Analyticity of Lyapunov Exponents for Mixed Markov Quasi-Periodic Cocycles

What the math gives to ML

The paper provides a constructive regularity mechanism for the top Lyapunov exponent of a Markov-switching quasi-periodic linear system: under a primitive transition matrix, non-resonant base dynamics, and a simple top exponent, the exponent is analytic in the transition probabilities. The proof maintains a vector of state-conditioned projective measures and obtains uniform contraction of a Markov transfer operator, rather than treating the switching process as iid. This transfers naturally to Markov-gated recurrent networks, stochastic-depth networks, or mixture-of-experts routers whose mode transitions affect the recurrent Jacobian. The most useful implementation is to optimize routing probabilities while monitoring the stationary projective distribution and testing the predicted smoothness boundary: gradients should remain stable away from transition-matrix degeneracy, resonance, or Lyapunov-exponent collisions.