Arkiv för Matematik

, Volume 38, Issue 2, pp 281–317

Ergodic properties of fibered rational maps

  • Mattias Jonsson
Article

Abstract

We study the ergodic properties of fibered rational maps of the Riemann sphere. In particular we compute the topological entropy of such mappings and construct a measure of maximal relative entropy. The measure is shown to be the unique one with this property. We apply the results to selfmaps of ruled surfaces and to certain holomorphic mapping of the complex projective planeP2.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AR]Abramov, L. M. andRokhlin, V. A., The entropy of a skew product of measurepreserving transformations,Vestnik Leningrad. Univ. 17:7 (1962), 5–13 (Russian). English transl.:Amer. Math. Soc. Transl. 48 (1966), 255–265.Google Scholar
  2. [B]Bogenschütz, T., Entropy, pressure, and a variational principle for random dynamical systems,Random Comput. Dynam. 1 (1992/93), 99–116.MathSciNetGoogle Scholar
  3. [Bo]Bowen, R., Entropy for group endomorphisms and homogeneous spaces,Trans. Amer. Math. Soc. 153 (1971), 401–414.MATHMathSciNetGoogle Scholar
  4. [Br]Brolin, H., Invariant sets under iteration of rational functions,Ark. Mat. 6 (1965), 103–144.MATHMathSciNetGoogle Scholar
  5. [D]Dabija, M., Böttcher divisors,Preprint, 1998.Google Scholar
  6. [FS1]Fornæss, J. E. andSibony, N., Random iterations of rational functions,Ergodic Theory Dynam. Systems 11 (1991), 687–708.MathSciNetGoogle Scholar
  7. [FS2]Fornæss, J. E. andSibony, N., Critically finite rational maps onP 2, inThe Madison Symposium on Complex Analysis (Nagel, A. and Stout, E. L., eds.), Contemp. Math.137, pp. 245–260, Amer. Math. Soc., Providence, R. I., 1992.Google Scholar
  8. [FS3]Fornæss, J. E. andSibony, N., Complex dynamics in higher dimension, inComplex Potential Theory (Gauthier, P. M. and Sabidussi, G., eds.), pp. 131–186, Kluwer, Dordrecht, 1994.Google Scholar
  9. [FS4]Fornæss, J. E. andSibony, N., Complex dynamics in higher dimension II, inModern Methods in Complex Analysis (Bloom, T., Catlin, D., D'Angelo, J. P. and Siu, Y.-T., eds.), Ann. of Math. Stud.137, pp. 135–182, Princeton Univ. Press, Princeton, N. J., 1995.Google Scholar
  10. [FW]Fornæss, J. E. andWeickert, B., Random iterations inP k, to appear inErgodic Theory Dynam. Systems.Google Scholar
  11. [FLM]Freire, A., Lopez, A. andMañé, R., An invariant measure for rational maps,Bol. Soc. Brasil. Mat. 14 (1983), 45–62.MathSciNetGoogle Scholar
  12. [G]Gromov, M., Entropy, homology and semialgebraic geometry,Astérisque 145–146 (1987), 225–240.MathSciNetGoogle Scholar
  13. [H1]Heinemann, S.-M., Dynamische Aspekte holomorpher Abbildungen inC n,Ph. D. Thesis, Göttingen, 1994.Google Scholar
  14. [H2]Heinemann S.-M., Julia sets for holomorphic endomorphisms ofC n Ergodic Theory Dynam. Systems 16 (1996), 1275–1296.MATHMathSciNetGoogle Scholar
  15. [H3]Heinemann, S.-M., Julia sets of skew products inC 2,Kyushu J. Math. 52 (1998), 299–329.MATHMathSciNetGoogle Scholar
  16. [HP]Hubbard, J. H. andPapadopol, P., Superattractive fixed points inC n,Indiana Univ. Math. J. 43 (1994), 321–365.CrossRefMathSciNetGoogle Scholar
  17. [J]Jonsson, M., Dynamics of polynomial skew products onC 2,Math. Ann. 314 (1999), 403–447.CrossRefMATHMathSciNetGoogle Scholar
  18. [JW]Jonsson, M. andWeickert, B., A nonalgebraic attractor inP 2, to appear inProc. Amer. Math. Soc. Google Scholar
  19. [K]Kifer, Y.,Ergodic Theory of Radom Transformations, Progress in Probability and Statististics10, Birkhäuser, Boston, Mass., 1986.Google Scholar
  20. [KS]Kolyada, S. andSnoha, L., Topological entropy of nonautonomous dynamical systems,Random Comput. Dynam. 4 (1996), 205–233.MathSciNetGoogle Scholar
  21. [LW]Ledrappier, F. andWalters, P., A relativised variational principle for continuous transformations,J. London Math. Soc. 16 (1977), 568–576.MathSciNetGoogle Scholar
  22. [L]Lyubich, M., Entropy properties of rational endomorphisms of the Riemann sphere,Ergodic Theory Dynam. Systems 3 (1983), 351–385.MATHMathSciNetGoogle Scholar
  23. [M]Mañé, R., On the uniqueness of the maximizing measure for rational maps,Bol. Soc. Brasil. Mat. 14 (1983), 27–43.MATHMathSciNetGoogle Scholar
  24. [MP]Misiurewicz, M. andPrzytycki, F., Topological entropy and degree of smooth mappings,Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 573–574.MathSciNetGoogle Scholar
  25. [R]Rokhlin, V. A., Lectures on the theory of entropy of transformations with invariant measures,Uspekhi Mat. Nauk 22:5 (137) (1967), 3–56 (Russian). English transl.:Russian Math. Surveys 22 (1967) 1–52.MATHGoogle Scholar
  26. [S1]Sester, O., Étude dynamique des polynômes fibrés,Ph. D. thesis, Université de Paris-Sud, 1997.Google Scholar
  27. [S2]Sester, O., Hyperbolicité des polynômes fibrés,Bull. Soc. Math. France 127 (1999), 393–428.MATHMathSciNetGoogle Scholar
  28. [Su]Sumi, H., Skew product maps related to finitely generated rational semigroups,Preprint, 1999.Google Scholar
  29. [U]Ueda, T., Complex dynamical systems on projective spaces,Preprint, 1994.Google Scholar
  30. [W]Walters, P.,An Introduction to Ergodic Theory, Graduate Texts in Math.79, Springer-Verlag, New York-Berlin, 1982.Google Scholar
  31. [Y]Young, L. S., Ergodic theory of differentiable dynamical systems, inReal and Complex Dynamical Systems (Branner, B. and Hjorth, P., eds.), pp. 293–336, Kluwer, Dordrecht, 1995.Google Scholar

Copyright information

© Institut Mittag-Leffler 2000

Authors and Affiliations

  • Mattias Jonsson
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations