi-periodic Schr & ouml;dinger equation. Comm. Math. Phys. 146 (1992), 447-482], and Hou and You [Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math. 190 (2012), 209-260] 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 Kolmogorov-Arnold-Moser (KAM) iterations. This, in return, makes various spectral applications of quasi-periodic Schr & ouml;dinger operators wide open.