Transitivity of Euclidean-type extensions of hyperbolic systems

摘要

Let f : X -> X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class of C-r(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 proportional to R-n where G is a compact connected Lie group acting linearly on R-n. When Fix G = {0}, it is again the case that the transitive cocycles are residual. When Fix G not equal {0}, the same result holds upon restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.