Log-Höder continuity at zero Lyapunov gap for finite state Markov $GL(2)$-cocycles

What the math gives to ML

The paper develops robustness bounds for Lyapunov exponents of finite-state Markov products of 2x2 matrices at the difficult point where the top and bottom exponents coincide. Its transferable asset is an explicit inverse-logarithmic sensitivity law: small perturbations of mode matrices or transition probabilities produce controlled changes in long-run growth rates, with a stronger modulus for conformal cocycles. This suggests a trust-region controller for recurrent and state-space networks whose linear dynamics switch according to a learned finite-state Markov chain. The controller can use empirical Lyapunov estimates to prevent optimizer steps from moving the recurrent dynamics into an unstable regime.