Iteration of meromorphic functions

Bergweiler, Walter

doi:10.1090/s0273-0979-1993-00432-4

Cited by 530 publications

(591 citation statements)

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“…An equivalent definition was given in [9]: J(f ) = ∂I(f ) where I(f ) = {z : f n (z) → ∞} is the set of escaping points; see [3] for an introduction to the dynamics of entire and meromorphic functions.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Dynamics of a higher dimensional analog of the trigonometric functions

Bergweiler¹,

Erëmenko²

2011

Ann. Acad. Sci. Fenn. Math.

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Abstract. We introduce a quasiregular analog F of the sine and cosine function such that, for a sufficiently large constant λ, the map x → λF (x) is locally expanding. We show that the dynamics of this map define a representation of R d , d ≥ 2, as a union of simple curves γ : [0, ∞) → R d which tend to ∞ and whose interiors γ * = γ ((0, ∞)) are disjoint such that the union of all γ * has Hausdorff dimension 1.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Dynamics of a higher dimensional analog of the trigonometric functions

Bergweiler¹,

Erëmenko²

2011

Ann. Acad. Sci. Fenn. Math.

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“…Functions of finite type have other nice properties; see [3,9,12]. As for polynomials, Sullivan's classification of the components of the Fatou set holds, and in particular there are no wandering domains or Baker domains (where {f •n } converges uniformly to ∞).…”

Section: Background and Motivationmentioning

confidence: 99%

“…However, even though Fatou studied the iteration of entire transcendental functions, the transcendental case has only recently received much attention; see the expositions [1,3,8,9]. The theory develops along the lines of the rational case, but often other and generally more complicated methods of proof need to be used.…”

Section: Introductionmentioning

confidence: 99%

Kernel convergence of hyperbolic components

Krauskopf¹,

Kriete²

1997

Ergod. Th. Dynam. Sys.

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Abstract. We study families G(λ, ·) of entire functions that are approximated by a sequence of families G n (λ, ·) of entire functions, where λ ∈ C is a parameter. In order to control the dynamics, the families are assumed to be of the same constant finite type. In this setting we prove the convergence of the hyperbolic components in parameter space as kernels in the sense of Carathéodory. IntroductionIn iteration theory, founded by Fatou and Julia [11,13], there has been much progress on the iteration of rational functions in the last few decades; for an overview see [4,21]. However, even though Fatou studied the iteration of entire transcendental functions, the transcendental case has only recently received much attention; see the expositions [1, 3, 8, 9]. The theory develops along the lines of the rational case, but often other and generally more complicated methods of proof need to be used. There is also a variety of phenomena that do not occur in the rational case. One question is: which results carry over from the rational case and which do not?An interesting approach to this question was suggested in [6] and is illustrated in [5, 15, 16]. The polynomials P n (λ, z) = λ(1+(z/n)) n converge uniformly on compact sets of the complex plane C to the exponentials E(λ, z) = λe z as n tends to infinity. In [6] a combinatorial description is given how external rays to the connectedness loci of the P n (λ, ·) converge to certain 'hairs' in the parameter plane for E(λ, ·). Furthermore, the pointwise convergence of hyperbolic components is shown in [5,6,15]. In the dynamical plane the convergence of Julia sets with respect to the Hausdorff metric is shown in [15] for the above families for suitable values of the parameter λ. The convergence of Julia sets was obtained in [7] for polynomials of constant degree, in [19] for rational functions, and has now been generalized to larger classes of functions; see [14,16,17,20].In this paper we are interested in convergence of hyperbolic components in parameter space for entire functions of the same constant finite type (for the exact definition see §3).

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“…The complement, J ( f ), of F( f ) is called the Julia set of f . An introduction to the properties of these sets can be found in, for example, [2] for rational functions and [3] and [6] for transcendental meromorphic functions.…”

Section: Introductionmentioning

confidence: 99%

“…If U p = U for some minimal p ∈ N, then we say that U is a periodic component of period p. There are five possible types of periodic components (see [3,Theorem 6]). In particular, U is called a Baker domain if there exists z 0 ∈ ∂U such that f np (z) → z 0 as n → ∞, for z ∈ U , where f p (z 0 ) is not defined (see [7] for a survey article on Baker domains).…”

Section: Introductionmentioning

confidence: 99%