Lyapunov exponents for random products of non-negative matrices
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918358587277312 |
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| author | Alibabaei, Nima |
| author_facet | Alibabaei, Nima |
| contents | We first study i.i.d. products of finitely many invertible $2 \times 2$ matrices with positive entries, and prove that the top Lyapunov exponent admits an explicit, rapidly convergent Neumann-series-type representation involving an infinite matrix. We further show that non-negative invertible $2 \times 2$ matrices are simultaneously conjugate to positive matrices if and only if ``generalized'' heteroclinic connections do not occur among products of length at most $2$.
These results yield a series formula for the Hausdorff dimension of the intersection of the middle-$n$th Cantor set with a random translate of itself, for every natural number $n$ except $4$. Furthermore, our method applies to the intersection of thick Cantor sets under random translation. We also determine the almost sure growth rate of i.i.d. three-term recurrences with finitely many positive coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23317 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lyapunov exponents for random products of non-negative matrices Alibabaei, Nima Dynamical Systems We first study i.i.d. products of finitely many invertible $2 \times 2$ matrices with positive entries, and prove that the top Lyapunov exponent admits an explicit, rapidly convergent Neumann-series-type representation involving an infinite matrix. We further show that non-negative invertible $2 \times 2$ matrices are simultaneously conjugate to positive matrices if and only if ``generalized'' heteroclinic connections do not occur among products of length at most $2$. These results yield a series formula for the Hausdorff dimension of the intersection of the middle-$n$th Cantor set with a random translate of itself, for every natural number $n$ except $4$. Furthermore, our method applies to the intersection of thick Cantor sets under random translation. We also determine the almost sure growth rate of i.i.d. three-term recurrences with finitely many positive coefficients. |
| title | Lyapunov exponents for random products of non-negative matrices |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2602.23317 |