Arkiv för Matematik

  • Ark. Mat.
  • Volume 38, Number 2 (2000), 281-317.

Ergodic properties of fibered rational maps

Mattias Jonsson

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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 plane P2.

Note

Supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT).

Article information

Source
Ark. Mat., Volume 38, Number 2 (2000), 281-317.

Dates
Received: 17 December 1998
Revised: 17 September 1999
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898687

Digital Object Identifier
doi:10.1007/BF02384321

Mathematical Reviews number (MathSciNet)
MR1785403

Zentralblatt MATH identifier
1021.37019

Rights
2000 © Institut Mittag-Leffler

Citation

Jonsson, Mattias. Ergodic properties of fibered rational maps. Ark. Mat. 38 (2000), no. 2, 281--317. doi:10.1007/BF02384321. https://projecteuclid.org/euclid.afm/1485898687


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References

  • [AR] Abramov, L. M. and Rokhlin, 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.
  • [B] Bogenschütz, T., Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dynam. 1 (1992/93), 99–116.
  • [Bo] Bowen, R., Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414.
  • [Br] Brolin, H., Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144.
  • [D] Dabija, M., Böttcher divisors, Preprint, 1998.
  • [FS1] Fornæss, J. E. and Sibony, N., Random iterations of rational functions, Ergodic Theory Dynam. Systems 11 (1991), 687–708.
  • [FS2] Fornæss, J. E. and Sibony, N., Critically finite rational maps on P2, in The 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.
  • [FS3] Fornæss, J. E. and Sibony, N., Complex dynamics in higher dimension, in Complex Potential Theory (Gauthier, P. M. and Sabidussi, G., eds.), pp. 131–186, Kluwer, Dordrecht, 1994.
  • [FS4] Fornæss, J. E. and Sibony, N., Complex dynamics in higher dimension II, in Modern 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.
  • [FW] Fornæss, J. E. and Weickert, B., Random iterations in Pk, to appear in Ergodic Theory Dynam. Systems.
  • [FLM] Freire, A., Lopez, A. and Mañé, R., An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 45–62.
  • [G] Gromov, M., Entropy, homology and semialgebraic geometry, Astérisque 145–146 (1987), 225–240.
  • [H1] Heinemann, S.-M., Dynamische Aspekte holomorpher Abbildungen in Cn, Ph. D. Thesis, Göttingen, 1994.
  • [H2] Heinemann S.-M., Julia sets for holomorphic endomorphisms of CnErgodic Theory Dynam. Systems 16 (1996), 1275–1296.
  • [H3] Heinemann, S.-M., Julia sets of skew products in C2, Kyushu J. Math. 52 (1998), 299–329.
  • [HP] Hubbard, J. H. and Papadopol, P., Superattractive fixed points in Cn, Indiana Univ. Math. J. 43 (1994), 321–365.
  • [J] Jonsson, M., Dynamics of polynomial skew products on C2, Math. Ann. 314 (1999), 403–447.
  • [JW] Jonsson, M. and Weickert, B., A nonalgebraic attractor in P2, to appear in Proc. Amer. Math. Soc.
  • [K] Kifer, Y., Ergodic Theory of Radom Transformations, Progress in Probability and Statististics 10, Birkhäuser, Boston, Mass., 1986.
  • [KS] Kolyada, S. and Snoha, L., Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), 205–233.
  • [LW] Ledrappier, F. and Walters, P., A relativised variational principle for continuous transformations, J. London Math. Soc. 16 (1977), 568–576.
  • [L] Lyubich, M., Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351–385.
  • [M] Mañé, R., On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 27–43.
  • [MP] Misiurewicz, M. and Przytycki, F., Topological entropy and degree of smooth mappings, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 573–574.
  • [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.
  • [S1] Sester, O., Étude dynamique des polynômes fibrés, Ph. D. thesis, Université de Paris-Sud, 1997.
  • [S2] Sester, O., Hyperbolicité des polynômes fibrés, Bull. Soc. Math. France 127 (1999), 393–428.
  • [Su] Sumi, H., Skew product maps related to finitely generated rational semigroups, Preprint, 1999.
  • [U] Ueda, T., Complex dynamical systems on projective spaces, Preprint, 1994.
  • [W] Walters, P., An Introduction to Ergodic Theory, Graduate Texts in Math. 79, Springer-Verlag, New York-Berlin, 1982.
  • [Y] Young, L. S., Ergodic theory of differentiable dynamical systems, in Real and Complex Dynamical Systems (Branner, B. and Hjorth, P., eds.), pp. 293–336, Kluwer, Dordrecht, 1995.