An improvement of j. Rivera-letelier result on weak hyperbolicity on periodic orbits for polynomials

Authors

  • Feliks Przytycki Polish Academy of Sciences.

DOI:

https://doi.org/10.4067/S0716-09172005000300006

Abstract

We prove that for f :img06-01.jpg a rational mapping of the Riemann sphere of degree at least 2 and Ω a simply connected immediate basin of attraction to an attracting fixed point, if |(f n)'(p)| ≥ Cn3+ξ for constants ξ > 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 : img06-02.jpg extends continuously to img06-03.jpg and FrΩ is locally connected. This improves a result proved by J. Rivera-Letelier for Ω the basin of infinity for polynomials, and 5 + ξ rather than 3 + ξ.

Author Biography

Feliks Przytycki, Polish Academy of Sciences.

Institute of Mathematics.

References

[1] J. Graczyk, S. Smirnov, Weak expansion and geometry of Julia sets, preprint 1999.

[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).

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

[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).

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

[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).

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

[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).

[9] F. Przytycki, M. Urbanski. 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

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

[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).

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

Published

2017-04-20

How to Cite

[1]
F. Przytycki, “An improvement of j. Rivera-letelier result on weak hyperbolicity on periodic orbits for polynomials”, Proyecciones (Antofagasta, On line), vol. 24, no. 3, pp. 277-286, Apr. 2017.

Issue

Section

Artículos