Quadrature domains and the real quadratic family

Lazebnik, Kirill

doi:10.1090/ecgd/361

Quadrature domains and the real quadratic family
HTML articles powered by AMS MathViewer

by Kirill Lazebnik PDF

Conform. Geom. Dyn. 25 (2021), 104-125 Request permission

Abstract:

We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component of the Mandelbrot set can be conformally mated with a congruence subgroup of $\mathrm {PSL}(2,\mathbb {Z})$, and that this conformal mating is the Schwarz function of a simply connected quadrature domain.

References

Dov Aharonov and Harold S. Shapiro, Domains on which analytic functions satisfy quadrature identities, J. Analyse Math. 30 (1976), 39–73. MR 447589, DOI 10.1007/BF02786704
Lipman Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257–300. MR 348097, DOI 10.1112/blms/4.3.257
Bodil Branner and Núria Fagella, Quasiconformal surgery in holomorphic dynamics, Cambridge Studies in Advanced Mathematics, vol. 141, Cambridge University Press, Cambridge, 2014. With contributions by Xavier Buff, Shaun Bullett, Adam L. Epstein, Peter Haïssinsky, Christian Henriksen, Carsten L. Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky. MR 3445628
Shaun Bullett and Luna Lomonaco, Mating quadratic maps with the modular group II, Invent. Math. 220 (2020), no. 1, 185–210. MR 4071411, DOI 10.1007/s00222-019-00927-9
Shaun Bullett and Christopher Penrose, Mating quadratic maps with the modular group, Invent. Math. 115 (1994), no. 3, 483–511. MR 1262941, DOI 10.1007/BF01231770
Ethan M. Coven and William L. Reddy, Positively expansive maps of compact manifolds, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 96–110. MR 591178
Peter Ebenfelt, Björn Gustafsson, Dmitry Khavinson, and Mihai Putinar (eds.), Quadrature domains and their applications, Operator Theory: Advances and Applications, vol. 156, Birkhäuser Verlag, Basel, 2005. The Harold S. Shapiro anniversary volume; Papers from the conference held at the University of California at Santa Barbara, Santa Barbara, CA, March 2003. MR 2129731, DOI 10.1007/b137105
S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee, Dynamics of Schwarz reflections: the mating phenomena, arXiv:1811.04979, 2018.
S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee, Schwarz reflections and the Tricorn, arXiv:1812.01573, 2018.
Seung-Yeop Lee, Mikhail Lyubich, Nikolai G. Makarov, and Sabyasachi Mukherjee, Schwarz reflections and anti-holomorphic correspondences, Adv. Math. 385 (2021), Paper No. 107766, 88. MR 4250610, DOI 10.1016/j.aim.2021.107766
Russell Lodge, Mikhail Lyubich, Sergei Merenkov, and Sabyasachi Mukherjee, On dynamical gaskets generated by rational maps, kleinian groups, and schwarz reflections, arXiv:1912.13438, 2019.
Seung-Yeop Lee and Nikolai G. Makarov, Topology of quadrature domains, J. Amer. Math. Soc. 29 (2016), no. 2, 333–369. MR 3454377, DOI 10.1090/jams828
Kirill Lazebnik, Nikolai G. Makarov, and Sabyasachi Mukherjee, Univalent polynomials and Hubbard trees, Trans. Amer. Math. Soc. 374 (2021), no. 7, 4839–4893. MR 4273178, DOI 10.1090/tran/8387
Kirill Lazebnik, Nikolai G. Makarov, and Sabyasachi Mukherjee, Bers slices in families of univalent maps, arXiv:2007.02429, 2020.
M. Lyubich, Conformal geometry and dynamics of quadratic polynomials, vol I-II, in preparation. http://www.math.stonybrook.edu/~mlyubich/book.pdf.
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
Carsten Lunde Petersen and Daniel Meyer, On the notions of mating, Ann. Fac. Sci. Toulouse Math. (6) 21 (2012), no. 5, 839–876 (English, with English and French summaries). MR 3088260, DOI 10.5802/afst.1355