Holomorphic motions commuting with semigroups

Zbigniew Słodkowski

Studia Mathematica (1996)

  • Volume: 119, Issue: 1, page 1-16
  • ISSN: 0039-3223

Abstract

top
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.

How to cite

top

Słodkowski, Zbigniew. "Holomorphic motions commuting with semigroups." Studia Mathematica 119.1 (1996): 1-16. <http://eudml.org/doc/216283>.

@article{Słodkowski1996,
abstract = {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 $ℂ̅̅̅̅̅̅ \ f_z(E)$, |z|<1 (under suitable assumptions). The classes of covering maps and maps with the path lifting property are discussed.},
author = {Słodkowski, Zbigniew},
journal = {Studia Mathematica},
keywords = {holomorphic motion},
language = {eng},
number = {1},
pages = {1-16},
title = {Holomorphic motions commuting with semigroups},
url = {http://eudml.org/doc/216283},
volume = {119},
year = {1996},
}

TY - JOUR
AU - Słodkowski, Zbigniew
TI - Holomorphic motions commuting with semigroups
JO - Studia Mathematica
PY - 1996
VL - 119
IS - 1
SP - 1
EP - 16
AB - 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 $ℂ̅̅̅̅̅̅ \ f_z(E)$, |z|<1 (under suitable assumptions). The classes of covering maps and maps with the path lifting property are discussed.
LA - eng
KW - holomorphic motion
UR - http://eudml.org/doc/216283
ER -

References

top
  1. [BR] L. Bers and H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), 259-286. Zbl0619.30027
  2. [Bi] E. Bishop, Mappings of partially analytic spaces, Amer. J. Math. 83 (1961), 209-242. Zbl0118.07701
  3. [DH] A. Douady et J. Hubbard, Itération de polynômes quadratiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 123-126. Zbl0483.30014
  4. [EF] C. J. Earle and R. S. Fowler, Holomorphic families of open Riemann surfaces, Math. Ann. 270 (1985), 249-273. Zbl0537.30036
  5. [EKK] C. J. Earle, I. Kra and S. L. Krushkal, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), 927-948. Zbl0812.30018
  6. [GR] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, N.J., 1965. Zbl0141.08601
  7. [Ma] B. Maskit, Kleinian Groups, Springer, 1988. 
  8. [MSS] R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Ecole Norm. Sup. 16 (1983), 193-217. Zbl0524.58025
  9. [Mu] C. McMullen, Private letter, 1990. 
  10. [Sł1] Z. Słodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), 347-355. Zbl0741.32009
  11. [Sł2] Z. Słodkowski, Invariant extensions of holomorphic motions, abstract No. 873-30-234, Abstracts Amer. Math. Soc. 13 (1992), p. 259. 
  12. [Sł3] Z. Słodkowski, Extensions of holomorphic motions, Ann. Scuola Norm. Sup. Pisa, to appear. 
  13. [Su] D. Sullivan, Quasiconformal homeomorphisms and dynamics, III: Topological conjugacy classes of analytic endomorphisms, preprint, 1985. 
  14. [ST] D. Sullivan and W. P. Thurston, Extending holomorphic motions, Acta Math. 157 (1986), 243-257. Zbl0619.30026

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.