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1998 | 157 | 2-3 | 161-173
Tytuł artykułu

Conformal measures for rational functions revisited

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Rocznik
Tom
157
Numer
2-3
Strony
161-173
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-08-28
poprawiono
1998-03-15
Twórcy
autor
Bibliografia
  • [ADU] J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495-548.
  • [BMO] A. M. Blokh, J. C. Mayer and L. G. Oversteegen, Limit sets and conformal measures, preprint, 1998.
  • [DNU] M. Denker, Z. Nitecki and M. Urbański, Conformal measures and S-unimodal maps, in: Dynamical Systems and Applications, World Sci. Ser. Appl. Anal. 4, World Sci., 1995, 169-212.
  • [DU] M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), 103-134.
  • [DU1] M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere, ibid. 4 (1991), 365-384.
  • [DU2] M. Denker and M. Urbański, On absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math. 3 (1991), 561-579.
  • [DU3] M. Denker and M. Urbański, Geometric measures for parabolic rational maps, Ergodic Theory Dynam. Systems 12 (1992), 53-66.
  • [LM] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geom. 47 (1997), 17-94.
  • [Ma] R. Ma né, On the Bernoulli property of rational maps, Ergodic Theory Dynam. Systems 5 (1985), 71-88.
  • [MU] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. 73 (1996), 105-154.
  • [MM] C. T. McMullen, Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps, preprint, Stony Brook, 1997.
  • [Pr1] F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), 309-317.
  • [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.
  • [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.
  • [Pr4] F. Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, preprint, 1997
  • [PU] F. Przytycki and M. Urbański, Fractal sets in the plane - ergodic theory methods, to appear.
  • [Sh] M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, preprint, 1991.
  • [U1] M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414.
  • [U2] M. Urbański, On some aspects of fractal dimensions in higher dimensional dynamics, preprint, 1995.
  • [U3] M. Urbański, Geometry and ergodic theory of conformal nonrecurrent dynamics, Ergodic Theory Dynam. Systems 17 (1997), 1449-1476.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv157i2p161bwm
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