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  • Ergodic Theory and Dynamical Systems, Volume 9, Issue 2
  • June 1989, pp. 321-328

Topology and growth of a special class of holomorphic self-maps of ℂ*

  • Linda Keen (a1)
  • DOI: http://dx.doi.org/10.1017/S0143385700004995
  • Published online: 01 September 2008
Abstract
Abstract

It is a general problem to find appropriate sets of moduli for families of functions that generate dynamical systems. In this paper we solve this problem for a specific family of holomorphic self-maps of ℂ* defined by

The main theorem states that any function topologically conjugate to a member of ℱ is holomorphically conjugate to some member of the family. It follows that the coefficients of the polynomials P(z) and Q(z) are a suitable set of moduli for the families of dynamical systems generated by these functions.

The moduli spaces of functions in ℱ are easy to study computationally and have been studied by many authors. (See references in the text.)

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[AB]L. V. Ahlfors and L. Bers . Riemann's mapping theorem for variable metrics. Ann. of Math. 72 (1960).

[Be2]L. Bers . Uniformization by Beltrami equations. Comm. Pure and Applied Math. 14 3 (1961), pp. 215228.

[Bl]P. Blanchard . Complex analytic dynamics on the Riemann Sphere. Bull. A.M.S. 11 (1984).

[Ku]H. Künzi . Quasikonforme Abbildungen, Springer-Verlag, Berlin-New York, 1960.

[LV]O. Lehto & K. Virtanen . Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin-New York, 1973.

[LeV]Le-Van Thiem . Über das Umkehrproblem der Wertverteilungslehre. Comment. math. Helv. 23 (1949).

[McM]C. McMullen . Area and Hausdorff dimension of Julia sets of entire functions. Trans. AMS 300 (1987), 1, 329342.

[N]R. Nevanlinna . Analytic Functions. Springer-Verlag, Berlin-New York, 1970.

[Po]K. Pöschl . Über die Wertverteilung der erzeugenden Funktionen Riemannscher Flächen mit endlich vielen periodischen Enden. Math. Ann. 123 (1951), Helv. 2 (1930).

[Wi1]H. Wittich . Zum Beweis eines Satzes uber quasikonforme Abbildungen. Math. Zeit. 51 (1948).

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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