A note on non-converging Julia sets

Krauskopf, Bernd; Kriete, Hartje

doi:10.1088/0951-7715/9/2/018

Cited by 6 publications

(9 citation statements)

References 7 publications

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“…The following lemma is well-known for rational maps on the Riemann sphere, but is more complicated to show in the current situation; see [14,16,17]. LEMMA 2.…”

Section: Proof Of the Main Theoremmentioning

confidence: 98%

“…, converging to G uniformly on compact subsets of C × C. In order to avoid the pathological case of approximating polynomials by transcendental functions (compare [17]) we assume that the G n are families of polynomials if G is a family of polynomials.…”

Section: Families Of Functions Consider a Familymentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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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 following lemma is well-known for rational maps on the Riemann sphere, but is more complicated to show in the current situation; see [14,16,17]. LEMMA 2.…”

Section: Proof Of the Main Theoremmentioning

confidence: 98%

Section: Families Of Functions Consider a Familymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Kernel convergence of hyperbolic components

Krauskopf¹,

Kriete²

1997

Ergod. Th. Dynam. Sys.

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“…Kisaka [14] extended this result as follows: Assume a sequence of polynomials P n converges uniformly on compact sets to a transcendental entire function f as n → ∞. If F (f ) contains all the singular values and consists only of basins of attracting cycles, then J(P n ) converges to J(f ) in the Hausdorff metric (see also [16] as remark). Krauskopf and Kriete [18] proved the similar results for meromorphic functions.…”

Section: Introductionmentioning

confidence: 97%

Dynamical convergence of a certain polynomial family to f_a(z) = z + e^z + a

Morosawa¹

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. A transcendental entire function f a (z) = z + e z + a may have a Baker domain or a wandering domain, which never appear in the dynamics of polynomials. We consider a sequence of polynomials+ a, which converges uniformly on compact sets to f a as d → ∞. We show its dynamical convergence under a certain assumption, even though f a has a Baker domain or a wandering domain. We also investigate the parameter spaces of f a and P a,d .

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“…We mention the work of A. Douady [5], who investigates the mapping f → J (f, C) from the class of polynomials of fixed degree to the set of nonempty plane compacta equipped with the Hausdorff metric d H (X, Y ) := max{∂(X, Y ), ∂(Y, X)}, ∂(X, Y ) := sup x∈X dist(x, Y ). We also mention subsequent papers [6,7,8,9,10,11] dealing with other classes of functions. Continuity of Julia sets is closely related to behaviour of connected components of the Fatou set containing periodic points.…”

Section: Introduction 1preliminariesmentioning

confidence: 99%

Carathéodory Convergence of Immediate Basins of Attraction to a Siegel Disk

Gumenyuk

2009

Analysis and Mathematical Physics

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Let f n be a sequence of analytic functions in a domain U with a common attracting fixed point z 0 . Suppose that f n converges to f 0 uniformly on each compact subset of U and that z 0 is a Siegel point of f 0 . We establish a sufficient condition for the immediate basins of attraction A * (z 0 , f n , U ) to form a sequence that converges to the Siegel disk of f 0 as to the kernel with respect to z 0 . The same condition is shown to imply the convergence of the Koenigs functions associated with f n to that of f 0 . Our method allows us also to obtain a kind of quantitative result for analytic one-parametric families.

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A note on non-converging Julia sets

Cited by 6 publications

References 7 publications

Kernel convergence of hyperbolic components

Kernel convergence of hyperbolic components

Dynamical convergence of a certain polynomial family to f_a(z) = z + e^z + a

Carathéodory Convergence of Immediate Basins of Attraction to a Siegel Disk

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