On the Julia set of analytic self-maps of the punctured plane
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Citations
A STUDY OF THE DYNAMICS OF λ sin z
Some examples of Baker domains
The escaping set of transcendental self-maps of the punctured plane
Boundaries of univalent Baker domains
Some examples of Baker domains
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].