Conformal measures for rational functions revisited

Manfred Denker; R. Mauldin; Z. Nitecki; Mariusz Urbański

Fundamenta Mathematicae (1998)

  • Volume: 157, Issue: 2-3, page 161-173
  • ISSN: 0016-2736

Abstract

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We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.

How to cite

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Denker, Manfred, et al. "Conformal measures for rational functions revisited." Fundamenta Mathematicae 157.2-3 (1998): 161-173. <http://eudml.org/doc/212283>.

@article{Denker1998,
abstract = {We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.},
author = {Denker, Manfred, Mauldin, R., Nitecki, Z., Urbański, Mariusz},
journal = {Fundamenta Mathematicae},
keywords = {rational function; Riemann sphere; conformal measure; Markov partition; thermodynamic formalism},
language = {eng},
number = {2-3},
pages = {161-173},
title = {Conformal measures for rational functions revisited},
url = {http://eudml.org/doc/212283},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Denker, Manfred
AU - Mauldin, R.
AU - Nitecki, Z.
AU - Urbański, Mariusz
TI - Conformal measures for rational functions revisited
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 2-3
SP - 161
EP - 173
AB - We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.
LA - eng
KW - rational function; Riemann sphere; conformal measure; Markov partition; thermodynamic formalism
UR - http://eudml.org/doc/212283
ER -

References

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  2. [BMO] A. M. Blokh, J. C. Mayer and L. G. Oversteegen, Limit sets and conformal measures, preprint, 1998. 
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  4. [DU] M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), 103-134. Zbl0718.58035
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  12. [Pr1] F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), 309-317. Zbl0787.58037
  13. [Pr2] F. Przytycki, Iterations of holomorphic Collet-Eckmann maps: Conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials, Trans. Amer. Math. Soc. 350 (1998), 717-742. Zbl0892.58063
  14. [Pr3] F. Przytycki, On measure and Hausdorff dimension of Julia sets for holomorphic Collet-Eckmann maps, in: Dynamical Systems (Montevideo, 1995), Pitman Res. Notes Math. 362, Longman, 1996, 167-181. Zbl0868.58063
  15. [Pr4] F. Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, preprint, 1997 
  16. [PU] F. Przytycki and M. Urbański, Fractal sets in the plane - ergodic theory methods, to appear. 
  17. [Sh] M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, preprint, 1991. 
  18. [U1] M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414. Zbl0807.58025
  19. [U2] M. Urbański, On some aspects of fractal dimensions in higher dimensional dynamics, preprint, 1995. 
  20. [U3] M. Urbański, Geometry and ergodic theory of conformal nonrecurrent dynamics, Ergodic Theory Dynam. Systems 17 (1997), 1449-1476. Zbl0894.58036

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