Transitivity of Euclidean-type extensions of hyperbolic systems

Ergodic Theory Dyn. Syst. 29 (2009) 1585-1602.

Ian Melbourne, Viorel Niţică and Andrei Török

Abstract

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 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>2 odd.

More generally, we consider Euclidean-type groups G \ltimes 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. Otherwise, the same result holds on restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.