Iteration of Quasiregular Mappings

Bergweiler, Walter

doi:10.1007/bf03321776

Cited by 28 publications

(53 citation statements)

References 62 publications

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“…Remarkably, it is still possible to say quite a lot about the dynamics of quasiregular mappings when there is not a uniform bound on the distortion of the iterates. See for example the recent works of Bergweiler [2,3].…”

Section: Introductionmentioning

confidence: 99%

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Fixed curves near fixed points

Fletcher¹

2015

Illinois J. Math.

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Let H be a composition of an R-linear planar mapping and z → z n . We classify the dynamics of H in terms of the parameters of the R-linear mapping and the degree by associating a certain finite Blaschke product. We apply this classification to this situation where z 0 is a fixed point of a planar quasiregular mapping with constant complex dilatation in a neighbourhood of z 0 . In particular we find how many curves there are that are fixed by f and that land at z 0 . arXiv:1504.05463v1 [math.DS]

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Section: Introductionmentioning

confidence: 99%

“…The dynamics of H with K = 2.25, θ = 0.75 and n = 6.are those on ∂I(H) that arise from the Julia set of B, which we recall is a Cantor set. This gives another class of examples where the boundary of the escaping set and the set with a blowing-up property do not agree, c.f [2]…”

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confidence: 99%

Fixed curves near fixed points

Fletcher¹

2015

Illinois J. Math.

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“…As in the case of rational functions, the Julia set of such a map is defined as the set of all points in S d where the iterates of f fail to be normal. Many results of the Fatou-Julia iteration theory of rational functions continue to hold in this more general setting; see [2,Section 4] and [15,Chapter 21] for surveys. In principle the corresponding iteration theory of transcendental entire functions could also be extended to uniformly quasiregular self-maps of R d with an essential singularity at ∞, but so far no examples of such maps are known for d ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

The Julia Set and the Fast Escaping Set of a Quasiregular Mapping

Bergweiler

Fletcher

Nicks

2014

Comput. Methods Funct. Theory

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“…still be studied in the absence of uniform quasiregularity. Recent papers in this direction include [2,3,4,9,11].…”

Section: Introductionmentioning

confidence: 99%

On Böttcher coordinates and quasiregular maps

Fletcher¹,

Fryer²

2012

Contemporary Mathematics

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Abstract. It is well-known that a polynomial f (z) = a d z d (1 + o(1)) can be conjugated by a holomorphic map φ to w → w d in a neighbourhood of infinity. This map φ is called a Böttcher coordinate for f near infinity. In this paper we construct a Böttcher type coordinate for compositions of affine mappings and polynomials, a class of mappings first studied in [9]. As an application, we prove that if h is affine and c ∈ C, then h(z) 2 + c is not uniformly quasiregular.MSC 2010: 30C65 (Primary), 30D05, 37F10, 37F45 (Secondary).

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Iteration of Quasiregular Mappings

Cited by 28 publications

References 62 publications

Fixed curves near fixed points

Fixed curves near fixed points

The Julia Set and the Fast Escaping Set of a Quasiregular Mapping

On Böttcher coordinates and quasiregular maps

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