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

top
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

top

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

top
  1. [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. Zbl0789.28010
  2. [BMO] A. M. Blokh, J. C. Mayer and L. G. Oversteegen, Limit sets and conformal measures, preprint, 1998. 
  3. [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. Zbl0856.58023
  4. [DU] M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), 103-134. Zbl0718.58035
  5. [DU1] M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere, ibid. 4 (1991), 365-384. Zbl0722.58028
  6. [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. Zbl0745.28008
  7. [DU3] M. Denker and M. Urbański, Geometric measures for parabolic rational maps, Ergodic Theory Dynam. Systems 12 (1992), 53-66. Zbl0737.58030
  8. [LM] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geom. 47 (1997), 17-94. Zbl0910.58032
  9. [Ma] R. Ma né, On the Bernoulli property of rational maps, Ergodic Theory Dynam. Systems 5 (1985), 71-88. 
  10. [MU] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. 73 (1996), 105-154. Zbl0852.28005
  11. [MM] C. T. McMullen, Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps, preprint, Stony Brook, 1997. 
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.