Reducibility Without KAM

Abstract We prove rotations-reducibility for close to constant quasi-periodic $$SL(2,\mathbb {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> cocycles in one frequency in the finite regularity and smooth cases, and derive some applications to quasi-periodic Schrödinger operators.