Transitivity of Euclidean-type extensions of hyperbolic systems

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 special Euclidean group SE(n) those that are transitive form a residual set (countable intersection of open dense sets). This result is new for n ≥ 3 odd. More generally, we consider Euclidean-type groups GnRn where G is a compact connected Lie group acting linearly on Rn. When FixG = {0}, it is again the case that the transitive cocycles are residual. When FixG 6 = {0}, the same result holds on restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity. 1