# On the Julia set of analytic self-maps of the punctured plane

## Summary (1 min read)

### 1 Introduction and main result

- 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).
- It is less obvious that the authors also have exp−1 J(g) ⊂ J(f)(4) and hence, together with (3), exp−1 J(g) = J(f).
- This result is stated in [12, Lemma 1.2] and [20, Lemma 2.2], but I have been unable to follow the arguments for (4) given there.
- The authors theorem may be useful to obtain results for analytic self-maps of C∗ from those for entire functions.
- The authors also mention that it was used in [4, 7, 18] that (5) holds for certain particular examples of functions f and g satisfying (1).

### 2 Lemmas

- The second claim is deduced from the first one by induction.
- Then all components of F (f) are simply connected.
- The conclusion now follows from [6, Theorem 10].

### 3 Points that tend to infinity under iteration

- Eremenko [10] considered for transcendental entire f the set I(f) = {z : lim n→∞ |fn(z)| =∞} and proved that J(f) = ∂I(f).
- 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].
- The authors summarize the above discussion as follows.

### 4 Proof of the theorem

- The authors have already shown in the introduction (and also after Lemma 1) that (3) holds.
- The conclusion then follows since J(f) is closed.
- Combining this with (7) the authors find that |fnk(z2)| = o(Mnk)(8) as n→∞.
- Hence <fn(z2)→∞ and <fn+1(z2)/<fn(z2)→∞ as n→∞ by (2).
- The authors deduce that |fn(z2)| ≥ <fn(z2) ≥Mn for all large n, contradicting (8).

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##### Citations

160 citations

### Cites background from "On the Julia set of analytic self-m..."

...It is also known that (4) holds if g(z) = e (see [8])....

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91 citations

61 citations

### Cites background or result from "On the Julia set of analytic self-m..."

...[11] W. Bergweiler and A. Hinkkanen, On semiconjugation of entire functions, Math....

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...[10] W. Bergweiler, An entire function with simply and multiply connected wandering domains, to appear in Pure Appl....

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...Since the function f and h(z) = z + sin z are both lifts under w = eiz of g(w) = w exp(1 2 (w − 1/w)), z ∈ C \ {0}, we have F (g) = exp(iF (f)) = exp(iF (h)), by a result of Bergweiler [9]....

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...[7] W. Bergweiler, Iteration of meromorphic functions, Bull....

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...1 For a transcendental entire function f , the fast escaping set was introduced by Bergweiler and Hinkkanen in [11]: A(f) = {z : there exists L ∈ N such that |fn+L(z)| > M(R, fn), for n ∈ N}....

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53 citations

52 citations

### Additional excerpts

...4 is from [12] and Example 6....

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##### References

936 citations

705 citations

### "On the Julia set of analytic self-m..." refers result in this paper

...The case that D = C so that / is entire was considered first by Fatou [16] in 1926 and since then by many other authors, see [6] and [11, §4] for surveys....

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479 citations

### "On the Julia set of analytic self-m..." refers methods in this paper

...Eremenko's proof that 1 ( f ) φ 0 is based on the theory of Wiman and Valiron on the behavior of entire functions near points of maximum modulus [17, 30]....

[...]

271 citations

### "On the Julia set of analytic self-m..." refers result in this paper

...This case was studied in long memoirs by Fatou [15] and Julia [19] between 1918 and 1920 and has been the object of much research in recent years, see the books [5, 9, 29] for an introduction....

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...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 ....

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...As another example we mention the results of Baker and Weinreich [4] on the boundary of unbounded invariant components of the Fatou set of transcendental entire functions....

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...The main objects studied in complex dynamics are the Fatou set F ( f ) which is defined as the set where the family { / " } of iterates of / is normal and the Julia set J ( f ) := D\F(f)....

[...]

...The case that D = C so that / is entire was considered first by Fatou [16] in 1926 and since then by many other authors, see [6] and [11, §4] for surveys....

[...]

261 citations

### "On the Julia set of analytic self-m..." refers methods in this paper

...Eremenko's proof that 1 ( f ) φ 0 is based on the theory of Wiman and Valiron on the behavior of entire functions near points of maximum modulus [17, 30]....

[...]