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On finite-dimensional attractors of homeomorphisms
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Robinson, James C. and Sánchez-Gabites, J. J. (2016) On finite-dimensional attractors of homeomorphisms. Bulletin of the London Mathematical Society, 48 (3). pp. 483-498. doi:10.1112/blms/bdw011
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Official URL: http://dx.doi.org/10.1112/blms/bdw011
Abstract
Let E be a linear space and suppose that A is the global attractor of either (i) a homeomorphism F:E→E or (ii) a semigroup S(⋅) on E that is injective on A. In both cases A has trivial shape, and the dynamics on A can be described by a homeomorphism F:A→A (in the second case we set F=S(t) for some t>0). If the topological dimension of A is finite we show that for any ϵ>0 there is an embedding e:A→Rk, with k∼dim(A), and a (dynamical) homeomorphism $f:\R^k\rightarrow\R^k$ such that F is conjugate to f on A (i.e.\ F|A=e−1∘f∘e) and f has an attractor Af with e(A)⊂Af⊂N(e(A),ϵ). In other words, we show that the dynamics on A is essentially finite-dimensional.
We characterise subsets of Rn that can be the attractors of homeomorphisms as cellular sets, give elementary proofs of various topological results connected to Borsuk's theory of shape and cellularity in Euclidean spaces, and prove a controlled homeomorphism extension theorem.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Homeomorphisms, Vector spaces | ||||||||
| Journal or Publication Title: | Bulletin of the London Mathematical Society | ||||||||
| Publisher: | Oxford University Press | ||||||||
| ISSN: | 0024-6093 | ||||||||
| Official Date: | 1 March 2016 | ||||||||
| Dates: |
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| Date of first compliant deposit: | 29 January 2016 | ||||||||
| Volume: | 48 | ||||||||
| Number: | 3 | ||||||||
| Page Range: | pp. 483-498 | ||||||||
| DOI: | 10.1112/blms/bdw011 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Funder: | Engineering and Physical Sciences Research Council (EPSRC), Spain. Ministerio de Ciencia e Innovación (MICINN) | ||||||||
| Grant number: | EP/G007470/1 (EPSRC), 2009-07030 (MICINN) |
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