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Conformal Geometry and Dynamics

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Potential theory and a characterization of polynomials in complex dynamics


Authors: Yûsuke Okuyama and Małgorzata Stawiska
Journal: Conform. Geom. Dyn. 15 (2011), 152-159
MSC (2010): Primary 37F10; Secondary 31A05
DOI: https://doi.org/10.1090/S1088-4173-2011-00230-X
Published electronically: October 4, 2011
MathSciNet review: 2846305
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Abstract: We obtain a characterization of polynomials among rational functions on $ \mathbb{P}^1$ from the point of view of complex dynamics and potential theory. This characterization generalizes a theorem of Lopes. Our proof applies both classical and (dynamically) weighted potential theory.


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  • 1. BAKER, M., A finiteness theorem for canonical heights attached to rational maps over function fields, J. Reine Angew. Math. 626 (2009), 205-233. MR 2492995 (2011c:14075)
  • 2. BERTELOOT, F. AND MAYER, V., Rudiments de dynamique holomorphe, Vol. 7 of Cours Spécialisés [Specialized Courses], Société Mathématique de France, Paris (2001). MR 1973050 (2005b:37087)
  • 3. BROLIN, H., Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. MR 0194595 (33:2805)
  • 4. DEMARCO, L., Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326, 1 (2003), 43-73. MR 1981611 (2004f:32044)
  • 5. FREIRE, A., LOPES, A. AND MAñé, R., An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14, 1 (1983), 45-62. MR 736568 (85m:58110b)
  • 6. HUBBARD, J. H. AND PAPADOPOL, P., Superattractive fixed points in $ {\bf C}\sp n$, Indiana Univ. Math. J. 43, 1 (1994), 321-365. MR 1275463 (95e:32025)
  • 7. LALLEY, S. P., Brownian motion and the equilibrium measure on the Julia set of a rational mapping, Ann. Probab. 20, 4 (1992), 1932-1967. MR 1188049 (94f:58079)
  • 8. LYUBICH, M. JU., Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3, 3 (1983), 351-385. MR 741393 (85k:58049)
  • 9. LOPES, A. O., Equilibrium measures for rational maps, Ergodic Theory Dynam. Systems 6, 3 (1986), 393-399. MR 863202 (88e:58055)
  • 10. MAñé, R. AND DA ROCHA, L. F., Julia sets are uniformly perfect, Proc. Amer. Math. Soc. 116, 3 (1992), 251-257. MR 1106180 (92k:58229)
  • 11. MINDA, C. D., Regular analytic arcs and curves, Colloq. Math., 38, 1 (1977), 73-82. MR 0507798 (58:22536)
  • 12. OBA, M. K. AND PITCHER, T. S., A new characterization of the $ F$ set of a rational function, Trans. Amer. Math. Soc. 166 (1972), 297-308. MR 0297978 (45:7030)
  • 13. POPOVICI, I. AND VOLBERG, A., Rigidity of harmonic measure, Fund. Math. 150, 3 (1996), 237-244. MR 1405045 (97g:30023)
  • 14. RANSFORD, T., Potential theory in the complex plane, Cambridge University Press, Cambridge (1995). MR 1334766 (96e:31001)
  • 15. UEDA, T., Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan 46, 3 (1994), 545-555. MR 1276837 (95d:32030)
  • 16. ZDUNIK, A., Harmonic measure on the Julia set for polynomial-like maps, Invent. Math. 128, 2 (1997), 303-327. MR 1440307 (98k:58149)

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Additional Information

Yûsuke Okuyama
Affiliation: Division of Mathematics, Graduate School of Science and Technology, Kyoto Institute of Technology, Kyoto 606-8585 Japan

Małgorzata Stawiska
Affiliation: Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd., Lawrence, Kansas 66045
Address at time of publication: Mathematical Reviews, 416 Fourth St., Ann Arbor, Michigan 48103
Email: stawiska@umich.edu

DOI: https://doi.org/10.1090/S1088-4173-2011-00230-X
Keywords: Balanced measure, harmonic measure, complex dynamics, Lopes’ theorem, Brolin’s theorem, weighted potential theory
Received by editor(s): December 14, 2010
Published electronically: October 4, 2011
Additional Notes: The first author was partially supported by JSPS Grant-in-Aid for Young Scientists (B), 21740096.
The second author thanks the Department of Mathematics of the University of Kansas for supporting her as a Robert D. Adams Visiting Assistant Professor in the years 2008–2011.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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