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Rational Misiurewicz maps are rare. (English) Zbl 1185.37103
Summary: We show that the set of Misiurewicz maps has Lebesgue measure zero in the space of rational functions for any fixed degree \(d \geq 2\).

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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[1] Aspenberg, M.: The Collet-Eckmann condition for rational maps on the Riemann sphere. Ph. D. thesis, Stockholm, 2004, http://www.diva-portal.org/diva/getDocument?Urn_nbn_se_kth-diva-3788-2fulltext.pdf · Zbl 1287.37032
[2] Aspenberg M., Graczyk J.: Dimension and measure for semi-hyperbolic rational maps of degree 2. C. R. Acad. Sci. Paris Ser. I 347, 395–400 (2009) · Zbl 1215.37030
[3] Benedicks M., Carleson L.: On iterations of 1 ax 2 on (, 1). Ann. of Math. (2) 122(1), 1–25 (1985) · Zbl 0597.58016
[4] Benedicks M., Carleson L.: The dynamics of the Hénon map. Ann. of Math. (2) 133(1), 73–169 (1991) · Zbl 0724.58042
[5] Carleson L., Jones P.W., Yoccoz J.-C.: Julia and John. Bol. Soc. Brasil. Mat. (N.S.) 25(1), 1–30 (1994) · Zbl 0804.30023
[6] Chirka, E.M.: Complex Analytic Sets, Volume 46 of Mathematics and its Applications (Soviet Series). Dordrecht: Kluwer Academic Publishers Group, 1989, Translated from the Russian by R. A. M. Hoksbergen
[7] de Melo, W., van Strien, S.: One-Dimensional Dynamics. Volume 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Berlin: Springer-Verlag, 1993 · Zbl 0791.58003
[8] Graczyk J., Światek G., Kotus J.: Non-recurrent meromorhic functions. Fund. Math. 182, 269–281 (2004) · Zbl 1079.37041
[9] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library, New York: John Wiley & Sons Inc., 1994, Reprint of the 1978 original · Zbl 0408.14001
[10] Lattés S.: Sur l’itération des substitutions rationnelles et les fonctions de Poincaré. C. R. Acad. Sci. Paris 166, 26–28 (1918) · JFM 46.0522.01
[11] Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane. New York: Springer-Verlag, Second edition, 1973, Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126 · Zbl 0267.30016
[12] Mañé R., Sad P., Sullivan D.: On the dynamics of rational maps. Ann. Scient. de l’Ec. Norm. Sup. 16(2), 193–217 (1983)
[13] Mañé R.: On a theorem of Fatou. Bol. Soc. Brasil. Mat. (N.S.) 24(1), 1–11 (1993) · Zbl 0781.30023
[14] McMullen, C.T.: Complex Dynamics and Renormalization. Volume 135 of Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, 1994 · Zbl 0807.30013
[15] Mihalache, N.: La condition de Collet-Eckmann pour les orbites critiques recurrents. Ph.D. thesis, 2006, Orsay
[16] Misiurewicz M.: Absolutely continuous invariant measures for certain maps of an interval. Publ. Math. de l’ IHÉS 53, 17–51 (1981) · Zbl 0477.58020
[17] Przytycki, F.: On measure and Hausdorff dimension of Julia sets of holomorphic Collet-Eckmann maps. In: International Conference on Dynamical Systems (Montevideo, 1995), Volume 362 of Pitman Res. Notes Math. Ser., Harlow: Longman, 1996, pp. 167–181 · Zbl 0868.58063
[18] Rudin, W.: Real and Complex Analysis. New York: McGraw-Hill Book Co., Third edition, 1987 · Zbl 0925.00005
[19] Sands D.: Misiurewicz maps are rare. Commun. Math. Phys. 197(1), 109–129 (1998) · Zbl 0921.58015
[20] Shishikura, M., Lei, T.: An alternative proof of Mañé’s theorem on non-expanding Julia sets. In: The Mandelbrot set, theme and variations, Volume 274 of London Math. Soc. Lecture Note Ser., Cambridge: Cambridge Univ. Press, 2000, pp. 265–279 · Zbl 1062.37046
[21] van Strien S.: Misiurewicz maps unfold generically (even if they are critically non-finite). Fund. Math. 163(1), 39–54 (2000) · Zbl 0965.37038
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