Invariant Probability Measures under $p$-adic Transformations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913868353110016 |
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| author | Maslyuchenko, Oleksandr V. Morawiec, Janusz Zürcher, Thomas |
| author_facet | Maslyuchenko, Oleksandr V. Morawiec, Janusz Zürcher, Thomas |
| contents | It is well-known that the Lebesgue measure is the unique absolutely continuous invariant probability measure under the $p$-adic transformation. The purpose of this paper is to characterize the family of all invariant probability measures under the $p$-adic transformation and to provide some description of them. In particular, we describe the subfamily of all atomic invariant measures under the $p$-adic transformation as well as the subfamily of all continuous and singular invariant probability measures under the $p$-adic transformation. Iterative functional equations play the base role in our considerations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_06406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant Probability Measures under $p$-adic Transformations Maslyuchenko, Oleksandr V. Morawiec, Janusz Zürcher, Thomas Classical Analysis and ODEs Primary 37E05, 39B12, Secondary 26A30, 28D05 It is well-known that the Lebesgue measure is the unique absolutely continuous invariant probability measure under the $p$-adic transformation. The purpose of this paper is to characterize the family of all invariant probability measures under the $p$-adic transformation and to provide some description of them. In particular, we describe the subfamily of all atomic invariant measures under the $p$-adic transformation as well as the subfamily of all continuous and singular invariant probability measures under the $p$-adic transformation. Iterative functional equations play the base role in our considerations. |
| title | Invariant Probability Measures under $p$-adic Transformations |
| topic | Classical Analysis and ODEs Primary 37E05, 39B12, Secondary 26A30, 28D05 |
| url | https://arxiv.org/abs/2412.06406 |