Dynamics of the family $\lambda \tan z$
HTML articles powered by AMS MathViewer
- by Linda Keen and Janina Kotus
- Conform. Geom. Dyn. 1 (1997), 28-57
- DOI: https://doi.org/10.1090/S1088-4173-97-00017-9
- Published electronically: August 13, 1997
- PDF | Request permission
Abstract:
We study the the tangent family ${\mathcal F} = \{\lambda \tan z, \lambda \in {\mathbb C} - \{0\}\}$ and give a complete classification of their stable behavior. We also characterize the the hyperbolic components and give a combinatorial description of their deployment in the parameter plane.References
- Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141
- I. N. Baker, J. Kotus, and Yi Nian Lü, Iterates of meromorphic functions. II. Examples of wandering domains, J. London Math. Soc. (2) 42 (1990), no. 2, 267–278. MR 1083445, DOI 10.1112/jlms/s2-42.2.267
- I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. I, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 241–248. MR 1116639, DOI 10.1017/S014338570000612X
- I. N. Baker, J. Kotus, and Yi Nian Lü, Iterates of meromorphic functions. III. Preperiodic domains, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 603–618. MR 1145612, DOI 10.1017/S0143385700006386
- I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. IV. Critically finite functions, Results Math. 22 (1992), no. 3-4, 651–656. MR 1189754, DOI 10.1007/BF03323112
- Lipman Bers and H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), no. 3-4, 259–286. MR 857675, DOI 10.1007/BF02392595
- Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
- Robert L. Devaney and MichałKrych, Dynamics of $\textrm {exp}(z)$, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 35–52. MR 758892, DOI 10.1017/S014338570000225X
- A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 (French). MR 762431
- Robert L. Devaney and Linda Keen, Dynamics of tangent, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 105–111. MR 970550, DOI 10.1007/BFb0082826
- Robert L. Devaney and Linda Keen, Dynamics of meromorphic maps: maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 55–79. MR 985854, DOI 10.24033/asens.1575
- P. Fatou, Sur les équations fonctionnelles: (Troisième mémoire), Bull. Sci. Math. France, 48 (1920), 208–314.
- W. H. Jiang, Dynamics of $\lambda \tan z$, Ph. D. thesis, CUNY 1991, unpublished.
- L. Keen and J. Kotus, Ergodicity of Julia sets of meromorphic functions with compact postcritical set, manuscript.
- Curtis T. McMullen, Frontiers in complex dynamics, Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 2, 155–172. MR 1260523, DOI 10.1090/S0273-0979-1994-00519-1
- C. T. McMullen and D. Sullivan, Quasiconformal homeomorphisms and dynamics III: The TeichmĂĽller space of a holomorphic dynamical system, Preprint, 1995.
- R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343, DOI 10.24033/asens.1446
- J. Milnor, Dynamics in one complex variable: Introductory lectures, MSI-SUNY Stonybrook preprint series.
- R. Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math., 58 (1932).
Bibliographic Information
- Linda Keen
- Affiliation: Department of Mathematics, CUNY Lehman College, Bronx, New York 10468
- MR Author ID: 99725
- Email: linda@alpha.lehman.cuny.edu
- Janina Kotus
- Affiliation: Institute of Mathematics, Technical University of Warsaw, 00-661 Warsaw, Poland
- Email: janinak@snowman.impan.gov.pl
- Received by editor(s): April 28, 1997
- Published electronically: August 13, 1997
- Additional Notes: The first author was supported in part by NSF Grant DMS-9205433, PSC-CUNY Award, I.B.M. and M.S.R.I
The second author was supported in part by Polish KBN Grant No 21046910 “Iteracje i fraktale” and KBN Grant No 2 P03A 025 12 “Iteracje funkcji holomorficznych", the Fulbright Foundation, I.M.S. Stony Brook and M.S.R.I - © Copyright 1997 American Mathematical Society
- Journal: Conform. Geom. Dyn. 1 (1997), 28-57
- MSC (1991): Primary 30D05, 58F23, 32H50; Secondary 58F08, 34C35
- DOI: https://doi.org/10.1090/S1088-4173-97-00017-9
- MathSciNet review: 1463839