Dominated splittings and periodic data for quasi-compact operator cocycles
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915522130477056 |
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| author | Backes, Lucas |
| author_facet | Backes, Lucas |
| contents | For infinite-dimensional quasi-compact cocycles over a map satisfying a certain closing condition, we show that periodic orbits carry enough information to guarantee the existence of a dominated splitting. More precisely, we establish that if the moduli of the $(k+1)$-largest eigenvalues of the cocycle are $e^{λ_1n}\geq e^{λ_2n}\geq \ldots\geq e^{λ_kn}\geq e^{λ_{k+1}n}$ at every periodic point of period $n$, and $λ_k>λ_{k+1}$, then the cocycle admits a dominated splitting of index $k$. As a consequence, if $λ_k>0>λ_{k+1}$ then the cocycle is uniformly hyperbolic. Furthermore, we are able to obtain these same conclusions even when the eigenvalues are only close to constant, not strictly constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24822 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dominated splittings and periodic data for quasi-compact operator cocycles Backes, Lucas Dynamical Systems For infinite-dimensional quasi-compact cocycles over a map satisfying a certain closing condition, we show that periodic orbits carry enough information to guarantee the existence of a dominated splitting. More precisely, we establish that if the moduli of the $(k+1)$-largest eigenvalues of the cocycle are $e^{λ_1n}\geq e^{λ_2n}\geq \ldots\geq e^{λ_kn}\geq e^{λ_{k+1}n}$ at every periodic point of period $n$, and $λ_k>λ_{k+1}$, then the cocycle admits a dominated splitting of index $k$. As a consequence, if $λ_k>0>λ_{k+1}$ then the cocycle is uniformly hyperbolic. Furthermore, we are able to obtain these same conclusions even when the eigenvalues are only close to constant, not strictly constant. |
| title | Dominated splittings and periodic data for quasi-compact operator cocycles |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2509.24822 |