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  • Ergodic Theory and Dynamical Systems, Volume 3, Issue 3
  • September 1983, pp. 351-385

Entropy properties of rational endomorphisms of the Riemann sphere

  • M. Ju. Ljubich (a1)
  • DOI:
  • Published online: 01 September 2008

In this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.

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[2]R. Bowen , Entropy expansive mappings. Trans. Amer. Math. Soc. 164 (1972), 323332.

[3]R. Bowen . Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125136.

[6]H. Brolin . Invariant sets under iteration of rational functions. Arkiv Math. 6 (1965), 103144.

[8]M. Denker & T. Keane . Eine Bemerkung zur topologischen Entropie. Monatsh. Math. 85 (1978), 177183.

[9]E. J. Dinaburg . On the relations among various entropy characteristics of dynamical systems. Math. USSR Izvestija 5 (1971), #2, 337378.

[12]L. Goodwin . Topological entropy bounds measure-theoretic entropy. Proc. Amer. Math. Soc. 23 (1969), 679688.

[13]J. Guckenheimer . Endomorphisms of the Riemann sphere. Proc. Symp. Pure Math. 14 (1970), 95123.

[14]K. Jacobs . Fastperiodizitätseigenschaften allgemeiner Halbgruppen in Banachräumen. Math. Z. 67 (1957), 8392.

[15]M. V. Jakobson . Structure of polynomial mappings on a singular set. Math. USSR Sbornik 6 (1968), #1, 97114.

[16]M. V. Jakobson . On the problem of classification of polynomial endomorphisms of the plane. Math. USSR Sbornik 9 (1969), #3, 345369.

[20]Yu. I. Lyubich . On the boundary spectrum of contractions in Minkowski spaces. Siberian Math. J. 11 #2, (1970), 271279.

[21]M. Yu. Lyubich . Entropy of analytic endomorphisms of the Riemannian sphere. Fund. Anal. Appl. 15 #4, (1981), 300302.

[25]V. A. Rokhlin . Lectures on the entropy theory of measure-preserving transformations. Russian Math. Surveys 22 (1967), #5, 152.

[26]D. Ruelle . Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9 (1968), 267278.

[27]P. Walters . Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121153.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
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