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Finite-dimensional negatively invariant subsets of banach spaces
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Carvalho, Alexandre N., Cunha, Arthur C., Langa, José A. and Robinson, James C. (2022) Finite-dimensional negatively invariant subsets of banach spaces. Journal of Mathematical Analysis and Applications, 509 (2). 125945. doi:10.1016/j.jmaa.2021.125945 ISSN 0022-247X.
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Official URL: https://doi.org/10.1016/j.jmaa.2021.125945
Abstract
We give a simple proof of a result due to Mañé (1981) [17] that a compact subset of a Banach space that is negatively invariant for a map S is finite-dimensional if , where C is compact and L is a contraction (and both are linear). In particular, we show that if S is compact and differentiable then is finite-dimensional. We also prove some results (following Málek et al. (1994) [15] and Zelik (2000) [23]) that give bounds on the (box-counting) dimension of such sets assuming a ‘smoothing property’: in its simplest form this requires S to be Lipschitz from X into another Banach space Z that is compactly embedded in X. The resulting bounds depend on the Kolmogorov ε-entropy of the embedding of Z into X. We give applications to an abstract semilinear parabolic equation and the two-dimensional Navier–Stokes equations on a periodic domain.
Item Type: | Journal Article | ||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Journal or Publication Title: | Journal of Mathematical Analysis and Applications | ||||||||
Publisher: | Academic Press | ||||||||
ISSN: | 0022-247X | ||||||||
Official Date: | 15 May 2022 | ||||||||
Dates: |
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Volume: | 509 | ||||||||
Number: | 2 | ||||||||
Article Number: | 125945 | ||||||||
DOI: | 10.1016/j.jmaa.2021.125945 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 16 December 2021 | ||||||||
Date of first compliant Open Access: | 28 December 2022 | ||||||||
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