Keen, Linda; Kotus, Janina Dynamics of the family \(\lambda \tan z\). (English) Zbl 0884.30019 Conform. Geom. Dyn. 1, No. 4, 28-57 (1997). Summary: 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 hyperbolic components and give a combinatorial description of their deployment in the parameter plane. Cited in 5 ReviewsCited in 30 Documents MSC: 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 37F99 Dynamical systems over complex numbers 37B99 Topological dynamics 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Keywords:iteration; meromorphic function; tangent; holomorphic motion; Julia set; parameter plane PDFBibTeX XMLCite \textit{L. Keen} and \textit{J. Kotus}, Conform. Geom. Dyn. 1, No. 4, 28--57 (1997; Zbl 0884.30019) Full Text: DOI arXiv References: [1] Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385 – 404. · Zbl 0104.29902 [2] 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. · Zbl 0726.30022 [3] I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. I, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 241 – 248. · Zbl 0711.30024 [4] 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. · Zbl 0774.30023 [5] 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. · Zbl 0774.30024 [6] Lipman Bers and H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), no. 3-4, 259 – 286. · Zbl 0619.30027 [7] Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. · Zbl 0782.30022 [8] Robert L. Devaney and Michał Krych, Dynamics of \?\?\?(\?), Ergodic Theory Dynam. Systems 4 (1984), no. 1, 35 – 52. · Zbl 0567.58025 [9] 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). A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie II, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985 (French). With the collaboration of P. Lavaurs, Tan Lei and P. Sentenac. [10] 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. · Zbl 0662.30019 [11] 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. · Zbl 0666.30017 [12] P. Fatou, Sur les équations fonctionnelles: (Troisième mémoire), Bull. Sci. Math. France, 48 (1920), 208-314. [13] W. H. Jiang, Dynamics of \(\lambda \tan z\), Ph. D. thesis, CUNY 1991, unpublished. [14] L. Keen and J. Kotus, Ergodicity of Julia sets of meromorphic functions with compact postcritical set, manuscript. · Zbl 0914.30015 [15] Curtis T. McMullen, Frontiers in complex dynamics, Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 2, 155 – 172. · Zbl 0807.30013 [16] C. T. McMullen and D. Sullivan, Quasiconformal homeomorphisms and dynamics III: The Teichmüller space of a holomorphic dynamical system, Preprint, 1995. · Zbl 0926.30028 [17] 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. · Zbl 0524.58025 [18] J. Milnor, Dynamics in one complex variable: Introductory lectures, MSI-SUNY Stonybrook preprint series. · Zbl 0946.30013 [19] R. Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math., 58 (1932). · Zbl 0004.35504 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.