Holomorphic motions commuting with semigroups

ArticleOriginal scientific text

Title

Authors ¹

Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045, U.S.A.

A holomorphic family

f_{z}

, |z|<1, of injections of a compact set E into the Riemann sphere can be extended to a holomorphic family of homeomorphisms

F_{z}

, |z|<1, of the Riemann sphere. (An earlier result of the author.) It is shown below that there exist extensions

F_{z}

which, in addition, commute with some holomorphic families of holomorphic endomorphisms of

ℂ ̅ ̅ ̅ ̅ ̅ ̅ f_{z} (E)

, |z|<1 (under suitable assumptions). The classes of covering maps and maps with the path lifting property are discussed.

[BR] L. Bers and H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), 259-286.
[Bi] E. Bishop, Mappings of partially analytic spaces, Amer. J. Math. 83 (1961), 209-242.
[DH] A. Douady et J. Hubbard, Itération de polynômes quadratiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 123-126.
[EF] C. J. Earle and R. S. Fowler, Holomorphic families of open Riemann surfaces, Math. Ann. 270 (1985), 249-273.
[EKK] C. J. Earle, I. Kra and S. L. Krushkal, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), 927-948.
[GR] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, N.J., 1965.
[Ma] B. Maskit, Kleinian Groups, Springer, 1988.
[MSS] R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Ecole Norm. Sup. 16 (1983), 193-217.
[Mu] C. McMullen, Private letter, 1990.
[Sł1] Z. Słodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), 347-355.
[Sł2] Z. Słodkowski, Invariant extensions of holomorphic motions, abstract No. 873-30-234, Abstracts Amer. Math. Soc. 13 (1992), p. 259.
[Sł3] Z. Słodkowski, Extensions of holomorphic motions, Ann. Scuola Norm. Sup. Pisa, to appear.
[Su] D. Sullivan, Quasiconformal homeomorphisms and dynamics, III: Topological conjugacy classes of analytic endomorphisms, preprint, 1985.
[ST] D. Sullivan and W. P. Thurston, Extending holomorphic motions, Acta Math. 157 (1986), 243-257.