Equivalence of Collet–Eckmann conditions for slowly recurrent rational maps

Bylund, Mats

doi:10.1090/ecgd/378

Equivalence of Collet–Eckmann conditions for slowly recurrent rational maps
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by Mats Bylund

Conform. Geom. Dyn. 26 (2022), 235-242

DOI: https://doi.org/10.1090/ecgd/378

Published electronically: December 20, 2022

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Abstract:

In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet–Eckmann, second Collet–Eckmann, and topological Collet–Eckmann conditions are equivalent and also invariant under topological conjugacy.

References

Artur Avila and Carlos Gustavo Moreira, Statistical properties of unimodal maps: the quadratic family, Ann. of Math. (2) 161 (2005), no. 2, 831–881. MR 2153401, DOI 10.4007/annals.2005.161.831
Magnus Aspenberg, The Collet-Eckmann condition for rational functions on the Riemann sphere, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Tekn.dr.)–Kungliga Tekniska Hogskolan (Sweden). MR 2715770
Michael Benedicks and Lennart Carleson, On iterations of $1-ax^2$ on $(-1,1)$, Ann. of Math. (2) 122 (1985), no. 1, 1–25. MR 799250, DOI 10.2307/1971367
Michael Benedicks and Lennart Carleson, The dynamics of the Hénon map, Ann. of Math. (2) 133 (1991), no. 1, 73–169. MR 1087346, DOI 10.2307/2944326
P. Collet and J.-P. Eckmann, On the abundance of aperiodic behaviour for maps on the interval, Comm. Math. Phys. 73 (1980), no. 2, 115–160. MR 573469, DOI 10.1007/BF01198121
P. Collet and J.-P. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems 3 (1983), no. 1, 13–46. MR 743027, DOI 10.1017/S0143385700001802
Jacek Graczyk and Stas Smirnov, Collet, Eckmann and Hölder, Invent. Math. 133 (1998), no. 1, 69–96. MR 1626469, DOI 10.1007/s002220050239
Bing Gao and Weixiao Shen, Summability implies Collet-Eckmann almost surely, Ergodic Theory Dynam. Systems 34 (2014), no. 4, 1184–1209. MR 3227153, DOI 10.1017/etds.2012.173
Jacek Graczyk and Grzegorz Świątek, Lyapunov exponent and harmonic measure on the boundary of the connectedness locus, Int. Math. Res. Not. IMRN 16 (2015), 7357–7364. MR 3428965, DOI 10.1093/imrn/rnu161
Huaibin Li, Topological invariance of the Collet-Eckmann condition for one-dimensional maps, Nonlinearity 30 (2017), no. 5, 2010–2022. MR 3639298, DOI 10.1088/1361-6544/aa67a1
Stefano Luzzatto and Lanyu Wang, Topological invariance of generic non-uniformly expanding multimodal maps, Math. Res. Lett. 13 (2006), no. 2-3, 343–357. MR 2231123, DOI 10.4310/MRL.2006.v13.n3.a2
Feliks Przytycki and Steffen Rohde, Porosity of Collet-Eckmann Julia sets, Fund. Math. 155 (1998), no. 2, 189–199. MR 1606527
Feliks Przytycki and Steffen Rohde, Rigidity of holomorphic Collet-Eckmann repellers, Ark. Mat. 37 (1999), no. 2, 357–371. MR 1714763, DOI 10.1007/BF02412220
Feliks Przytycki and Juan Rivera-Letelier, Statistical properties of topological Collet-Eckmann maps, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 1, 135–178 (English, with English and French summaries). MR 2332354, DOI 10.1016/j.ansens.2006.11.002
Feliks Przytycki, Juan Rivera-Letelier, and Stanislav Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math. 151 (2003), no. 1, 29–63. MR 1943741, DOI 10.1007/s00222-002-0243-x
Feliks Przytycki, Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: on non-renormalizable quadratic polynomials, Trans. Amer. Math. Soc. 350 (1998), no. 2, 717–742. MR 1407501, DOI 10.1090/S0002-9947-98-01890-X
Feliks Przytycki, Hölder implies Collet-Eckmann, Astérisque 261 (2000), xiv, 385–403 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755448
Mary Rees, Positive measure sets of ergodic rational maps, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 383–407. MR 870689, DOI 10.24033/asens.1511
Juan Rivera-Letelier, The maximal entropy measure detects non-uniform hyperbolicity, Math. Res. Lett. 17 (2010), no. 5, 851–866. MR 2727614, DOI 10.4310/MRL.2010.v17.n5.a5