Abstract
We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψ and a unique exp(P(ψ) − ψ)-conformal measure, where P(ψ) denotes the topological pressure of ψ.
Similar content being viewed by others
References
Brück R.: Geometric properties of Julia sets of the composition of polynomials of the form z 2 + c n . Pacific J. Math. 198(2), 347–372 (2001)
Büger M.: Self-similarity of Julia sets of the composition of polynomials. Ergod. Th. Dyn. Syst. 17, 1289–1297 (1997)
Büger M.: On the composition of polynomials of the form z 2 + c n. Math. Ann. 310(4), 661–683 (1998)
Brück R., Büger M., Reitz S.: Random iterations of polynomials of the form z 2 + c n : connectedness of Julia sets. Ergod. Th. Dyn. Syst. 19(5), 1221–1231 (1999)
Denker M., Urbański M.: On the existence of conformal measures. Trans. AMS 328, 563–587 (1991)
Denker M., Urbański M.: Ergodic theory of equilibrium states for rational maps. Nonlinearity 4, 103–134 (1991)
Fornaess J.E., Sibony N.: Random iterations of rational functions. Ergod. Th. Dyn. Syst. 11, 687–708 (1991)
Hinkkanen A., Martin G.J.: The dynamics of semigroups of rational functions I. Proc. Lond. Math. Soc. 73(3), 358–384 (1996)
Lyubich M.: Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. Dyn. Syst. 3, 351–386 (1983)
Parry W.: Entropy and Generators in Ergodic Theory, Mathematics Lecture Note Series. Benjamin Inc., Reading (1969)
Ruelle D.: Thermodynamic formalism. Encyclopedia of Mathematics and Applications, vol. 5. Addison-Wesley, Reading (1979)
Stankewitz, R., Sumi, H.: Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups. http://arxiv.org/abs/0708.3187(preprint)
Sumi H.: Skew product maps related to finitely generated rational semigroups. Nonlinearity 13, 995–1019 (2000)
Sumi H.: Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. Dyn. Syst. 21, 563–603 (2001)
Sumi H.: Dimensions of Julia sets of expanding rational semigroups. Kodai Math. J. 28(2), 390–422 (2005)
Sumi H.: Semi-hyperbolic fibered rational maps and rational semigroups. Ergod. Th. Dyn. Syst. 26, 893–922 (2006)
Sumi, H.: Random dynamics of polynomials and devil’s-staircase-like functions in the complex plane. Appl. Math. Comput. 187 (2007), no. 1, pp. 489–500 (Proceedings paper)
Sumi, H.: Dynamics of postcritically bounded polynomial semigroups. http://arxiv.org/abs/math/0703591 (2007, preprint)
Sumi, H., Urbański, M.: Real analyticity of Hausdorff dimension for expanding rational semigroups. http://arxiv.org/abs/0707.2447 (2007, preprint)
Walters P.: An Introduction To Ergodic Theory. Springer, Heidelberg (1982)
Zhou W., Ren F.: The Julia sets of the random iteration of rational functions. Chin. Bull. 37(12), 969–971 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. G. Dani.
The research of M. Urbański was supported in part by the NSF Grant DMS 0400481. H. Sumi thanks University of North Texas for kind hospitality, during his stay there.
Rights and permissions
About this article
Cite this article
Sumi, H., Urbański, M. The equilibrium states for semigroups of rational maps. Monatsh Math 156, 371–390 (2009). https://doi.org/10.1007/s00605-008-0016-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-008-0016-8