Numerical bounds on the regularity of an invariant function: Probability of extinction of Galton-Watson processes in dynamical environments
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917078369304576 |
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| author | Morand, Thomas |
| author_facet | Morand, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical bounds on the regularity of an invariant function: Probability of extinction of Galton-Watson processes in dynamical environments Morand, Thomas Dynamical Systems 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. |
| title | Numerical bounds on the regularity of an invariant function: Probability of extinction of Galton-Watson processes in dynamical environments |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2511.10114 |