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Transitivity of Euclidean-type extensions of hyperbolic systems

Published online by Cambridge University Press:  03 February 2009

I. MELBOURNE ,
V. NIŢICĂ  and
A. TÖRÖK
I. MELBOURNE
Affiliation:
Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: ism@math.uh.edu)
V. NIŢICĂ
Affiliation:
Department of Mathematics, West Chester University, 323 Anderson Hall, West Chester, PA 19383, USA (email: vnitica@wcupa.edu) Institute of Mathematics of the Romanian Academy, PO Box 1–764, RO-70700 Bucharest, Romania
A. TÖRÖK
Affiliation:
Institute of Mathematics of the Romanian Academy, PO Box 1–764, RO-70700 Bucharest, Romania Department of Mathematics, University of Houston, 651 PGH, Houston, TX 77204-3008, USA (email: torok@math.uh.edu)
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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 G⋉ℝn where G is a compact connected Lie group acting linearly on ℝn. 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1585 - 1602
Copyright
Copyright © Cambridge University Press 2009

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