Transitivity of Heisenberg group extensions of hyperbolic systems (Q2884084)

In this paper, the authors continue their study on the transitivity of various non-compact Lie group extensions \((f,\beta):X\times \Gamma\to X\times \Gamma\) over a hyperbolic basic set \(f:X\to X\); see [the authors, Ann. Henri Poincaré 6, No. 4, 725--746 (2005; Zbl 1079.22005); Discrete Contin. Dyn. Syst. 14, No. 2, 355--363 (2006; Zbl 1098.37027); Ergodic Theory Dyn. Syst. 29, No. 5, 1585--1602 (2009; Zbl 1181.37030)].NEWLINENEWLINEThere is a trivial obstruction for a cocycle to be transitive. Namely, if a cocycle \((f,\beta)\) is cohomologous to one taking values in some half-space, then the cocycle is certainly not transitive. The authors show that this may be the only obstruction of transitivity among some nilpotent Lie group cocycles.NEWLINENEWLINEMore precisely, let \((X,f)\) be a hyperbolic basic set, \(\mathcal{H}_n\) be the \((2n+1)\)-dimensional Heisenberg group. Then for a \(C^r\)-generic cocycle \(\beta:X\to \mathcal{H}_n\), if \((f,\beta)\) is not cohomologous to one taking values in some half-space, then the cocycle is transitive.NEWLINENEWLINEThe authors obtain a stronger result if the nilpotent Lie group \(\Gamma\) has a compact commutator subgroup. That is, assuming \([\Gamma,\Gamma]\) is compact, then for an open and dense subset of the cocycles \(\beta:X\to \Gamma\), if \((f,\beta)\) avoids the above obstruction, then \((f,\beta)\) is transitive.