Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps (Q466195)

A nilmanifold \(\mathcal{H}\) is a manifold constructed as the quotient space of a 3-dimensional real Heisenberg group \(\mathbb{H}.\) An automorphism \(f: \mathcal{H}\rightarrow\mathcal{H}\) is called partially hyperbolic if the tangent bundle \(T\mathcal{H}\) admits a \(Df\)-invariant splitting \(T\mathcal{H}=E^{s}\oplus E^{c}\oplus E^{u},\) and there exist an integer \(k>0\) and a constant \(0<\mu<1\) such that, for any \(p\in\mathcal{H}\) and any unit vectors \(v^{s}\in E^{s}(p),v^{c}\in E^{c}(p),v^{u}\in E^{u}(p),\) we have NEWLINE\[NEWLINE\|Df^{k}(v^{s})\|<\mu<\|Df^{k}(v^{c})\|<\mu^{-1}<\|Df^{k}(v^{u})\|.NEWLINE\]NEWLINENEWLINENEWLINEIn the paper under review, the author shows that partially hyperbolic automorphisms on \(\mathcal{H}\) are not robustly transitive and gives a proof by introducing the Birkhoff section. The author provides examples of dynamical systems that are stably ergodic, but not robustly transitive, and gives a negative answer to the problem of whether minimality of stable and unstable foliations and accessibility imply robust transitivity for partially hyperbolic diffeomorphisms.NEWLINENEWLINEA corollary is given about the holonomy maps of the stable and unstable foliations of the diffeomorphisms \(\{f_{n}\}.\) This holonomy maps are twisted quasiperiodically forced circle homeomorphisms, which are transitive but non-minimal and satisfy certain fiberwise regularity properties.