We show that given n ≥ 3, q ≥ 1, and a finite set {y 1 , . . . , yq} in R n there exists a quasiregular mapping R n → R n omitting exactly points y 1 , . . . , yq.
Abstract. We consider decomposition spaces R 3 /G that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on R 3 /G constructed via modular embeddings of R 3 /G into a Euclidean space promote the controlled topology to a controlled geometry.The quasisymmetric parametrizability of the metric space R 3 /G×R m by R 3+m for any m ≥ 0 imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, on the defining sequences for R 3 /G. We give a necessary condition and a sufficient condition for the existence of such a parametrization.The necessary condition answers negatively a question of Heinonen and Semmes on quasisymmetric parametrizability of spaces associated to the Bing double. The sufficient condition gives new examples of quasispheres in S 4 .
Let f : M → M be a uniformly quasiregular self-map of a compact, connected, and oriented Riemannian n-manifold M without boundary, n 2. We show that, for k ∈ {0, . . . , n}, the induced homomorphism f * :is the kth singular cohomology of M , is complex diagonalizable and the eigenvalues of f * have absolute value (deg f ) k/n . As an application, we obtain a degree restriction for uniformly quasiregular self-maps of closed manifolds. In the proof of the main theorem, we use a Sobolev-de Rham cohomology based on conformally invariant differential forms and an induced push-forward operator.
We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism f of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure µ f , which is balanced and invariant under f and non-atomic, and whose support agrees with the Julia set of f . Furthermore, we show that f is strongly mixing with respect to the measure µ f . We also characterize the measure µ f using an approximation property by iterated pullbacks of points under f up to a set of exceptional initial points of Hausdorff dimension at most n − 1. These dynamical mixing and approximation results are reminiscent of the Mattila-Rickman equidistribution theorem for quasiregular mappings. Our methods are based on the existence of an invariant measurable conformal structure due to Iwaniec and Martin and the A-harmonic potential theory.where Df is the point-wise operator norm of the differential Df of the mapping f and J f is the Jacobian determinant of the differential. A quasiregular endomorphism f : M → M is said to be uniformly quasiregular if there exists K 1 so that all the iterates of f are K-quasiregular.Results of Peltonen [21] and Astola, Kangaslampi and Peltonen [1] show that the existence of branching uniformly quasiregular dynamics is not limited to S n . However, results of Martin, Meyer and Peltonen [16] show that, among space forms, only spherical space forms admit branching uniformly quasiregular endomorphisms. Furthermore, by results of Bridson, Hinkkanen and Martin [4, Corollary 5.3], all quasiregular endomorphisms of closed n-manifolds having a torsion-free, non-elementary, and word-hyperbolic fundamental group, are in fact quasiconformal. We refer the reader to [4,16] for the terminology and related results. Since the
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