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Uniformly quasiregular maps with toroidal Julia sets

Riikka Kangaslampi, Kirsi Peltonen, Jang-Mei Wu
2012 Conformal Geometry and Dynamics  
We construct the first examples of uniformly quasiregular mappings that have a 2-torus as the Julia set.  ...  The spaces supporting this type of mappings include the Hopf link complement and its lens space quotients.  ...  Acknowledgment The authors are grateful to Pekka Pankka for inspiration and valuable discussions related to this work.  ... 
doi:10.1090/s1088-4173-2012-00235-4 fatcat:6b6df7yt4zejjmgshtuwbduhnm

Uniformly quasiregular mappings of Lattès type

Volker Mayer
1997 Conformal Geometry and Dynamics  
Using an analogy of the Lattès' construction of chaotic rational functions, we show that there are uniformly quasiregular mappings of the nsphere R n whose Julia set is the whole sphere.  ...  Moreover there are analogues of power mappings, uniformly quasiregular mappings whose Julia set is S n−1 and its complement in S n consists of two superattracting basins.  ...  The dynamical behaviour of such a mapping splits the sphere into two parts: the Fatou set F (f ), which is the set of points x for which {f k } is a normal family in a neighborhood of x, and the Julia  ... 
doi:10.1090/s1088-4173-97-00013-1 fatcat:ipw74ickmzapdbksvz5cn7nmpa

Strongly automorphic mappings and Julia sets of uniformly quasiregular mappings [article]

Alastair Fletcher, Doug Macclure
2018 arXiv   pre-print
In this paper, we prove the analogous statement in the setting of strongly automorphic quasiregular mappings and uniformly quasiregular mappings in R^n.  ...  We further give a classification of the behaviour of uniformly quasiregular mappings on their Julia set when the Julia set is a quasisphere, quasidisk or all of R^n and the Julia set coincides with the  ...  Given f, x0 and ϕ, there are infinitely many possibilities for L. In the plane, a linearizer is normalized via L0 (0) = 1. Note that such a normalization is unavailable for quasiregular mappings.  ... 
arXiv:1703.02455v5 fatcat:v7ek5e2o5bdsjjddl7aihugg5e

Endomorphisms of mapping tori [article]

Christoforos Neofytidis
2020 arXiv   pre-print
This generalises and gives a new proof of the analogous classification for fibered 3-manifolds. Various applications are given.  ...  In particular, we deduce that rigidity results for Gromov hyperbolic groups hold for the above mapping tori with trivial center.  ...  Any quasiregular map is open and discrete, i.e., the preimage of any point consists of isolated points. In particular, the preimage of any point under a proper quasiregular map is a finite set.  ... 
arXiv:2010.12490v1 fatcat:xyode2jpjndevplg433txddtga

Locally injective automorphic mappings in $R^n$

O. Martio, U. Srebro
1999 Mathematica Scandinavica  
The authors wish to thank Bill Abikoff for pointing an error in an earlier version of the proof of 2.10.  ...  The research of the second author was partially supported by the Fund for the Promotion of Research at the Technion and by grants from the Academy of Finland.  ...  Let nz denote the unit normal vector to dT 1 at the point F z. Then for !  ... 
doi:10.7146/math.scand.a-13884 fatcat:iloshsdbbjgrjmurquf4surdxe

Isometry group of Sasaki-Einstein metric [article]

Weiyong He
2013 arXiv   pre-print
We prove that the identity component of the holomorphic isometry group of a Sasaki-Einstein metric is the identity component of a maximal compact subgroup of its automorphism group.  ...  Acknowledgement: We thank Sun Song for valuable discussions on Sasaki geometry. The author is partially supported by an NSF grant.  ...  Now for any ζ ∈ t ′ , we suppose ζ satisfies the normalized condition such that for any normalized Reeb vector fieldξ,ξ + tζ is still a normalized Reeb vector field for (small) real number t.  ... 
arXiv:1303.3059v1 fatcat:myqsqfm3onchfksn6jpesjtofu

Page 6687 of Mathematical Reviews Vol. , Issue 89M [page]

1989 Mathematical Reviews  
Automorphism groups of X correspond to extensions I’ normalizing I. Let X be the torus or the projective plane minus k disks (and no ramification points). Let ® be an automorphism of X of order N.  ...  of G, and they carry this out in detail for several choices of the point group G.  ... 

Lattès-type mappings on compact manifolds

Laura Astola, Riikka Kangaslampi, Kirsi Peltonen
2010 Conformal Geometry and Dynamics  
A uniformly quasiregular mapping acting on a compact Riemannian manifold distorts the metric by a bounded amount, independently of the number of iterates.  ...  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 planeC.  ...  Acknowledgment We are deeply grateful to Gaven Martin for his continuous enthusiasm, encouragement, and generosity in sharing mathematics.  ... 
doi:10.1090/s1088-4173-2010-00220-1 fatcat:coc63m353neb5db74rdm3nxgk4

Quasiregular semigroups with examples [article]

A. Fletcher
2018 arXiv   pre-print
In this paper, we study properties of the Julia and Fatou sets of quasiregular semigroups and, equally as importantly, give several families of examples illustrating some of the behaviours that can arise  ...  While there is a completely viable theory for the iteration of uniformly quasiregular maps, it is a highly non-trivial matter to construct them.  ...  Uqr maps and normal families. For m ≥ 1, we write f m for the m-fold iterate of f .  ... 
arXiv:1801.02545v3 fatcat:zmls3nles5dulf6utf7xdbjd7u

Immersion theorem for Vaisman manifolds [article]

L. Ornea, M. Verbitsky
2003 arXiv   pre-print
We prove a non-Kaehler analogue of Kodaira embedding theorem: any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold.  ...  A slice of the action of G C exists at any point x ∈ M where G C acts smoothly, that is, for all points x ∈ M which are not critical points of f : M −→ Q.  ...  Denote byG the group of automorphisms of the pair (M , M ) and mapping to G under the natural forgetful map Aut(M , M ) −→ Aut(M ) (see Subsection 4.1).  ... 
arXiv:math/0306077v3 fatcat:fmiwpkt55rhl7b26ckjj3dhsvi

An immersion theorem for Vaisman manifolds

Liviu Ornea, Misha Verbitsky
2005 Mathematische Annalen  
A slice of the action of G C exists at any point x ∈ M where G C acts smoothly, that is, for all points x ∈ M which are not critical points of f : M −→ Q.  ...  Denote byG the group of automorphisms of the pair (M, M) and mapping to G under the natural forgetful map Aut(M, M) −→ Aut(M) (see Subsection 4.1).  ...  By Claim 4.4,G K is a subgroup of the group of isometries of M, and therefore it is compact.  ... 
doi:10.1007/s00208-004-0620-4 fatcat:o2qzjghdfrglpngotl7h52u5ia

Quasiregular semigroups with examples

Alastair Fletcher
2019 Discrete and Continuous Dynamical Systems. Series A  
In this paper, we study properties of the Julia and Fatou sets of quasiregular semigroups and, equally as importantly, give several families of examples illustrating some of the behaviours that can arise  ...  While there is a completely viable theory for the iteration of uniformly quasiregular maps, it is a highly non-trivial matter to construct them.  ...  The chordal distance is normalized so that χ(x, y) ≤ 1 for all x, y ∈ S n with equality if and only if x and y are antipodal points.  ... 
doi:10.3934/dcds.2019090 fatcat:vhb2qvnqpfcdxc67zd4m74ohgu

Quasiregular Mappings, Curvature & Dynamics

Gaven J. Martin
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
Because of Rickman's version of Montel's theorem there is a close analogy between the dynamics of rational endomorphisms of closed manifolds and the classical Fatou-Julia theory of iteration of rational  ...  We survey recent developments in the area of geometric function theory and nonlinear analysis and in particular those that pertain to recent developments linking these areas to dynamics and rigidity theory  ...  It follows that the elements of the generalized derivative are either all constant, all elliptic, or all loxodromic, and this allows for a classification of the fixed points of a uniformly quasiregular  ... 
doi:10.1142/9789814324359_0105 fatcat:vjtvdjahknam7bjehyvwwg2wi4

Page 1150 of Mathematical Reviews Vol. 52, Issue 4 [page]

1976 Mathematical Reviews  
The proof is based on the method of Dirichlet series and probabilistic limit theorems. J. Kubilius (Vilnius) Sigmund, Karl 8073 Normal and quasiregular points for automorphisms of the torus. Math.  ...  The present author gives a very elegant proof for T? for the case when A and Bare automorphisms. _P. Gerl (Salzburg) Hansen, Rodney T. A characterization of complementing pairs by distance sets.  ... 

Book Review: Convex bodies and algebraic geometry: An introduction to toric varieties

Miles Reid
1989 Bulletin of the American Mathematical Society  
Ri2] , The analogue of Picard's theorem for quasiregular mappings in dimension three, Acta Math. 154 (1985), 195-242. [RR] H. Reiman and T.  ...  For example, Riemann-Roch and Serre duality for complex projective varieties apply directly to questions on the number of lattice points in the interior of a convex cone, and the Hodge index theorem to  ...  The main idea here is that the automorphism group of a toric variety X is an algebraic group G with T c X as its maximal torus, and the cones in the character lattice of T used to construct X also describe  ... 
doi:10.1090/s0273-0979-1989-15864-3 fatcat:fj4lhxq2rrbj3csrqzoc3w46me
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