Transitivity of Euclidean-type extensions of hyperbolic systems

Abstract

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 ≥ 3 odd.More generally, we consider Euclidean-type groups G 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 = {0}, the same result holds on restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.

Transitivity of Heisenberg group extensions of hyperbolic systems

“…The proof of Theorem 1.4 develops further the techniques in [8]. In addition, results from the classical theory of Diophantine approximation come into play.…”

“…In order to prove Theorem 1.4, it is enough to show that S r (X, H n ) contains a dense set of cocycles that are transitive (see for example the introduction in [8]). …”

“…As was the case in [8] for certain Euclidean-type groups (e.g. SE(3)), replacing generic by open and dense in Theorem 1.4 remains an unsolved problem.…”

“…In Section 4 we prove Theorem 1.5. In Section 5 we recall a technical result from [8]. In Section 6 we specialize to the setting of nilpotent Lie groups and prove Theorem 1.4.…”

“…In [8] we showed that for SE(n), n ≥ 3 odd, the transitive C r cocycles form a residual subset of the space of all C r cocycles for all r > 0. In other words, transitivity is generic for such extensions.…”