Computation of Lyapunov exponents of matrix products
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912197116952576 |
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| author | Fan, Aihua Verbitskiy, Evgeny |
| author_facet | Fan, Aihua Verbitskiy, Evgeny |
| contents | For $m$ given square matrices $A_0, A_1, \cdots, A_{m-1}$ ($m\ge 2$), one of which is assumed to be of rank $1$, and for a given sequence $(ω_n)$ in $\{0,1, \cdots, m-1\}^\mathbb{N}$, the following limit, if it exists, $$L(ω):=\lim_{n\to \infty} \frac 1n \log \|A_{ω_0} A_{ω_2}\cdots A_{ω_{n-1}}\|$$ defines the Lyapunov exponent of the sequence of matrices $(A_{ω_n})_{n\ge 0}$. It is proved that the Lyapunov exponent $L(ω)$ has a closed-form expression under certain conditions. One special case arises when $A_j$'s are non-negative and $ω$ is generic with respect to some shift-invariant measure; a second special case occurs when $A_j$'s (for $1\le j<m$) are invertible and $ω$ is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of $\mathcal{B}$-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11941 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computation of Lyapunov exponents of matrix products Fan, Aihua Verbitskiy, Evgeny Dynamical Systems Probability 37H15, 37B10 For $m$ given square matrices $A_0, A_1, \cdots, A_{m-1}$ ($m\ge 2$), one of which is assumed to be of rank $1$, and for a given sequence $(ω_n)$ in $\{0,1, \cdots, m-1\}^\mathbb{N}$, the following limit, if it exists, $$L(ω):=\lim_{n\to \infty} \frac 1n \log \|A_{ω_0} A_{ω_2}\cdots A_{ω_{n-1}}\|$$ defines the Lyapunov exponent of the sequence of matrices $(A_{ω_n})_{n\ge 0}$. It is proved that the Lyapunov exponent $L(ω)$ has a closed-form expression under certain conditions. One special case arises when $A_j$'s are non-negative and $ω$ is generic with respect to some shift-invariant measure; a second special case occurs when $A_j$'s (for $1\le j<m$) are invertible and $ω$ is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of $\mathcal{B}$-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages. |
| title | Computation of Lyapunov exponents of matrix products |
| topic | Dynamical Systems Probability 37H15, 37B10 |
| url | https://arxiv.org/abs/2501.11941 |