scholarly journals Asymptotic values of meromorphic functions of smooth growth

1981 ◽  
Vol 11 (1) ◽  
pp. 195-214
Author(s):  
Hidenobu Yoshida
Keyword(s):  
Meromorphic Functions ◽  
Asymptotic Values

Asymptotic values of continuous functions in Euclidean space

1992 ◽  
Vol 111 (2) ◽  
pp. 309-318 ◽  
Author(s):  
P. J. Rippon
Keyword(s):  
Euclidean Space ◽  
Meromorphic Functions ◽  
Continuous Functions ◽  
Asymptotic Values

In this paper we generalize a result of Hayman 4, lemma 4 on asymptotic values of meromorphic functions, which can be stated as follows.

Ergodic properties of semi-hyperbolic functions with polynomial Schwarzian derivative

2010 ◽  
Vol 53 (2) ◽  
pp. 471-502
Author(s):  
Volker Mayer ◽  
Mariusz Urbański
Keyword(s):  
Schwarzian Derivative ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Hyperbolic Functions ◽  
Absolutely Continuous ◽  
Ergodic Properties ◽  
Asymptotic Values ◽  
Conformal Measure ◽  
Finite Invariant Measure

AbstractThe ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative are investigated under the condition that the function is semi-hyperbolic, i.e. the asymptotic values of the Fatou set are in attracting components and the asymptotic values in the Julia set are boundedly non-recurrent. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure m with minimal exponent h; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a σ-finite invariant measure μ absolutely continuous with respect to m. Our main result states that μ is finite if and only if the order ρ of the function f satisfies the condition h > 3ρ/(ρ+1). When finite, this measure is shown to be metrically exact. We also establish a version of Bowen's Formula, showing that the exponent h equals the Hausdorff dimension of the Julia set of f.

scholarly journals A Construction of Meromorphic Functions with Prescribed Asymptotic Behavior

1971 ◽  
Vol 41 ◽  
pp. 75-87 ◽  
Author(s):  
J.L. Stebbins
Keyword(s):  
Asymptotic Behavior ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
The Other ◽  
Asymptotic Values ◽  
Other Hand ◽  
Extended Complex ◽  
The Given ◽  
Prescribed Asymptotic Behavior ◽  
Extended Complex Plane

Although there are several constructions of meromorphic functions with prescribed asymptotic sets [e.g., 5,6], it is usually difficult to determine or prescribe the nature of the asymptotic paths used in these constructions. On the other hand, there are several other constructions of meromorphic functions with prescribed asymptotic paths [e.g., 1, 10, 12], but the extent of the asymptotic values for these functions cannot always be restricted to the values approached along the given paths. Gross [3] has accomplished both results by prescribing paths for every value in the extended complex plane.

scholarly journals Spiral Asymptotic Values of Functions Meromorphic in the Unit Disk

1967 ◽  
Vol 30 ◽  
pp. 247-262 ◽  
Author(s):  
J.L. Stebbins
Keyword(s):  
Unit Disk ◽  
Meromorphic Functions ◽  
Science And Technology ◽  
Extensive Study ◽  
State University ◽  
Wayne State University ◽  
Asymptotic Values

This paper contains part of the author’s Ph.D. dissertation directed by Frederick Bagemihl at Wayne State University. The research was supported by a grant from the Michigan Institute of Science and Technology.Alice Roth has made an extensive study of entire meromorphic functions with prescribed behavior along half rays emanating from the origin (6). The question arose whether analogous results could be found for functions meromorphic in the unit disk with the same behavior prescribed along an exhaustive class of spirals emanating from the origin. In this paper, I present a class of spirals which satisfactorily fills this role. However, I make no claim to the effect that only this class will suffice.

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