Linked solenoid mappings and the non-transversality locus invariant (Q2758111)

A homeomorphism \(f:S^{3}\to S^{3}\) is called a linked solenoid mapping of degree \(m\) if there exists an imbedded torus \(T\subset S^{3}\) cutting \(S^{3}\) into two linked unknotted solid tori \(T^{+},\) \(T^{-},\) such that both \(f:T^{+}\to T^{+}\) and \(f^{-1}:T^{-}\to T^{-}\) are solenoidal of degree \(m.\) Such maps arise in the study of Hénon mappings in \(\mathbb{C}^{2}.\) These maps are structurally stable, and thus linked solenoidal maps are structurally stable on their nonwandering sets, but they are not structurally stable.NEWLINENEWLINEThe authors define the normalized angle mappings \(\Phi^{\pm}=(m-1)\pi^{\pm},\) where \(\pi^{\pm}:T^{\pm}\to T^{\pm}\) are continuous mappings making \(T^{+}\) and \(T^{-}\) into bundles of disks over the circle such that certain diagrams commute. Then \(X_{k}\subset U_{k}=T_{k}^{+}\cap T_{k+1}^{-}\) is defined to be the locus where the foliations \(\Phi_{k}^{+}\) and \(\Phi _{k+1}^{-}\) are not transverse. Finally, a conjugacy invariant \(\text{ntl}(f)\subset( \mathbb{R}/2\pi\mathbb{Z}) ^{2}\) is defined as the image of \(X_{k}\) by the mapping \(( \Phi_{k}^{+} ,\Phi_{k+1}^{-}) .\)NEWLINENEWLINEA conjugacy invariant \(\text{ntl}(f)\) for some particular linked solenoid maps is computed, and it is shown that it can take on an infinite-dimensional set of values. Furthermore, it is proved that in an open set of solenoid mappings a conjugacy invariant \(\text{ntl}(f)\) is a complete invariant and classifies these mappings up to topological conjugacy. The paper concludes with a list of open problems.