On an extension of a theorem by Ruelle to long-range potentials
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910405982420992 |
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| author | van Enter, Aernout C. D. Fernández, Roberto Makhmudov, Mirmukhsin Verbitskiy, Evgeny |
| author_facet | van Enter, Aernout C. D. Fernández, Roberto Makhmudov, Mirmukhsin Verbitskiy, Evgeny |
| contents | Ruelle's transfer operator plays an important role in understanding thermodynamic and probabilistic properties of dynamical systems. In this work, we develop a method of finding eigenfunctions of transfer operators based on comparing Gibbs measures on the half-line $\mathbb Z_+$ and the whole line $\Z$. For a rather broad class of potentials, including both the ferromagnetic and antiferromagnetic long-range Dyson potentials, we are able to establish the existence of integrable, but not necessarily continuous, eigenfunctions. For a subset thereof we prove that the eigenfunction is actually continuous. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07326 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On an extension of a theorem by Ruelle to long-range potentials van Enter, Aernout C. D. Fernández, Roberto Makhmudov, Mirmukhsin Verbitskiy, Evgeny Dynamical Systems Probability 37D35, 37A60, 37C30, 82B26, 82B20 Ruelle's transfer operator plays an important role in understanding thermodynamic and probabilistic properties of dynamical systems. In this work, we develop a method of finding eigenfunctions of transfer operators based on comparing Gibbs measures on the half-line $\mathbb Z_+$ and the whole line $\Z$. For a rather broad class of potentials, including both the ferromagnetic and antiferromagnetic long-range Dyson potentials, we are able to establish the existence of integrable, but not necessarily continuous, eigenfunctions. For a subset thereof we prove that the eigenfunction is actually continuous. |
| title | On an extension of a theorem by Ruelle to long-range potentials |
| topic | Dynamical Systems Probability 37D35, 37A60, 37C30, 82B26, 82B20 |
| url | https://arxiv.org/abs/2404.07326 |