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arXiv · 2511.10114

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

Thomas Morand
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Abstract

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 $ϕ$, analytic, increasing, and convex in the fibre variable, we consider the smallest invariant function q satisfying q(x) = $ϕ$(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.

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Thomas Morand. 2025-11-13. Numerical bounds on the regularity of an invariant function: Probability of extinction of Galton-Watson processes in dynamical environments. https://arxiv.org/abs/2511.10114

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