A uniformly quasiregular mapping acting on a compact Riemannian manifold distorts the metric by a bounded amount, independently of the number of iterates. 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. We study a rich subclass of uniformly quasiregular mappings that can be produced using an analogy of classical Lattès' construction of chaotic rational functions acting on the extended plane . We show that there is a plenitude of compact manifolds that support these mappings. Moreover, we find that in some cases there are alternative ways to construct this type of mapping with different Julia sets.
title = "Latt{\`e}s-type mappings on compact manifolds",
abstract = "A uniformly quasiregular mapping acting on a compact Riemannian manifold distorts the metric by a bounded amount, independently of the number of iterates. 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. We study a rich subclass of uniformly quasiregular mappings that can be produced using an analogy of classical Latt{\`e}s' construction of chaotic rational functions acting on the extended plane . We show that there is a plenitude of compact manifolds that support these mappings. Moreover, we find that in some cases there are alternative ways to construct this type of mapping with different Julia sets.",
author = "L.J. Astola and R. Kangaslampi and K. Peltonen",
Research output: Contribution to journal › Article › Academic › peer-review
TY - JOUR
T1 - Lattès-type mappings on compact manifolds
AU - Astola, L.J.
AU - Kangaslampi, R.
AU - Peltonen, K.
PY - 2010
Y1 - 2010
N2 - A uniformly quasiregular mapping acting on a compact Riemannian manifold distorts the metric by a bounded amount, independently of the number of iterates. 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. We study a rich subclass of uniformly quasiregular mappings that can be produced using an analogy of classical Lattès' construction of chaotic rational functions acting on the extended plane . We show that there is a plenitude of compact manifolds that support these mappings. Moreover, we find that in some cases there are alternative ways to construct this type of mapping with different Julia sets.
AB - A uniformly quasiregular mapping acting on a compact Riemannian manifold distorts the metric by a bounded amount, independently of the number of iterates. 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. We study a rich subclass of uniformly quasiregular mappings that can be produced using an analogy of classical Lattès' construction of chaotic rational functions acting on the extended plane . We show that there is a plenitude of compact manifolds that support these mappings. Moreover, we find that in some cases there are alternative ways to construct this type of mapping with different Julia sets.