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Gaven J. Martin Volker Mayer 《Transactions of the American Mathematical Society》2003,355(11):4349-4363
We consider rigidity phenomena for holomorphic functions and then more generally for uniformly quasiregular maps.
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Gaven J. Martin 《Mathematische Annalen》2002,324(2):329-340
A uniformly quasiregular mapping, is a mapping of the m-sphere with the property that it and all its iterates have uniformly bounded distortion. Such maps are rational with respect to
some bounded measurable conformal structure and there is a Fatou-Julia type theory associated with the dynamical system obtained
by iterating these mappings. We begin by investigating the analogue of Siegel's theorem on the local conjuga cy of rotational
dynamics. We are led to consider the analytic continuation properties of solutions to the highly nonlinear first order Beltrami
systems. We reduce these problems to a central and well known conjecture in the theory of transformation groups; namely the
Hilbert-Smith conjecture, which roughly asserts that effective transformation groups of manifolds are Lie groups. Our affirmative
solution to this problem then implies unique analytic continuation and Siegel's theorem.
Received: 14 September 2000 / Revised version: 23 November 2001 / Published online: 5 September 2002
RID="*"
ID="*" Research supported in part by grants from the Marsden Fund and Royal Society (NZ). 相似文献
3.
Quasiregular mappings in even dimensions 总被引:39,自引:0,他引:39
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Tomáš Gavenčiak 《Discrete Mathematics》2010,310(10-11):1557-1563
5.
Petra Bonfert-Taylor Gaven Martin 《Proceedings of the American Mathematical Society》2001,129(7):2019-2029
We study Fuchsian quasiconformal groups with small dilatation. For this class of groups we establish a Jørgensen-type inequality in all dimensions. We show discreteness persists to the limit under algebraic convergence and that such groups are discrete if and only if every two generator subgroup is discrete.
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Petra?Bonfert-Taylor Richard D.?CanaryEmail author Gaven?Martin Edward?Taylor 《Mathematische Annalen》2005,331(2):281-295
We exhibit strong constraints on the geometry and topology of a uniformly quasiconformally homogeneous hyperbolic manifold. In particular, if n3, a hyperbolic n-manifold is uniformly quasiconformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold. Moreover, if n3, we show that there is a constant Kn>1 such that if M is a hyperbolic n-manifold, other than which is K–quasiconformally homogeneous, then KKn.Mathematics Subject Classification (2000): 30C60Research supported in part by NSF grant 070335 and 0305704.Research supported in part by NSF grant 0203698.Research supported in part by the NZ Marsden Fund and the Royal Society (NZ).Research supported in part by NSF grant 0305704. 相似文献
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Gaven Martin Volker Mayer Kirsi Peltonen 《Proceedings of the American Mathematical Society》2006,134(7):2091-2097
A uqr mapping of an -manifold is a mapping which is rational with respect to a bounded measurable conformal structure on . Remarkably, the only closed manifolds on which locally (but not globally) injective uqr mappings act are Euclidean space forms. We further characterize space forms admitting uniformly quasiregular self mappings and we show that the space forms admitting branched uqr maps are precisely the spherical space forms. We further show that every non-injective uqr map of a Euclidean space form is a quasiconformal conjugate of a conformal map. This is not true if the non-injective hypothesis is removed.
10.
For a self mapping f: D→D of the unit disk in C which has finite distortion, we give a separation condition on the components of the set where the distortion is very large - say greater than a given constant - which implies that f still extends homeomorphically and quasisymmetrically to the boundary S = ?D. Thus f shares its boundary values with a quasiconformal mapping whose distortion we explicitly estimate in terms of the data. This condition, uniformly separated in modulus, allows the set where the distortion is large to accumulate on the entire boundary S, but it does not allow a component to run out to the boundary - a necessary restriction. The lift of a Jordan domain in a Riemann surface to its universal cover D is always uniformly separated in modulus, and this allows us to apply these results in the theory of Riemann surfaces to identify an interesting link between the support of the high distortion of a map between surfaces and their geometry - again with explicit estimates. As part of our investigations, we study mappings ?: S → S which are the germs of a conformal mapping and give good bounds on the distortion of a quasiconformal extension of ? to the disk D. We then extend these results to the germs of quasisymmetric mappings. These appear of independent interest and identify new geometric invariants. 相似文献