Transitivity of Heisenberg group extensions of hyperbolic systems

Ergodic Theory Dyn. Syst. 32 (2012) 223-235

Ian Melbourne, Viorel Niţică and Andrei Török

Abstract

We show that among C^r extensions (r>0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group H_n of dimension 2n+1, those that avoid an obvious obstruction to topological transitivity are generically topological transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with compact commutator subgroup (for example, H_n/Z), among those that avoid the obvious obstruction, topological transitivity is open and dense.