Examples of Uniformly Quasiregular Mappings

In this paper we construct examples of uniformly quasiregular (uqr) mappings. These provide counterexamples for a rigidity conjecture in quasiregular dynamics. It states that a closed manifold of dimension at least three, which admits a branched uqr mapping, is quasiconformally equivalent to an ordinary sphere Sn. 1. Statements and results In this paper we construct examples of a certain type of quasiregular mappings. Topologically, quasiregular mappings are branched coverings, that is, covering mappings without the local injectivity requirement. This is a consequence of the completely analytic definition demanding that the mapping distorts the metric only by a bounded amount. More precisely: Definition 1. A continuous mapping f : G → R of a domain G ⊂ R is quasiregular, if it belongs to W 1 n,loc(G) and if there exists a number K, 1 ≤ K <∞, such that |f ′(x)|n ≤ KJf(x) holds almost everywhere in G. In the above definition W 1 n,loc(G) is the Sobolev space of mappings with weak first order partial derivatives which are locally L integrable. For the above mappings the partial derivatives in the ordinary sense exist almost everywhere. We can thus define the formal derivative of f in terms of partial derivatives. The Jacobian determinant det f ′(x) of f at x is denoted by Jf (x). By the norm of the linear mapping f ′(x) we mean here the operator norm |f ′(x)| = sup |h|=1 |f ′(x)h|. The above definition generalizes immediately to Riemannian manifolds. The branch set Bf is the set of points for which f is not locally homeomorphic. For the basic properties and theory of quasiregular mappings see [R]. Received by the editors July 26, 1999 and, in revised form, October 27, 1999. 1991 Mathematics Subject Classification. Primary 30C65, 58F; Secondary 53C.