Upper bound on the regularity of the Lyapunov exponent for random products of matrices

The paper proves that for finitely supported measures on SL2(R) with L(µ)>0 that are not uniformly hyperbolic, the Lyapunov exponent cannot be locally α-Hölder around µ for any α>H(µ)/L(µ). In its tangency-based formulation (Theorem A), it also exhibits an analytic one-parameter family through µ witnessing the failure of α-Hölder regularity when µ has a heteroclinic tangency . The model’s solution invokes exactly this theorem and the standard “unfolding a tangency” mechanism, giving a correct argument that aligns with the paper’s core strategy. Minor overreach: the model asserts a specific cusp lower bound |E|^{H/L}·|log|E|| along a rotation family; the paper’s method builds analytic Schrödinger-type families and proves the non-α-Hölder threshold via matchings/IDS counting and Thouless’ formula, without explicitly establishing that precise cusp for an arbitrary rotation path . Overall, both are consistent; the model’s proof is essentially an application of the paper’s main result, with consistent mechanism and assumptions.