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Linear response for random dynamical systems
We study for the first time linear response for random compositions of maps, chosen independently according to a distribution P. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when P changes smoothly to Pε? For a wid...
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| Format: | Default Article |
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2020
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| Online Access: | https://hdl.handle.net/2134/11710419.v1 |
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| _version_ | 1818167784352251904 |
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| author | Wael Bahsoun Marks Ruziboev Benoit Saussol |
| author_facet | Wael Bahsoun Marks Ruziboev Benoit Saussol |
| author_sort | Wael Bahsoun (1258407) |
| collection | Figshare |
| description | We study for the first time linear response for random compositions of maps, chosen independently according to a distribution P. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when P changes smoothly to Pε? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to ε; moreover, we obtain a linear response formula. Our results cover random maps whose transfer operator does not necessarily admit a spectral gap. We apply our results to iid compositions, with respect to various distributions Pε, of uniformly expanding circle maps, Gauss-R´enyi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations. |
| format | Default Article |
| id | rr-article-11710419 |
| institution | Loughborough University |
| publishDate | 2020 |
| record_format | Figshare |
| spelling | rr-article-117104192020-02-05T00:00:00Z Linear response for random dynamical systems Wael Bahsoun (1258407) Marks Ruziboev (8356209) Benoit Saussol (7162388) General Mathematics Pure Mathematics We study for the first time linear response for random compositions of maps, chosen independently according to a distribution P. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when P changes smoothly to Pε? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to ε; moreover, we obtain a linear response formula. Our results cover random maps whose transfer operator does not necessarily admit a spectral gap. We apply our results to iid compositions, with respect to various distributions Pε, of uniformly expanding circle maps, Gauss-R´enyi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations. 2020-02-05T00:00:00Z Text Journal contribution 2134/11710419.v1 https://figshare.com/articles/journal_contribution/Linear_response_for_random_dynamical_systems/11710419 CC BY-NC-ND 4.0 |
| spellingShingle | General Mathematics Pure Mathematics Wael Bahsoun Marks Ruziboev Benoit Saussol Linear response for random dynamical systems |
| title | Linear response for random dynamical systems |
| title_full | Linear response for random dynamical systems |
| title_fullStr | Linear response for random dynamical systems |
| title_full_unstemmed | Linear response for random dynamical systems |
| title_short | Linear response for random dynamical systems |
| title_sort | linear response for random dynamical systems |
| topic | General Mathematics Pure Mathematics |
| url | https://hdl.handle.net/2134/11710419.v1 |