Repulsive Fixpoints of Analytic Functions with Applications to Complex Dynamics

Essén, Matts; Wu, Shengjian

doi:10.1112/s0024610700001149

Cited by 16 publications

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“…M. Essén and S. J. Wu [13], [14] proved the following result, thereby answering affirmatively a question of Yang [20,Problem 8].…”

Section: §1 Introduction and Main Resultsmentioning

confidence: 53%

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Normality, Quasinormality and Periodic Points

Chang¹

2009

Nagoya Mathematical Journal

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Abstract. Let M ≥ 1 be a positive number. Let F be a family of holomorphic functions f in some domain D ⊂ C for which there exists an integerWe show first that F is quasinormal of order at most one in D. This strengthens a result of W. Bergweiler. Secondly, for the case M = 1, we prove that F is normal in D if there exists a positive number K < 3 such that |f ′ (η)| ≤ K for each f ∈ F and every fixed point η of f in D. This improves a result of M. Essén and S. J. Wu. We also construct an example which shows that the condition |f ′ (η)| ≤ K < 3 can not be replaced by |f ′ (η)| < 3.

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“…M. Essén and S. J. Wu [13], [14] proved the following result, thereby answering affirmatively a question of Yang [20,Problem 8].…”

Section: §1 Introduction and Main Resultsmentioning

confidence: 53%

“…Then the iterates f n : D n → C of f are defined inductively by D 1 = D, f 1 = f and D n = f −1 (D n−1 ) = {z ∈ D : f (z) ∈ D n−1 }, f n = f n−1 • f for n ≥ 2. Note that D n+1 ⊂ D n ⊂ D for all n ∈ N. See [3], [7], [8], [11], [13], [14].…”

Section: §1 Introduction and Main Resultsmentioning

confidence: 99%

Normality, Quasinormality and Periodic Points

Chang¹

2009

Nagoya Mathematical Journal

View full text Add to dashboard Cite

show abstract

“…Note that D n+1 ⊂ D n ⊂ D for all n ∈ N. See [2,3,11,14,15]. Let z 0 ∈ D. If there exists a smallest integer p ∈ N such that z 0 ∈ D p , f p (z 0 ) = z 0 , then z 0 is said to be a periodic point of period p of f and the corresponding cycle {z 0 , f(z 0 ), · · · , f p−1 (z 0 )} is said to be a periodic cycle of period p of f in D. A periodic point of period 1 is said to be a fixed point.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%