Lyapunov exponents for uniformly hyperbolic random matrix products
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918445764837376 |
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| author | Alibabaei, Nima |
| author_facet | Alibabaei, Nima |
| contents | We consider a finite family of invertible $2 \times 2$ real matrices and a transitive Markov shift on the index set. Let $λ$ be the top Lyapunov exponent for random matrix products driven by the Markov shift. We prove that, if the matrices are projectively uniformly hyperbolic with respect to the Markov shift, then $λ$ admits an explicit representation in terms of an infinite matrix. This rapidly convergent representation yields a polynomial-time algorithm for approximating $λ$: only $O\big( (\log(1/\varepsilon))^3 \big)$ arithmetic operations are needed to achieve error $\varepsilon$. Furthermore, $λ$ depends real analytically on the matrix entries and the transition probabilities near a projectively uniformly hyperbolic system, and each Taylor coefficient can be approximated in polynomial time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12244 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lyapunov exponents for uniformly hyperbolic random matrix products Alibabaei, Nima Dynamical Systems We consider a finite family of invertible $2 \times 2$ real matrices and a transitive Markov shift on the index set. Let $λ$ be the top Lyapunov exponent for random matrix products driven by the Markov shift. We prove that, if the matrices are projectively uniformly hyperbolic with respect to the Markov shift, then $λ$ admits an explicit representation in terms of an infinite matrix. This rapidly convergent representation yields a polynomial-time algorithm for approximating $λ$: only $O\big( (\log(1/\varepsilon))^3 \big)$ arithmetic operations are needed to achieve error $\varepsilon$. Furthermore, $λ$ depends real analytically on the matrix entries and the transition probabilities near a projectively uniformly hyperbolic system, and each Taylor coefficient can be approximated in polynomial time. |
| title | Lyapunov exponents for uniformly hyperbolic random matrix products |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2604.12244 |