Rational Maps

Beardon, Alan F.

doi:10.1007/978-1-4612-4422-6_2

Cited by 45 publications

(127 citation statements)

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“…So, only three of them are free critical points. It is a known fact that any basin of attraction must hold, at least, one critical point (see, e.g., [10] or [11]). Then, the knowledge of the critical points and their behavior as a starting point of the iterative process is of vital importance.…”

Section: Study Of the Parameter Spacementioning

confidence: 99%

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Study of a Biparametric Family of Iterative Methods

Campos

Cordero

Magreñán

et al. 2014

Abstract and Applied Analysis

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The dynamics of a biparametric family for solving nonlinear equations is studied on quadratic polynomials. This biparametric family includes the -iterative methods and the well-known Chebyshev-Halley family. We find the analytical expressions for the fixed and critical points by solving 6-degree polynomials. We use the free critical points to get the parameter planes and, by observing them, we specify some values of ( , ) with clear stable and unstable behaviors.

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Section: Study Of the Parameter Spacementioning

confidence: 99%

“…The natural space to study the dynamics of a rational map is the Riemann sphereĈ, :Ĉ →Ĉ (see [10,11], e.g.). The dynamics of this rational map induces a subdivision of the complex sphere into two sets, namely, the Fatou set F( ) and the Julia set J( ) of .…”

Section: Introductionmentioning

confidence: 99%

Study of a Biparametric Family of Iterative Methods

Campos

Cordero

Magreñán

et al. 2014

Abstract and Applied Analysis

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“…The Julia set is the complement of the Fatou set, and it is the locus of chaos; small errors may become arbitrarily large after many iterations. For a deep review on iteration of rational maps see [2]. See [1] for a wide study of complex dynamics.…”

Section: Introductionmentioning

confidence: 99%

Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family

et al. 2015

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“…We shall use the usual definitions of complex dynamics (see, e.g., [4], [5], [10], [14]). We denote the iterates of f by f 1 = f , and f n = f •f n−1 for n ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

Growth conditions for entire functions with only bounded Fatou components

Hinkkanen

Miles

2009

J Anal Math

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Abstract. Let f be a transcendental entire function of order < 1/2. We denote the maximum and minimum modulus of f by M (r, f ) = max{|f (z)| : |z| = r} and m(r, f ) = min{|f (z)| : |z| = r}. We obtain a minimum modulus condition satisfied by many f of order zero that implies all Fatou components are bounded. A special case of our result is that if log log M (r, f ) = O(log r/(log log r) K ) for some K > 1, then there exist α > 1 and C > 0 such that for all large R, there exists r ∈ (R, R α ] with log m(r, f ) log M (R, f ) ≥ α 1 − C (log log R) K , and this in turn implies boundedness of all Fatou components.The condition on m(r, f ) is a refined form of a minimum modulus conjecture formulated by the first author. We also show that there are some functions of order zero, and there are functions of any positive order, for which even refined forms of the minimum modulus conjecture fail. Our results and counterexamples indicate rather precisely the limits of the method of using the minimum modulus to rule out the existence of unbounded Fatou components.

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Rational Maps

Cited by 45 publications

References 0 publications

Study of a Biparametric Family of Iterative Methods

Study of a Biparametric Family of Iterative Methods

Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family

Growth conditions for entire functions with only bounded Fatou components

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