On the Julia set of analytic self-maps of the punctured plane
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Citations
On semiconjugation of entire functions
Dynamics of transcendental meromorphic functions
Slow escaping points of meromorphic functions
Iteration of entire functions
Dynamics of Transcendental Functions
References
Iteration of meromorphic functions
The Local Growth of Power Series: A Survey of the Wiman-Valiron Method
Related Papers (5)
Frequently Asked Questions (6)
Q2. What is the main object of the Fatou set?
The main objects studied in complex dynamics are the Fatou set F (f) which is defined as the set where the family {fn} of iterates of f is normal and the Julia set J(f) := D\\F (f).
Q3. What is the conclusion of Lemma 4?
By Lemma 1 and Lemma 4 there exist z1, z2 ∈ U such that w1 = exp z1 is a repelling periodic point of g, say gk(w1) = w1, and w2 = exp z2 ∈ The author′(g).
Q4. What is the result of Lemma 3?
By a result of Baker ([3, Lemma 1], see also [6, Lemma 7]) and Lemma 3 there exists a constant C such that |fn(z2)| ≤ C|fn(z1)|(7)for all large n.Since gk(w1) = w1 the authors have exp f k(z1) = exp z1 and hence f k(z1) = z1 + m2πi for some m ∈ Z. By Lemma 4 the authors havef 2k(z1) = f k(z1 +m2πi) = f k(z1) + ` km2πi = z1 +m(1 + ` k)2πiand induction shows thatfnk(z1) = z1 +m n−1∑ j=0 `jk 2πi.
Q5. What is the corresponding proof of the Lebesgue measure?
Recall that z0 is called a repelling periodic point of f if f n(z0) = z0 and |(fn)′(z0)| > 1 for some n ∈ N, with a slight modification if f is rational and z0 = ∞. Lemma 1 is due to Fatou [15, §30, p. 69] and Julia [19, p. 99, p. 118] for rational functions, Baker [1] for entire functions, and Bhattacharyya [8, Theorem 5.2] for analytic self-maps of C∗. A different proof (that applies to all three cases) has recently been given by Schwick [28].
Q6. What is the main difficulty of proving that I(f) 6=?
Eremenko’s proof that I(f) 6= ∅ is based on the theory of Wiman and Valiron on the behavior of entire functions near points of maximum modulus [17, 30].