Entire functions arising from trees

Cui, Weiwei

doi:10.1007/s11425-019-1644-0

Cited by 3 publications

(10 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 1.2. Speiser graphs and their dual graphs and Stoïlow's theorem are used in the study of the existence of entire functions with given tree structures in [2], which motivates our work for this paper. As well-known examples, the sine function sin(z) is a totally (also regularly) ramified transcendental entire function and the Weierstrass elliptic function is a totally (also regularly) ramified transcendental meromorphic function.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof. By definition, it is clear that possible signatures of critical values are (2,5), (3,4) and (2,2,3). By Theorem 3.5, the first two cases cannot happen.…”

Section: Some Results On Counting Totally Ramified Rational Mapsmentioning

confidence: 99%

“…By Theorem 3.5, the first two cases cannot happen. Thus, the only possible signature pattern for a totally ramified rational map of degree 7 is (2, 2, 3), (2, 2, 3), (2,2,3). If such a rational map f exists, then the Speiser graph Γ f has two 4-gon faces and one 6-gon face labeled with 0, 1 and ∞ respectively.…”

Section: Some Results On Counting Totally Ramified Rational Mapsmentioning

confidence: 99%

“…If i 1 = 3, there are three possible signature patterns in the sense of Definition 2.1. They are: (2, 2, 4), (2,2,4), (2, 2, 2, 2); (2,3,3), (2,3,3), (2, 2, 2, 2); (2,3,3), (2,2,4), (2,2,2,2). The first two signature patterns are realized by unique Speiser graphs of valence 3 and having no 2-gon face under the equivalence relation given in Definition 2.4, which are given in Figure 9.…”

Section: Some Results On Counting Totally Ramified Rational Mapsmentioning

confidence: 99%

“…; the right one is for a degree 10 totally but not regularly ramified rational map with signature pattern (2, 2, 3, 3), (2, 2, 2, 2, 2), (2,4,4).…”

Section: Existence Of Totally But Not Regularly Ramified Rational Mapmentioning

confidence: 99%

See 4 more Smart Citations

Untitled

2022

Conform. Geom. Dyn.

View full text Add to dashboard Cite

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.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof. By definition, it is clear that possible signatures of critical values are (2,5), (3,4) and (2,2,3). By Theorem 3.5, the first two cases cannot happen.…”

Section: Some Results On Counting Totally Ramified Rational Mapsmentioning

confidence: 99%

Section: Some Results On Counting Totally Ramified Rational Mapsmentioning

confidence: 99%

Section: Some Results On Counting Totally Ramified Rational Mapsmentioning

confidence: 99%

“…; the right one is for a degree 10 totally but not regularly ramified rational map with signature pattern (2, 2, 3, 3), (2, 2, 2, 2, 2), (2,4,4).…”

Section: Existence Of Totally But Not Regularly Ramified Rational Mapmentioning

confidence: 99%

See 3 more Smart Citations

Untitled

2022

Conform. Geom. Dyn.

View full text Add to dashboard Cite

show abstract

Hausdorff dimension of escaping sets of meromorphic functions II

Aspenberg¹,

Cui²

2022

Ergod. Th. Dynam. Sys.

Self Cite

View full text Add to dashboard Cite

A function which is transcendental and meromorphic in the plane has at least two singular values. On the one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either $2$ or $1/2$ . On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in $[0,2]$ (cf. [M. Aspenberg and W. Cui. Hausdorff dimension of escaping sets of meromorphic functions. Trans. Amer. Math. Soc.374(9) (2021), 6145–6178]). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than $4$ .

show abstract

Totally ramified rational maps

Cui

2022

Conform. Geom. Dyn.

Self Cite

View full text Add to dashboard Cite

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 d\leq 6 must be regularly ramified; (2) for any integer d > 6 d>6 , there exists a totally ramified rational map of degree d d which is not regularly ramified. Furthermore, we count totally ramified rational maps up to degree 10 10 . Finally, we present explicit formulas for all totally but not regularly ramified rational maps of degree 7 7 or 8 8 , up to pre- and post-composition by Möbius transformations.

show abstract

Entire functions arising from trees

Cited by 3 publications

References 28 publications

Untitled

Untitled

Hausdorff dimension of escaping sets of meromorphic functions II

Totally ramified rational maps

Contact Info

Product

Resources

About