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Asymmetric unimodal maps with non-universal period-doubling scaling laws
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Kozlovski, Oleg and van Strien, Sebastian (2020) Asymmetric unimodal maps with non-universal period-doubling scaling laws. Communications in Mathematical Physics, 379 . 103-143 . doi:10.1007/s00220-020-03835-9 ISSN 0010-3616.
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Official URL: https://doi.org/10.1007/s00220-020-03835-9
Abstract
We consider a family of strongly-asymmetric unimodal maps \{f_t\}_{t\in [0,1]} of the form f_t=t\cdot f where f:[0,1]\rightarrow [0,1] is unimodal, f(0)=f(1)=0, f(c)=1 is of the form and
\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}
where we assume that \beta >1. We show that such a family contains a Feigenbaum–Coullet–Tresser 2^\infty map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the 2^\infty map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Probability measures , Cantor sets , Invariant sets | ||||||||
| Journal or Publication Title: | Communications in Mathematical Physics | ||||||||
| Publisher: | Springer | ||||||||
| ISSN: | 0010-3616 | ||||||||
| Official Date: | October 2020 | ||||||||
| Dates: |
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| Volume: | 379 | ||||||||
| Page Range: | 103-143 | ||||||||
| DOI: | 10.1007/s00220-020-03835-9 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||
| Date of first compliant deposit: | 26 May 2020 | ||||||||
| Date of first compliant Open Access: | 27 August 2020 | ||||||||
| RIOXX Funder/Project Grant: |
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