Local dynamics of uniformly quasiregular mappings

Hinkkanen, A.; Martin, Gaven; Mayer, Volker

doi:10.7146/math.scand.a-14450

Cited by 40 publications

(73 citation statements)

References 23 publications

(17 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now, the proof of [8,Theorem 6.3] shows that L, M are injective in neighborhoods of 0 U, V respectively. Since L and M are linearizers for f at the repelling fixed point x 0 , then L(U) ∩ M(V ) = E ∋ x 0 , which is a subset of the domain of f for which the linearization is valid.…”

Section: Relationship Between Linearizersmentioning

confidence: 99%

“…For such mappings, the Julia set and Fatou set can be defined and they partition space, just as in the complex case. The fixed points of uqr mappings have been classified [8] in a similar fashion to the classification of fixed points of holomorphic functions. In particular, there are repelling fixed points of uniformly quasiregular mappings.…”

Section: Quasiregular Linearizersmentioning

confidence: 99%

“…Theorem 2.1 (Theorem 6.3, [8]). Let f be uqr, x 0 be a repelling fixed point of f and ψ ∈ Df (x 0 ).…”

Section: 2mentioning

confidence: 99%

“…Let f be a uniformly quasiregular mapping. Hinkkanen, Martin and Mayer [8] define the set Df (x 0 ) of generalized derivatives of f at x 0 by the set of limits of…”

Section: 2mentioning

confidence: 99%

See 3 more Smart Citations

On Quasiregular Linearizers

Fletcher

Macclure

2015

Comput. Methods Funct. Theory

View full text Add to dashboard Cite

Abstract. Linearization is a well-known concept in complex dynamics. If p is a polynomial and z 0 is a repelling fixed point, then there is an entire function L which conjugates p to the linear map z → p ′ (z 0 )z. This notion of linearization carries over into the quasiregular setting, in the context of repelling fixed points of uniformly quasiregular mappings. In this article, we investigate how linearizers arising from the same uqr mapping and the same repelling fixed point are related. In particular, any linearizer arising from a uqr solution to a Schröder equation is shown to be automorphic with respect to some quasiconformal group.

show abstract

Section: Relationship Between Linearizersmentioning

confidence: 99%

Section: Quasiregular Linearizersmentioning

confidence: 99%

See 2 more Smart Citations

On Quasiregular Linearizers

Fletcher

Macclure

2015

Comput. Methods Funct. Theory

View full text Add to dashboard Cite

show abstract

“…Behaviour near the fixed points of uniformly quasiregular mappings was studied in detail in [11]. There it was shown that if a uniformly quasiregular mapping is locally injective at a fixed point, then it is in fact bi-Lipschitz there, whereas typically a quasiregular mapping is only locally Hölder continuous.…”

mentioning

confidence: 99%

Superattracting fixed points of quasiregular mappings

Fletcher¹,

Nicks²

2014

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

Abstract. We investigate the rate of convergence of the iterates of an n-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity.

show abstract