Uniformly quasiregular maps with toroidal Julia sets (Q2888651)

The authors construct examples of uniformly quasiregular maps on three-manifolds whose Julia set is a torus. A self map \(f\) of a Riemannian manifold is called uniformly quasiregular if it is \(W^{1,n}_{\text{loc}}\)-Sobolev and the differential of its iterates \(Df^k\) satisfies the bound \(| Df^k |< K \text{det}(D f^k)\) almost everywhere, for a constant \(K\) independent of the number of iterates \(k\geq 1\). Such maps are rational with respect to some measurable conformal structure and there is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. Here the authors construct a uniformly quasiregular map on \(S^3\setminus H\), the complement of the Hopf link in the three sphere, without branching and its Julia set is a two-torus. There is a metric of Semmes type on \(S^3\setminus H\) such that the covering map from an Euclidean space is conformal, and the self map of \(S^3\setminus H\) is induced by a homothety of the Euclidean space via this covering; hence the map is of Lattès type.NEWLINENEWLINENEWLINEThe construction of the map is explicit. In a second step, the authors show that their construction descends to the quotients of \(S^3\setminus H\) by the action of the cyclic group on \(S^3\) preserving \(H\), that leads to lens spaces.