DENSITY OF NON-ZERO EXPONENT OF CONTRACTION FOR PINCHING COCYCLES IN Hom(S1)

The paper proves Theorem A via: (i) holonomies under su-domination and their continuity (Proposition 3.2, citing Avila–Viana) ; (ii) an invariance principle showing that zero contraction exponent(s) force s-/u-invariant disintegrations (Theorems 5.2 and 5.1) after cohomologizing to a one-sided cocycle and applying Malicet’s theorem ; (iii) a localized rotation perturbation near a homoclinic loop that destroys su-states (Equation (8) and the ensuing contradiction) ; and (iv) openness of the absence of su-states by closedness of s/u-states under weak-* limits (Proposition 4.3 and the final argument in Section 6) . The candidate solution follows the same architecture: stable/unstable holonomies, the same invariance principle, the same homoclinic rotation idea, and the same openness argument. Minor stylistic differences (e.g., describing two compensating rotations versus a single bump-function rotation) do not affect correctness, and the non-positivity of λ_con justifies the “≥ 0 implies 0” step (Definition 2.3) .