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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 ISSN 0024-6093 .

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Official URL: http://dx.doi.org/10.1112/blms/bdw011

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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, Engineering and Medicine > 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:
DateEvent
1 March 2016Available
26 January 2016Accepted
26 September 2013Submitted
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
Date of first compliant deposit: 29 January 2016
Date of first compliant Open Access: 29 January 2016
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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