On difference Riccati equation and continued fractions
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913540118413312 |
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| author | Ivanov, Alexey V. |
| author_facet | Ivanov, Alexey V. |
| contents | We study a difference Riccati equation $Φ(x) + ρ(x)/Φ(x-ω) = v(x)$ with $1-$periodic continuos coefficients. Using continued fraction theory we investigate a problem of existence of continuos solutions for this equation. It is shown that convergence of a continued fraction representing a solution of the Riccati equation can be expressed in terms of hyperbolicity of a cocycle naturally associated to this continued fraction. We apply the critical set method to establish the uniform hyperbolicity of the cocycle and to obtain sufficient conditions for the convergence of a continued fraction giving a representation for a solution of the Riccati equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07730 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On difference Riccati equation and continued fractions Ivanov, Alexey V. Dynamical Systems Functional Analysis 37C55, 37D25, 37C40 We study a difference Riccati equation $Φ(x) + ρ(x)/Φ(x-ω) = v(x)$ with $1-$periodic continuos coefficients. Using continued fraction theory we investigate a problem of existence of continuos solutions for this equation. It is shown that convergence of a continued fraction representing a solution of the Riccati equation can be expressed in terms of hyperbolicity of a cocycle naturally associated to this continued fraction. We apply the critical set method to establish the uniform hyperbolicity of the cocycle and to obtain sufficient conditions for the convergence of a continued fraction giving a representation for a solution of the Riccati equation. |
| title | On difference Riccati equation and continued fractions |
| topic | Dynamical Systems Functional Analysis 37C55, 37D25, 37C40 |
| url | https://arxiv.org/abs/2410.07730 |