Attractor sets and Julia sets in low dimensions

Fletcher, A.

doi:10.1090/ecgd/334

Attractor sets and Julia sets in low dimensions
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Conform. Geom. Dyn. 23 (2019), 117-129 Request permission

Abstract:

If $X$ is the attractor set of a conformal IFS (iterated function system) in dimension two or three, we prove that there exists a quasiregular semigroup $G$ with a Julia set equal to $X$. We also show that in dimension two, with a further assumption similar to the open set condition, the same result can be achieved with a semigroup generated by one element. Consequently, in this case the attractor set is quasiconformally equivalent to the Julia set of a rational map.

References

Israel Berstein and Allan L. Edmonds, On the construction of branched coverings of low-dimensional manifolds, Trans. Amer. Math. Soc. 247 (1979), 87–124. MR 517687, DOI 10.1090/S0002-9947-1979-0517687-9
Christopher J. Bishop and Kevin M. Pilgrim, Dynamical dessins are dense, Rev. Mat. Iberoam. 31 (2015), no. 3, 1033–1040. MR 3420484, DOI 10.4171/RMI/862
Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590
Kenneth Falconer, Fractal geometry, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications. MR 3236784
Alastair Fletcher, Quasiregular semigroups with examples, Discrete Contin. Dyn. Syst. 39 (2019), no. 4, 2157–2172. MR 3927507, DOI 10.3934/dcds.2019090
A. Fletcher and D. Macclure, Strongly automorphic mappings and Julia sets of uniformly quasiregular mappings, to appear in J. Anal. Math.
Alastair N. Fletcher and Daniel A. Nicks, Quasiregular dynamics on the $n$-sphere, Ergodic Theory Dynam. Systems 31 (2011), no. 1, 23–31. MR 2755919, DOI 10.1017/S0143385709001072
Alastair N. Fletcher and Daniel A. Nicks, Julia sets of uniformly quasiregular mappings are uniformly perfect, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 3, 541–550. MR 2838349, DOI 10.1017/S0305004111000478
Alastair Fletcher and Jang-Mei Wu, Julia sets and wild Cantor sets, Geom. Dedicata 174 (2015), 169–176. MR 3303046, DOI 10.1007/s10711-014-0010-3
Kevin G. Hare and Nikita Sidorov, Two-dimensional self-affine sets with interior points, and the set of uniqueness, Nonlinearity 29 (2016), no. 1, 1–26. MR 3460748, DOI 10.1088/0951-7715/29/1/1
A. Hinkkanen, Uniformly quasiregular semigroups in two dimensions, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 1, 205–222. MR 1375517
A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions. I, Proc. London Math. Soc. (3) 73 (1996), no. 2, 358–384. MR 1397693, DOI 10.1112/plms/s3-73.2.358
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
Tadeusz Iwaniec and Gaven Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 241–254. MR 1404085
Tadeusz Iwaniec and Gaven Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001. MR 1859913
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
Kathryn A. Lindsey, Shapes of polynomial Julia sets, Ergodic Theory Dynam. Systems 35 (2015), no. 6, 1913–1924. MR 3377290, DOI 10.1017/etds.2014.8
Kathryn A. Lindsey and Malik Younsi, Fekete polynomials and shapes of Julia sets, Trans. Amer. Math. Soc. 371 (2019), no. 12, 8489–8511. MR 3955554, DOI 10.1090/tran/7440
Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
Rich Stankewitz, Uniformly perfect analytic and conformal attractor sets, Bull. London Math. Soc. 33 (2001), no. 3, 320–330. MR 1817771, DOI 10.1017/S0024609301007950
Dennis Sullivan, Hyperbolic geometry and homeomorphisms, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp. 543–555. MR 537749
Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465–496. MR 624833
Pekka Tukia, On two-dimensional quasiconformal groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 73–78. MR 595178, DOI 10.5186/aasfm.1980.0530
P. Tukia and J. Väisälä, Lipschitz and quasiconformal approximation and extension, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 303–342 (1982). MR 658932, DOI 10.5186/aasfm.1981.0626
Feng Xie, Yongcheng Yin, and Yeshun Sun, Uniform perfectness of self-affine sets, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3053–3057. MR 1993212, DOI 10.1090/S0002-9939-03-06976-4