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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Lopes, A.O. and Mengue, J.K. 2014. Selection of measure and a large deviation principle for the general one-dimensionalXYmodel. Dynamical Systems, Vol. 29, Issue. 1, p. 24.

    Lopes, Artur O 1990. Dimension spectra and a mathematical model for phase transition. Advances in Applied Mathematics, Vol. 11, Issue. 4, p. 475.

    Lopes, Artur O. 1989. An analogy of the charge distribution on Julia sets with the Brownian motion. Journal of Mathematical Physics, Vol. 30, Issue. 9, p. 2120.

    Lopes, Artur O. 1989. The Dimension Spectrum of the Maximal Measure. SIAM Journal on Mathematical Analysis, Vol. 20, Issue. 5, p. 1243.

  • Ergodic Theory and Dynamical Systems, Volume 6, Issue 3
  • September 1986, pp. 393-399

Equilibrium measures for rational maps

  • Artur Oscar Lopes (a1)
  • DOI:
  • Published online: 01 September 2008

For a polynomial map the measure of maximal entropy is the equilibrium measure for the logarithm potential in the Julia set [1], [4].

Here we will show that in the case where f is a rational map such that f(∞) = ∞ and the Julia set is bounded, then the two measures mentioned above are equal if and only if f is a polynomial.

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[1]M. F. Barnsley , J. S. Geronimo & A. N. Harrington . Orthogonal polynomials associated with invariant measure on the Julia set. Bull. Amer. Math. Soc. 7 (1982).

[2]D. Bessis , D. Mehta & P. Moussa . Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings. Lett. Math. Phys. 6 (1982).

[4]A. Freire , A. Lopes & R. Mane . An invariant measure for rational maps. Bol. Soc. Bras. Mat. 14 (1) (1983).

[6]R. Mañé . On the uniqueness of the maximizing measure for rational maps. Bol.Soc. Bras. Mat. 14 (1) (1983).

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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