Hostname: page-component-7dd5485656-bvgqh Total loading time: 0 Render date: 2025-10-31T03:37:15.865Z Has data issue: false hasContentIssue false
  • English
  • Français

Rational functions with no recurrent critical points

Published online by Cambridge University Press:  19 September 2008

Mariusz Urbański
Mariusz Urbański
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-5116, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Let h be the Hausdorff dimension of the Julia set of a rational map with no nonperiodic recurrent critical points. We give necessary and sufficient conditions for h-dimensional Hausdorff measure and h-dimensional packing measure of the Julia set to be positive and finite. We also show that either the Julia set is the whole Riemann sphere or h < 2 and that if a rational map (not necessarily with no nonperiodic recurrent critical points!) has a rationally indifferent periodic point, then h > 1/2.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 2 , June 1994 , pp. 391 - 414
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[ADU]Aaronson, J., Denker, M. & Urbański, M.. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (1993), 495–548.CrossRefGoogle Scholar
[BI]Blanchard, P.. Complex analytic dynamics of the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85–141.CrossRefGoogle Scholar
[Br]Brolin, H.. Invariant sets under iteration of rational functions. Ark. Mat. 6 (1965), 103–144.CrossRefGoogle Scholar
[DU1]Denker, M. & Urbański, M.. Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point. J. London Math. Soc. 43 (1991), 107–118.CrossRefGoogle Scholar
[DU2]Denker, M. & Urbański, M.. On absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points. Forum Math. 3 (1991), 561–579.CrossRefGoogle Scholar
[DU3]Denker, M. & Urbański, M.. On Sullivan's conformal measures for rational maps of the Riemann sphere. Nonlinearity 4 (1991), 365–384.CrossRefGoogle Scholar
[DU4]Denker, M. & Urbański, M.. On Hausdorff measures on Julia sets of subexpanding rational maps. Israel J. Math. 76 (1992), 193–214.CrossRefGoogle Scholar
[DU5]Denker, M. & Urbański, M.. Geometric measures for parabolic rational maps. Ergod. Th. and Dynam. Sys. 12 (1992), 53–66.CrossRefGoogle Scholar
[DU6]Denker, M. & Urbański, M.. The capacity of parabolic Julia sets. Math. Zeitsch. 211 (1992), 73–86.CrossRefGoogle Scholar
[F]Federer, H.F.. Geometric Measure Theory Springer: New York, 1969.Google Scholar
[G]Guzman, M.. Differentiation of integrals in ℝn. Springer Lecture Notes in Mathematics 481. Springer: New York, 1975.CrossRefGoogle Scholar
[H]Hille, E.. Analytic Function Theory. Ginn and Company: Boston-New York-Chicago-Atlanta-Dallas- Palo Alto-Toronto, 1962.Google Scholar
[Ma]Mañé, R.. On a theorem of Fatou. Preprint.Google Scholar
[Mi]Milnor, J.. Dynamics in one complex variable: Introductory lectures. Preprint. (1990).Google Scholar
[P]Przytycki, F.. Lyapunov characteristic exponents are non-negative. Proc. Amer. Math. Soc. 119(1) (1993), 309–317.CrossRefGoogle Scholar
[S1]Sullivan, D.. Conformal dynamical systems. In: Geometric Dynamics. Springer Lecture Notes in Mathematics 1007. Springer: New York, 1983. pp. 725–752.CrossRefGoogle Scholar
[S2]Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259–277.CrossRefGoogle Scholar