Numerical bounds on the regularity of an invariant function: Probability of extinction of Galton-Watson processes in dynamical environments

We study the Lyapunov exponents of models that are close to skew product systems over a C__ uniformly expanding transformation of the circle. For a continuous fibre map $\phi$, analytic, increasing, and convex in the fibre variable, we consider the smallest invariant function q satisfying q(x) = $\phi$(x, q(T x)). We provide rigorous numerical bounds on two Lyapunov exponents (the fibre exponent and the base exponent), and present algorithms to compute these bounds effectively. We then apply this framework to Galton-Watson processes in dynamical environments in the uniformly supercritical case. The probability of extinction q of the process is the invariant function of the associated system. Using the previously computed Lyapunov exponents, we control the H{ö}lder regularity and differentiability class of the probability of extinction.