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Uniformly quasiregular maps with toroidal Julia sets

2012
*
Conformal Geometry and Dynamics
*

We construct

doi:10.1090/s1088-4173-2012-00235-4
fatcat:6b6df7yt4zejjmgshtuwbduhnm
*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. ...##
###
Uniformly quasiregular mappings of Lattès type

1997
*
Conformal Geometry and Dynamics
*

Using an analogy

doi:10.1090/s1088-4173-97-00013-1
fatcat:ipw74ickmzapdbksvz5cn7nmpa
*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 ...##
###
Strongly automorphic mappings and Julia sets of uniformly quasiregular mappings
[article]

2018
*
arXiv
*
pre-print

In this paper, we prove

arXiv:1703.02455v5
fatcat:v7ek5e2o5bdsjjddl7aihugg5e
*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. ...##
###
Endomorphisms of mapping tori
[article]

2020
*
arXiv
*
pre-print

This generalises

arXiv:2010.12490v1
fatcat:xyode2jpjndevplg433txddtga
*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. ...##
###
Locally injective automorphic mappings in $R^n$

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*! ...

##
###
Isometry group of Sasaki-Einstein metric
[article]

2013
*
arXiv
*
pre-print

We prove that

arXiv:1303.3059v1
fatcat:myqsqfm3onchfksn6jpesjtofu
*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. ...##
###
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

2010
*
Conformal Geometry and Dynamics
*

A uniformly

doi:10.1090/s1088-4173-2010-00220-1
fatcat:coc63m353neb5db74rdm3nxgk4
*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. ...##
###
Quasiregular semigroups with examples
[article]

2018
*
arXiv
*
pre-print

In this paper, we study properties

arXiv:1801.02545v3
fatcat:zmls3nles5dulf6utf7xdbjd7u
*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 . ...##
###
Immersion theorem for Vaisman manifolds
[article]

2003
*
arXiv
*
pre-print

We prove a non-Kaehler analogue

arXiv:math/0306077v3
fatcat:fmiwpkt55rhl7b26ckjj3dhsvi
*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). ...##
###
An immersion theorem for Vaisman manifolds

2005
*
Mathematische Annalen
*

A slice

doi:10.1007/s00208-004-0620-4
fatcat:o2qzjghdfrglpngotl7h52u5ia
*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. ...##
###
Quasiregular semigroups with examples

2019
*
Discrete and Continuous Dynamical Systems. Series A
*

In this paper, we study properties

doi:10.3934/dcds.2019090
fatcat:vhb2qvnqpfcdxc67zd4m74ohgu
*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*. ...##
###
Quasiregular Mappings, Curvature & Dynamics

2011
*
Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)
*

Because

doi:10.1142/9789814324359_0105
fatcat:vjtvdjahknam7bjehyvwwg2wi4
*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*...##
###
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

1989
*
Bulletin of the American Mathematical Society
*

Ri2] ,

doi:10.1090/s0273-0979-1989-15864-3
fatcat:fj4lhxq2rrbj3csrqzoc3w46me
*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 ...
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