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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.24 no.3 Antofagasta Dec. 2005

http://dx.doi.org/10.4067/S0716-09172005000300006 

 

Proyecciones
Vol. 24, No 3, pp. 277-286, December 2005.
Universidad Católica del Norte
Antofagasta - Chile

 

AN IMPROVEMENT OF J. RIVERA-LETELIER RESULT ON WEAK HYPERBOLICITY ON PERIODIC ORBITS FOR POLYNOMIALS

 

FELIKS PRZYTYCKI *

Polish Academy of Sciences, Polonia


Abstract

We prove that for f : a rational mapping of the Riemann sphere of degree at least 2 and W a simply connected immediate basin of attraction to an attracting fixed point, if |(fn)'(p)| ³ Cn3+x for constants x > 0, C > 0 all positive integers n and all repelling periodic points p of period n in Julia set for f , then a Riemann mapping R : extends continuously to and FrW is locally connected. This improves a result proved by J. Rivera-Letelier for W the basin of infinity for polynomials, and 5 + x rather than 3 + x.

 

*Supported by Polish KBN grant 2P03A 03425

 

References

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[2] F. Przytycki, J. Rivera-Letelier, S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Inventiones Mathematicae 151.1, pp. 29-63, (2003).         [ Links ]

[3] F. Przytycki, J. Rivera-Letelier, S. Smirnov, Equality of pressures for rational functions, Ergodic Theory and Dynamical Systems 23, pp. 891- 914, (2004).         [ Links ]

[4] F. Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math. 80, pp. 161-179, (1985).         [ Links ]

[5] F. Przytycki, Riemann map and holomorphic dynamics, Invent. Math. 85, pp. 439-455, (1986).         [ Links ]

[6] F. Przytycki, Iterations of holomorphic Collet-Eckmann maps: Conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials, Transactions of the AMS 350.2, pp. 717—742, (1998).         [ Links ]

[7] F. Przytycki, Expanding repellers in limit sets for iteration of holomorphic functions, Fundamenta Math. 186. 1, pp. 85-96, (2005).         [ Links ]

[8] F. Przytycki, Hyperbolic Hausdorff dimension is equal to the minimal exponent of conformal measure on Julia set. A simple proof, Proceedings of Kyoto Conference, Feb. (2004).         [ Links ]

[9] F. Przytycki, M. Urba´nski. Fractals in the Plane, Ergodic Theory Methods. to appear in Cambridge University Press. Available on http://www.math.unt.edu/~urbanski and http://www.impan.gov.pl/~feliksp         [ Links ]

[10] F. Przytycki, M. Urba´nski, Porosity of Julia sets of non-recurrent and parabolic Collet-Eckmann rational functions, Annales Academiae Scientiarum Fennicae 26, pp. 125-154, (2001).         [ Links ]

[11] F. Przytycki, A. Zdunik, Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees techniques, Fundamenta Math. 145, pp. 65-77, (1994).         [ Links ]

[12] J. Rivera-Letelier. Weak hyperbolicity on periodic orbits for polynomials, C. R. Acad. Sci. Paris 334, pp. 1113-1118, (2002).         [ Links ]

 

Feliks Przytycki
Institute of Mathematics
Polish Academy of Sciences
ul. Šniadeckich 8
00-956 Warszawa
Poland
e-mail: feliksp@impan.gov.pl

Received : January 2005. Accepted : November 2005

 

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