Computability of Brjuno-like functions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909989191286784 |
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| author | Shevchenko, Ivan O. Yampolsky, Michael |
| author_facet | Shevchenko, Ivan O. Yampolsky, Michael |
| contents | In his seminal paper from 1936, Alan Turing introduced the concept of non-computable real numbers and presented examples based on the algorithmically unsolvable Halting problem. We describe a different, analytically natural mechanism for the appearance of non-computability. Namely, we show that additive sampling of orbits of certain skew products over expanding dynamics produces Turing non-computable reals. We apply this framework to Brjuno-type functions to demonstrate that they realize bijections between computable and lower-computable numbers, generalizing previous results of M. Braverman and the second author for the Yoccoz-Brjuno function to a wide class of examples, including Wilton's functions and generalized Brjuno functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04195 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computability of Brjuno-like functions Shevchenko, Ivan O. Yampolsky, Michael Dynamical Systems Logic In his seminal paper from 1936, Alan Turing introduced the concept of non-computable real numbers and presented examples based on the algorithmically unsolvable Halting problem. We describe a different, analytically natural mechanism for the appearance of non-computability. Namely, we show that additive sampling of orbits of certain skew products over expanding dynamics produces Turing non-computable reals. We apply this framework to Brjuno-type functions to demonstrate that they realize bijections between computable and lower-computable numbers, generalizing previous results of M. Braverman and the second author for the Yoccoz-Brjuno function to a wide class of examples, including Wilton's functions and generalized Brjuno functions. |
| title | Computability of Brjuno-like functions |
| topic | Dynamical Systems Logic |
| url | https://arxiv.org/abs/2501.04195 |