Backward stability for polynomial maps with locally connected Julia sets

Blokh, Alexander; Oversteegen, Lex

doi:10.1090/S0002-9947-03-03415-9

Backward stability for polynomial maps with locally connected Julia sets
HTML articles powered by AMS MathViewer

by Alexander Blokh and Lex Oversteegen PDF

Trans. Amer. Math. Soc. 356 (2004), 119-133 Request permission

Abstract:

We study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial $f$ with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure $\mu$ this easily implies that one of the following holds: 1. for $\mu$-a.e. $x\in J(f)$, $\omega (x)=J(f)$; 2. for $\mu$-a.e. $x\in J(f)$, $\omega (x)=\omega (c(x))$ for a critical point $c(x)$ depending on $x$.

References

P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
Alexander M. Blokh, The “spectral” decomposition for one-dimensional maps, Dynamics reported, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 4, Springer, Berlin, 1995, pp. 1–59. MR 1346496
A. Blokh and G. Levin, Growing trees, laminations and the dynamics on the Julia set, IHES Preprint IHES/M/99/77 (1999), pp. 1–40.
A. Blokh and G. Levin, On dynamics of vertices of locally connected polynomial Julia sets, Proc. Amer. Math. Soc. 130 (2002), no. 11, 3219–3230. MR 1912999, DOI 10.1090/S0002-9939-02-06698-4
A. Blokh, G. Levin, An inequality for laminations, Julia sets and “growing trees”, Erg. Th. and Dyn. Sys. 22 (2002), pp. 63–97.
A. M. Blokh and M. Yu. Lyubich, Measurable dynamics of $S$-unimodal maps of the interval, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 5, 545–573. MR 1132757, DOI 10.24033/asens.1636
Alexander M. Blokh, John C. Mayer, and Lex G. Oversteegen, Recurrent critical points and typical limit sets of rational maps, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1215–1220. MR 1485461, DOI 10.1090/S0002-9939-99-04721-8
Alexander M. Blokh, John C. Mayer, and Lex G. Oversteegen, Recurrent critical points and typical limit sets of rational maps, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1215–1220. MR 1485461, DOI 10.1090/S0002-9939-99-04721-8
A. Blokh, M. Misiurewicz, Attractors for graph critical rational maps, Trans. Amer. Math. Soc. 354 (2002), pp. 3639–3661.
Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
M. Denker, R. D. Mauldin, Z. Nitecki, and M. Urbański, Conformal measures for rational functions revisited, Fund. Math. 157 (1998), no. 2-3, 161–173. Dedicated to the memory of Wiesław Szlenk. MR 1636885, DOI 10.4064/fm_{1}998_{1}57_{2}-3_{1}_{1}61_{1}73
Manfred Denker and Mariusz Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), no. 2, 563–587. MR 1014246, DOI 10.1090/S0002-9947-1991-1014246-4
Adrien Douady, Descriptions of compact sets in $\textbf {C}$, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 429–465. MR 1215973
P. Grzegorczyk, F. Przytycki, and W. Szlenk, On iterations of Misiurewicz’s rational maps on the Riemann sphere, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), no. 4, 431–444. Hyperbolic behaviour of dynamical systems (Paris, 1990). MR 1096102
K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
G. Levin, On backward stability of holomorphic dynamical systems, Fund. Math. 158 (1998), no. 2, 97–107. MR 1656942, DOI 10.4064/fm-158-2-97-107
M. Yu. Lyubich, Typical behavior of trajectories of the rational mapping of a sphere, Dokl. Akad. Nauk SSSR 268 (1983), no. 1, 29–32 (Russian). MR 687919
Ricardo Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 1, 1–11. MR 1224298, DOI 10.1007/BF01231694
Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
John Milnor, On the concept of attractor, Comm. Math. Phys. 99 (1985), no. 2, 177–195. MR 790735, DOI 10.1007/BF01212280
E. Prado, Ergodicity of conformal measures for unimodal polynomials, Tech. Report 6, SUNY–Stony Brook, 1996, Institute for Mathematical Sciences.
Feliks Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999), no. 5, 2081–2099. MR 1615954, DOI 10.1090/S0002-9947-99-02195-9
Dennis Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725–752. MR 730296, DOI 10.1007/BFb0061443
W. Thurston, The combinatorics of iterated rational maps, Preprint (1985).
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880