DISCONTINUITY OF LYAPUNOV EXPONENTS FOR SL(2,R)-VALUED COCYCLES

The paper proves Theorem A: for any α>0 and η<σ with 2^{3α}<η^2, one can approximate the locally constant diagonal cocycle A_{σ,η} by C^α cocycles with zero Lyapunov exponents, hence A_{σ,η} is a discontinuity point for Lyapunov exponents. The construction perturbs A_{σ,η} only on a single cylinder Z_n and two of its shifts via carefully chosen unipotent matrices R_{γ,n}, achieves a swap of the Oseledets directions across the n-block, and then uses an induced/projective ergodic argument to force λ^±=0; the Hölder closeness is verified by three explicit inequalities that reduce to 2^{3α}<η^2 when γ=4/3, see the statement of Theorem A and its proof outline and estimates in the paper. This is internally consistent and complete (A_{σ,η} and µ_p are set up in (2.4), Theorem A states 2^{3α}<η^2, the perturbation R_{γ,n} is defined in (3.2), and the Hölder estimates (3.5)–(3.8) lead to γ=4/3 and the required condition; the induced/projective argument then yields λ(B_n,µ_p)=0). By contrast, the model’s multiscale “tiny rotations in sparse windows” outline contains critical gaps: (i) a key conjugation estimate is incorrect in sign/size (the deviation of log‖H(b)R_θH(−a)‖ from |b−a| is not of order e^{-2 min{a,b}}|θ| as claimed, and the window product is only uniformly bounded when θ≈e^{-2a}, giving an O(1) correction, not a small one); and (ii) the covering/frequency argument is flawed—because μ_p(E_k^{01}∪E_k^{10})=[p(1−p)]^{m_k}, the series ∑_k(2m_k+1)[p(1−p)]^{m_k} always converges for 0<p<1, so the proposed window coverage cannot occupy a full proportion of time and cannot force λ^+=0. The paper’s approach avoids all such coverage issues by working with a single scale and an induced map. Therefore, the paper is correct while the candidate outline is not. See A_{σ,η} and µ_p setup and λ(A_{σ,η},µ_p) formula, the precise statement of Theorem A, the construction R_{γ,n} and Hölder bounds, and the ergodic/projective argument yielding λ(B_n,µ_p)=0 in the paper.