Dynamics of the family 𝜆tan𝑧

Keen, Linda; Kotus, Janina

doi:10.1090/S1088-4173-97-00017-9

Dynamics of the family $\lambda \tan z$
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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

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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 Teichmüller 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).