Ergodic Theory and Dynamical Systems

Research Article

Transitivity of Heisenberg group extensions of hyperbolic systems

IAN MELBOURNEa1, VIOREL NIŢICĂa2a3 and ANDREI TÖRÖKa3a4

a1 Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: ism@math.uh.edu)

a2 Department of Mathematics, West Chester University, West Chester, PA 19383, USA (email: vnitica@wcupa.edu)

a3 Institute of Mathematics of the Romanian Academy, PO Box 1–764, RO-70700 Bucharest, Romania

a4 Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA (email: torok@math.uh.edu)

Abstract

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 topologically transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with a compact commutator subgroup (for example ℋn/ℤ), among those that avoid the obvious obstruction, topological transitivity is open and dense.

(Received May 14 2010)

(Revised November 16 2010)

(Online publication April 05 2011)