No Herman rings for regularly ramified rational maps

Hu, Jun; Xiao, Yingqing

doi:10.1090/proc/14347

Cited by 6 publications

(7 citation statements)

References 9 publications

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“…Let us use the same notation introduced in the proof of the previous theorem. The proof is as the same as the one given in [9] to show the same conclusion for a totally ramified rational map, which goes as follows. We rewrite (3.1) as…”

Section: Speiser Graphs Of Branched Coverings and Applicationsmentioning

confidence: 60%

“…Generally, the trajectories of the critical points under the iteration of f determine the dynamical behaviors of f on C. Sometimes, certain pattern of critical points eliminates certain type of dynamical behavior. For example, it is proved in [9] that if all preimages of every critical value of f are critical points, then there is no Herman ring in the Fatou set of f . Motivated by this result, we investigate in this paper the existence of such rational maps for degree d ≥ 2.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…A natural problem is to classify all totally ramified rational maps, and a specific question is to ask how many distinct critical values a totally ramified rational map can have. By the Riemann-Hurwitz formula, it is proved in [9] that each totally ramified rational map has two or three distinct critical values. Again using the Riemann-Hurwitz formula, one can see that any rational map with two distinct critical values has only two distinct critical points.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Remark 1.1. The question on the existence of a totally ramified rational map of degree d ≥ 4 with three distinct critical values was first raised in [9]. Explicit formulas of five types of regularly ramified rational maps are constructed in [7] and dynamics of some one-parameter families of those maps are explored in [7] and studied in [8].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Untitled

2022

Conform. Geom. Dyn.

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Totally ramified rational maps and regularly ramified rational maps are defined and studied in this paper. We first give a complete classification of regularly ramified rational maps and show that our definition of a regularly ramified rational map is equivalent to a much stronger definition of a map of this kind given by Milnor [Dynamics in one complex variable, Princeton University Press, Princeton, NJ, 2006]. Then we show that (1) any totally ramified rational map of degree d ≤ 6 must be regularly ramified; (2) for any integer d > 6, there exists a totally ramified rational map of degree d which is not regularly ramified. Furthermore, we count totally ramified rational maps up to degree 10. Finally, we present explicit formulas for all totally but not regularly ramified rational maps of degree 7 or 8, up to pre-and post-composition by Möbius transformations.

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Section: Speiser Graphs Of Branched Coverings and Applicationsmentioning

confidence: 60%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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2022

Conform. Geom. Dyn.

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“…It is known that the Cantor circle Julia sets and Sierpiński carpet Julia sets can appear in McMullen family and the generalized McMullen family (see [5,30]). For the study of singular perturbation of unicritical polynomials, one may also refer to [27], [31], [32], [15] and [16].…”

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confidence: 99%

Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps

Wang¹,

Yang²,

Zhang³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we investigate the dynamics of the following family of rational maps f λ (z) = z 2n − λ 3n+1 z n (z 2n − λ n−1 ) with one parameter λ ∈ C * − {λ : λ 2n+2 = 1}, where n ≥ 2. This family of rational maps is viewed as a singular perturbation of the bi-critical map P −n (z) = z −n if λ = 0 is small. It is proved that the Julia set J(f λ ) is either a quasicircle, a Cantor set of circles, a Sierpiński carpet or a degenerate Sierpiński carpet provided the free critical orbits of f λ are attracted by the super-attracting cycle 0 ↔ ∞. Furthermore, we prove that there exists suitable λ such that J(f λ ) is a Cantor set of circles but the dynamics of f λ on J(f λ ) is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of J(f λ ) is also proved if the free critical orbits are not attracted by the cycle 0 ↔ ∞. Finally we give an estimate of the Hausdorff dimension of the Julia set of f λ in some special cases.

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On the formulas of meromorphic functions with periodic Herman rings

Yang

2021

Math. Ann.

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No Herman rings for regularly ramified rational maps

Cited by 6 publications

References 9 publications

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Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps

On the formulas of meromorphic functions with periodic Herman rings

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