Ulam meets Turing: constructing quadratic maps with non-computable SRB measures
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917881939230720 |
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| author | Rojas, Cristóbal Yampolsky, Michael |
| author_facet | Rojas, Cristóbal Yampolsky, Michael |
| contents | In 1946, S. Ulam invented Monte Carlo method, which has since become the standard numerical technique for making statistical predictions for long-term behaviour of dynamical systems. We show that this, or in fact any other numerical approach can fail for the simplest non-linear discrete dynamical systems given by the logistic maps $f_{a}(x)=ax(1-x)$ of the unit interval. We show that there exist computable real parameters $a\in (0,4)$ for which almost every orbit of $f_a$ has the same asymptotical statistical distribution in $[0,1]$, but this limiting distribution is not Turing computable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00006 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ulam meets Turing: constructing quadratic maps with non-computable SRB measures Rojas, Cristóbal Yampolsky, Michael Dynamical Systems 68Q17 and 37E05 In 1946, S. Ulam invented Monte Carlo method, which has since become the standard numerical technique for making statistical predictions for long-term behaviour of dynamical systems. We show that this, or in fact any other numerical approach can fail for the simplest non-linear discrete dynamical systems given by the logistic maps $f_{a}(x)=ax(1-x)$ of the unit interval. We show that there exist computable real parameters $a\in (0,4)$ for which almost every orbit of $f_a$ has the same asymptotical statistical distribution in $[0,1]$, but this limiting distribution is not Turing computable. |
| title | Ulam meets Turing: constructing quadratic maps with non-computable SRB measures |
| topic | Dynamical Systems 68Q17 and 37E05 |
| url | https://arxiv.org/abs/2501.00006 |