ANOSOV-KATOK CONSTRUCTIONS FOR QUASI-PERIODIC SL(2,R) COCYCLES

The uploaded paper proves exactly the statement the model claims is likely open: for every 1/2<κ<1 and Diophantine α, there is a set S dense in AR_α\UH_α where the Lyapunov exponent is exactly κ-Hölder at each point, measured in the C^0 norm on the fiber map (Theorem 1.1) . The proof proceeds in two steps: (i) a local Anosov–Katok construction yields, for any ε>0, a cocycle A with ‖A−Id‖ small at which L is exactly κ-Hölder (Proposition 4.1, with the matching upper bound Lemma 4.1 and lower bound Lemma 4.2) ; (ii) a quantitative almost-reducibility normal form with controlled errors (Proposition 4.2, proved in the appendix) is then used to transplant this property to a dense set inside AR_α\UH_α, after a careful elliptic/parabolic/hyperbolic case analysis that rules out the hyperbolic obstruction or reduces to the elliptic case (see the proof steps around (4.22)–(4.25)) . The model’s contrary argument rests on invoking a universal 1/2-Hölder upper bound and square-root behavior at certain boundaries; however, the paper constructs points where a stronger upper bound with exponent κ>1/2 holds uniformly for all perturbation directions (Lemma 4.1), so the liminf cannot be forced down to 1/2 by any “square-root” direction. Moreover, the paper explicitly positions the result within AR_α\UH_α (not only at boundary points where square-root phenomena occur), and shows density via the fibered Anosov–Katok scheme . Background remarks in the paper about sharp 1/2-Hölder continuity in the subcritical regime and at gap edges are acknowledged, but they do not preclude exact κ>1/2 behavior at specially constructed points; indeed, the authors’ construction gives a strictly stronger local modulus where available .