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Transitivity of Euclidean-type extensions of hyperbolic systems

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Melbourne, Ian, Nitica, V. and Torok, Andrei (2009) Transitivity of Euclidean-type extensions of hyperbolic systems. Ergodic Theory and Dynamical Systems, Vol.29 (No.5). pp. 1585-1602. doi:10.1017/S0143385708000886

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Official URL: http://dx.doi.org/10.1017/S0143385708000886

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Abstract

Let f:X→X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class of Cr(r>0) cocycles with fiber the special Euclidean group SE(n), those that are transitive form a residual set (countable intersection of open dense sets). This result is new for odd values of n≥3. More generally, we consider Euclidean-type groups Gn where G is a compact connected Lie group acting linearly on n. When Fix G={0}, it is again the case that the transitive cocycles are residual. When Fix G≠{0}, the same result holds upon restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.

Item Type: Journal Article
Divisions: Faculty of Science > Mathematics
Journal or Publication Title: Ergodic Theory and Dynamical Systems
Publisher: Cambridge University Press
ISSN: 0143-3857
Official Date: 2009
Dates:
DateEvent
2009Published
Volume: Vol.29
Number: No.5
Page Range: pp. 1585-1602
DOI: 10.1017/S0143385708000886
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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