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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.06511 |
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| _version_ | 1866917503550095360 |
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| author | Cornean, Horia D. Herbst, Ira W. Marcelli, Giovanna |
| author_facet | Cornean, Horia D. Herbst, Ira W. Marcelli, Giovanna |
| contents | We consider certain non-integer base $β$-expansions of Parry's type and we study various properties of the transfer (Perron-Frobenius) operator $\mathcal{P}:L^p([0,1])\mapsto L^p([0,1])$ with $p\geq 1$ and its associated composition (Koopman) operator, which are induced by a discrete dynamical system on the unit interval related to these $β$-expansions. We show that if $f$ is Lipschitz, then the iterated sequence $\{\mathcal{P}^N f\}_{N\geq 1}$ converges exponentially fast (in the $L^1$ norm) to an invariant state corresponding to the eigenvalue $1$ of $\mathcal{P}$. This "attracting" eigenvalue is not isolated: for $1\leq p\leq 2$ we show that the point spectrum of $\mathcal{P}$ also contains the whole open complex unit disk and we explicitly construct some corresponding eigenfunctions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral and dynamical results related to certain non-integer base expansions on the unit interval Cornean, Horia D. Herbst, Ira W. Marcelli, Giovanna Spectral Theory Mathematical Physics Dynamical Systems 37A30, 37A50 We consider certain non-integer base $β$-expansions of Parry's type and we study various properties of the transfer (Perron-Frobenius) operator $\mathcal{P}:L^p([0,1])\mapsto L^p([0,1])$ with $p\geq 1$ and its associated composition (Koopman) operator, which are induced by a discrete dynamical system on the unit interval related to these $β$-expansions. We show that if $f$ is Lipschitz, then the iterated sequence $\{\mathcal{P}^N f\}_{N\geq 1}$ converges exponentially fast (in the $L^1$ norm) to an invariant state corresponding to the eigenvalue $1$ of $\mathcal{P}$. This "attracting" eigenvalue is not isolated: for $1\leq p\leq 2$ we show that the point spectrum of $\mathcal{P}$ also contains the whole open complex unit disk and we explicitly construct some corresponding eigenfunctions. |
| title | Spectral and dynamical results related to certain non-integer base expansions on the unit interval |
| topic | Spectral Theory Mathematical Physics Dynamical Systems 37A30, 37A50 |
| url | https://arxiv.org/abs/2502.06511 |