A STUDY OF THE DYNAMICS OF λ<font>sin</font>z

Domínguez, Patricia; Sienra, Guillermo

doi:10.1142/s0218127402006199

Cited by 14 publications

(11 citation statements)

References 13 publications

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“…Since then a number of papers concerning the dynamics of trigonometric functions have appeared where various aspects were discussed, e.g. connectedness properties and buried points of the Julia set of sin(z) by Patricia Domínguez [3], the dynamics of λ sin(z) by Patricia Domínguez and Guillermo Sienra [4], the set of accessible points of the Julia set of λ sin(z) by Boguslawa Karpińska [7,8], the dynamical fine structure of iterated cosine maps by Dierk Schleicher [12] or the set of escaping points of the cosine family by Günther Rottenfußer and Dierk Schleicher [11].…”

Section: Introductionmentioning

confidence: 99%

Area of Fatou sets of trigonometric functions

Schubert

2007

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Area of Fatou sets of trigonometric functions

Schubert

2007

Proc. Amer. Math. Soc.

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“…Here our result implies that all Fatou components are bounded when f is hyperbolic, except for |a| < 1; this was already stated by Zhang [Zha09, p. 2, third paragraph], who mentions that it can be proved using polynomiallike mappings. Some special instances can also implicitly be found already in [DS02]. The same statement is established in [DF08, Prop.…”

Section: Theorem (Bounded Fatou Components)mentioning

confidence: 54%

Hyperbolic entire functions with bounded Fatou components

Bergweiler¹,

Fagella²,

Rempe³

2015

Comment. Math. Helv.

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We show that an invariant Fatou component of a hyperbolic transcendental entire function is a Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou components and local connectivity of Julia sets for hyperbolic entire functions, and give examples that demonstrate that our results are optimal. A particularly strong dichotomy is obtained in the case of a function with precisely two critical values.

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“…We then prove that f θ (z) = T θ (z). This is proved by using a topological rigidity property of the sine family which was proved in [6]. Theorem 1 then follows.…”

Section: Outline Of the Proof Of Theoremmentioning

confidence: 90%

“…Let f be an entire function. If f is topologically equivalent to the map z → sin(z), then f (z) = a + b sin(cz + d) where a, b, c, d ∈ C, and b, c = 0.For a proof of Lemma 7.3, see[6].…”

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confidence: 99%

On PZ type Siegel disks of the sine family

Zhang¹

2014

Ergod. Th. Dynam. Sys.

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We prove that for typical rotation numbers 0 < θ < 1, the boundary of the Siegel disk of f θ (z) = e 2πiθ sin(z) centered at the origin is a Jordan curve which passes through exactly two critical points π/2 and −π/2. Statement of the main resultLet 0 < θ < 1 be an irrational number and [a 1 , . . . , a n , . . .] be its continued fraction. We say that θ is of bounded type if sup{a n } < ∞. The set of bounded type irrational numbers has zero Lebesgue measure. The following theorem was proved in [10]. THEOREM 1.1. Let 0 < θ < 1 be a bounded type irrational number. Then the Siegel disk of e 2πiθ sin(z) centered at the origin is a quasi-disk with boundary containing exactly two critical points π/2 and −π/2. Theorem 1.1 was proved by performing quasi-conformal (qc) surgery on an appropriate premodel. In [7] the authors there proposed a surgery technique to construct Jordan Siegel disks with typical rotation numbers. This surgery technique was pioneered by Haissinsky, who used it to transform an attracting basin into a parabolic basin, and is now called transqc surgery. Petersen asked if Theorem 1.1 can be extended to the case of typical rotation numbers through trans-qc surgery. The main purpose of this paper is to prove the following theorem.MAIN THEOREM. For typical irrational numbers 0 < θ < 1, the Siegel disk of e 2πiθ sin(z) centered at the origin is a Jordan domain with boundary containing exactly two critical points π/2 and −π/2. Compared with qc surgery, the main difficulty in performing trans-qc surgery is to verify the integrability of certain degenerate Beltrami differentials. This often requires some delicate area estimates. In [7] the authors used Petersen puzzles developed in [6] to obtain the desired estimate for the Douady-Ghys Blaschke model. Such puzzle construction does not exist in constructing Siegel disks of entire functions. In fact it is not known if the

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A STUDY OF THE DYNAMICS OF λsinz

Cited by 14 publications

References 13 publications

Area of Fatou sets of trigonometric functions

Area of Fatou sets of trigonometric functions

Hyperbolic entire functions with bounded Fatou components

On PZ type Siegel disks of the sine family

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