Trajectories escaping to infinity in finite time
Author:
J. K. Langley
Journal:
Proc. Amer. Math. Soc. 145 (2017), 2107-2117
MSC (2010):
Primary 30D30
DOI:
https://doi.org/10.1090/proc/13377
Published electronically:
January 11, 2017
MathSciNet review:
3611324
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: If the function $f$ is transcendental and meromorphic in the plane, and either $f$ has finitely many poles or its inverse function has a logarithmic singularity over $\infty$, then the equation $\dot z = f(z)$ has infinitely many trajectories tending to infinity in finite increasing time.
- Walter Bergweiler and Alexandre Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), no. 2, 355–373. MR 1344897, DOI https://doi.org/10.4171/RMI/176
- Walter Bergweiler, Philip J. Rippon, and Gwyneth M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. Lond. Math. Soc. (3) 97 (2008), no. 2, 368–400. MR 2439666, DOI https://doi.org/10.1112/plms/pdn007
- Louis Brickman and E. S. Thomas, Conformal equivalence of analytic flows, J. Differential Equations 25 (1977), no. 3, 310–324. MR 447674, DOI https://doi.org/10.1016/0022-0396%2877%2990047-X
- Kevin A. Broughan, The structure of sectors of zeros of entire flows, Proceedings of the 17th Summer Conference on Topology and its Applications, 2003, pp. 379–394. MR 2077797
- Antonio Garijo, Armengol Gasull, and Xavier Jarque, Local and global phase portrait of equation $\dot z=f(z)$, Discrete Contin. Dyn. Syst. 17 (2007), no. 2, 309–329. MR 2257435, DOI https://doi.org/10.3934/dcds.2007.17.309
- Gary G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), no. 1, 88–104. MR 921748, DOI https://doi.org/10.1112/jlms/s2-37.121.88
- Otomar Hájek, Notes on meromorphic dynamical systems. I, Czechoslovak Math. J. 16(91) (1966), 14–27 (English, with Russian summary). MR 194661
- Otomar Hájek, Notes on meromorphic dynamical systems. II, Czechoslovak MAth. J. 16 (91) (1966), 28–35 (English, with Russian summary). MR 0194662
- Otomar Hájek, Notes on meromorphic dynamical systems. III, Czechoslovak Math. J. 16(91) (1966), 36–40 (English, with Russian summary). MR 194663
- W. K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull. 17 (1974), no. 3, 317–358. MR 385095, DOI https://doi.org/10.4153/CMB-1974-064-0
- W. K. Hayman, Subharmonic functions. Vol. 2, London Mathematical Society Monographs, vol. 20, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989. MR 1049148
- D. J. Needham and A. C. King, On meromorphic complex differential equations, Dynam. Stability Systems 9 (1994), no. 2, 99–122. MR 1287510, DOI https://doi.org/10.1080/02681119408806171
- Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30D30
Retrieve articles in all journals with MSC (2010): 30D30
Additional Information
J. K. Langley
Affiliation:
School of Mathematical Sciences, University of Nottingham, NG7 2RD, United Kingdom
MR Author ID:
110110
Email:
james.langley@nottingham.ac.uk
Received by editor(s):
May 11, 2016
Received by editor(s) in revised form:
July 4, 2016
Published electronically:
January 11, 2017
Communicated by:
Jeremy Tyson
Article copyright:
© Copyright 2017
American Mathematical Society