Furstenberg counterexamples over Diophantine rotations

We construct cocycles in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T times upper S upper U left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mo> × </mml:mo> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {T} \times SU(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over Diophantine rotations that are minimal and not uniquely ergodic. Such cocycles are dense in an open subset of cocycles over the fixed Diophantine rotation. By a standard argument, they are dense in the whole set of such cocycles if the rotation in the basis satisfies a full-measure arithmetic condition. Contrary to the examples existing in the literature, ours have vanishing Lyapunov exponents, like in the original construction by Furstenberg. The construction is explicit, without any need for a parameter exclusion argument, and can be carried out for any prescribed fibered rotation number, provided that the latter satisfies a necessary generic arithmetic condition with respect to the rotation in the basis <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .