UNIFORM POSITIVITY OF THE LYAPUNOV EXPONENT FOR C1 MONOTONE POTENTIALS GENERATED BY THE CAT MAP

The paper proves that for C1, bounded potentials on the 2-torus generated by the cat map with directional derivative along the unstable direction bounded away from zero, there is C0(v) with L(E;λ) > log λ − C0 for all E and λ > 0. The main theorem and its proof structure are explicit: a polar-decomposition reduction to a cocycle of the form Λ(Tω)R_{θ(ω,t)} (with t = E/λ) is stated, then Fubini is used to reduce to local unstable leaves Wz, and a leafwise lower bound is obtained by controlling ∑j log|cos θj| via transversality and measure estimates near θ = π/2; this yields L ≥ log λ − C0 uniformly in E (Theorem 1; reduction via (5); Proposition 1; inequality (22)) . The argument along leaves, including the growth of D_u θn and the decomposition into sets Ji with exponentially decaying measure, is presented in detail and culminates in the required lower bound . A minor presentational issue is that the main text displays a λ-independent R_{θ(ω,t)} in (5), while Appendix A clarifies that the rotation depends on λ and only converges to that form as λ → ∞; this should be stated explicitly (but does not invalidate the leafwise lower-bounding scheme) . By contrast, the model’s solution hinges on a claimed general inequality “Craig–Simon/Thouless-type lower bound” L(E) ≥ ∫ log|E − λ v(ω)| dμ(ω) − log 2. This is not a standard consequence of Craig–Simon: the Thouless formula relates L to the integrated density of states, not directly to the single-step potential distribution. The subharmonicity approach does not yield that particular bound in general, and the step is neither justified nor correct in the stated generality. Without that step, the rest of the model’s argument (which includes a plausible transversality estimate) does not establish the desired Lyapunov lower bound.