Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-27T19:41:40.108Z Has data issue: false hasContentIssue false
  • English
  • Français

Entropy properties of rational endomorphisms of the Riemann sphere

Published online by Cambridge University Press:  19 September 2008

M. Ju. Ljubich
M. Ju. Ljubich
Affiliation:
Department of Mathematics, Kharkov State University, Kharkov, USSR
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 3 , September 1983 , pp. 351 - 385
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Avez, A.. Proprietes ergodiques des endomorphisms dilatants de variétés compactes. C.R. Acad. Paris, ser A, 266 (1968), 610–612.Google Scholar
[2]Bowen, R., Entropy expansive mappings. Trans. Amer. Math. Soc. 164 (1972), 323–332.CrossRefGoogle Scholar
[3]Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.CrossRefGoogle Scholar
[4]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes Math. 470. Springer-Verlag: Berlin-Heidelberg-New York, 1975.CrossRefGoogle Scholar
[5]Bowen, R.. On axiom A diffeomorphisms. Conf. Board Math. Sci. 35 (1978).Google Scholar
[6]Brolin, H.. Invariant sets under iteration of rational functions. Arkiv Math. 6 (1965), 103–144.CrossRefGoogle Scholar
[7]Courant, R.. Geometrische Funktionen. Springer-Verlag: Berlin-Göttingen-Heidelberg-New York, 1964.Google Scholar
[8]Denker, M. & Keane, T.. Eine Bemerkung zur topologischen Entropie. Monatsh. Math. 85 (1978), 177–183.CrossRefGoogle Scholar
[9]Dinaburg, E. J.. On the relations among various entropy characteristics of dynamical systems. Math. USSR Izvestija 5 (1971), #2, 337–378.CrossRefGoogle Scholar
[10]Fatou, P.. Sur les équations fonctionneles. Bull. Soc. Math. France 47 (1919), 161–271; 48 (1920), 33–94 & 208–314.Google Scholar
[11]Goodman, T.. Relating topological entropy and measure entropy. Bull. London Math. Soc. 3 (1971), 176–180.Google Scholar
[12]Goodwin, L.. Topological entropy bounds measure-theoretic entropy. Proc. Amer. Math. Soc. 23 (1969), 679–688.CrossRefGoogle Scholar
[13]Guckenheimer, J.. Endomorphisms of the Riemann sphere. Proc. Symp. Pure Math. 14 (1970), 95–123.CrossRefGoogle Scholar
[14]Jacobs, K.. Fastperiodizitätseigenschaften allgemeiner Halbgruppen in Banachräumen. Math. Z. 67 (1957), 83–92.CrossRefGoogle Scholar
[15]Jakobson, M. V.. Structure of polynomial mappings on a singular set. Math. USSR Sbornik 6 (1968), #1, 97–114.CrossRefGoogle Scholar
[16]Jakobson, M. V.. On the problem of classification of polynomial endomorphisms of the plane. Math. USSR Sbornik 9 (1969), #3, 345–369.CrossRefGoogle Scholar
[17]Jakobson, M. V.. On the problem of classification of rational maps of the Riemann sphere. Uspehi Math. Nauk. 28 (1973), 247–248.Google Scholar
[18]Julia, G.. Mémoire sur l'iteration des fonctions rationneles. J. Math. Pure Appl. 8 (1918), 47–245.Google Scholar
[19]Lyubich, Yu. I.. Almost periodic functions in the spectral analysis of operators. Soviet Math. Doklady 1 (1960), 593–595.Google Scholar
[20]Lyubich, Yu. I.. On the boundary spectrum of contractions in Minkowski spaces. Siberian Math. J. 11 #2, (1970), 271–279.CrossRefGoogle Scholar
[21]Lyubich, M. Yu.. Entropy of analytic endomorphisms of the Riemannian sphere. Fund. Anal. Appl. 15 #4, (1981), 300–302.CrossRefGoogle Scholar
[22]Misiurewicz, M.. Diffeomorphisms without any measure with maximal entropy. Bull. Acad. Pol. Sci. 21 (1973), 903–910.Google Scholar
[23]Misiurewicz, M. & Przytycki, F.. Topological entropy and degree of smooth mappings. Bull. Acad. Pol. Sci. 25 (1977), 573–574.Google Scholar
[24]Montel, P.. Lecons sur les families normales de fonctions analytique et leurs applications. Gauthier-Villars: Paris, 1927.Google Scholar
[25]Rokhlin, V. A.. Lectures on the entropy theory of measure-preserving transformations. Russian Math. Surveys 22 (1967), #5, 1–52.CrossRefGoogle Scholar
[26]Ruelle, D.. Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9 (1968), 267–278.CrossRefGoogle Scholar
[27]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121–153.CrossRefGoogle Scholar