Ergodic Theory and Dynamical Systems

Research Article

Transitivity of Euclidean-type extensions of hyperbolic systems

I. MELBOURNEa1, V. NIŢICĂa2a3 and A. TÖRÖKa3a4

a1 Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: ism@math.uh.edu)

a2 Department of Mathematics, West Chester University, 323 Anderson Hall, West Chester, PA 19383, USA (email: vnitica@wcupa.edu)

a3 Institute of Mathematics of the Romanian Academy, PO Box 1–764, RO-70700 Bucharest, Romania

a4 Department of Mathematics, University of Houston, 651 PGH, Houston, TX 77204-3008, USA (email: torok@math.uh.edu)

Abstract

Let f:XX 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 Gxs22C9xs211Dn where G is a compact connected Lie group acting linearly on xs211Dn. 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.

(Received October 25 2007)

(Revised October 06 2008)