Transitivity of Heisenberg group extensions of hyperbolic systems

We show that among Cr extensions (r > 0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group Hn 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, Hn/Z), among those that avoid the obvious obstruction, topological transitivity is open and dense. ∗Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK. [email protected] †Department of Mathematics, West Chester University, West Chester, PA 19383, USA, and Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO-70700 Bucharest, Romania. [email protected] ‡Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA, and Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO-70700 Bucharest, Romania. [email protected]