Master teapots and entropy algorithms for the Mandelbrot set
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- by Kathryn Lindsey, Giulio Tiozzo and Chenxi Wu;
- Trans. Amer. Math. Soc. 378 (2025), 3297-3348
- DOI: https://doi.org/10.1090/tran/9346
- Published electronically: March 5, 2025
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Abstract:
We construct an analogue of W. Thurston’s “Master teapot” for each principal vein in the Mandelbrot set, and generalize geometric properties known for the corresponding object for real maps. In particular, we show that eigenvalues outside the unit circle move continuously, while we show “persistence” for roots inside the unit circle. As an application, this shows that the outside part of the corresponding “Thurston set” is path connected. In order to do this, we define a version of kneading theory for principal veins, and we prove the equivalence of several algorithms that compute the core entropy.References
- J. F. Alves and J. Sousa Ramos, Kneading theory for tree maps, Ergodic Theory Dynam. Systems 24 (2004), no. 4, 957–985. MR 2085386, DOI 10.1017/S014338570400015X
- Viviane Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, vol. 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1793194, DOI 10.1142/9789812813633
- Karen M. Brucks and Henk Bruin, Topics from one-dimensional dynamics, London Mathematical Society Student Texts, vol. 62, Cambridge University Press, Cambridge, 2004. MR 2080037, DOI 10.1017/CBO9780511617171
- Bodil Branner and Adrien Douady, Surgery on complex polynomials, Holomorphic dynamics (Mexico, 1986) Lecture Notes in Math., vol. 1345, Springer, Berlin, 1988, pp. 11–72. MR 980952, DOI 10.1007/BFb0081395
- Mathieu Baillif and André de Carvalho, Piecewise linear model for tree maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11 (2001), no. 12, 3163–3169. MR 1885262, DOI 10.1142/S0218127401004108
- Harrison Bray, Diana Davis, Kathryn Lindsey, and Chenxi Wu, The shape of Thurston’s master teapot, Adv. Math. 377 (2021), Paper No. 107481, 32. MR 4182912, DOI 10.1016/j.aim.2020.107481
- Andries E. Brouwer and Willem H. Haemers, Spectra of graphs, Universitext, Springer, New York, 2012. MR 2882891, DOI 10.1007/978-1-4614-1939-6
- Thierry Bousch, Paires de similtudes, Preprint, available from author’s webpage, 1988.
- Thierry Bousch, Connexité locale et par chemins hölderiens pour les systèmes itérés de fonctions, Preprint, available from the author’s webpage, 1992.
- Danny Calegari, Sarah Koch, and Alden Walker, Roots, Schottky semigroups, and a proof of Bandt’s conjecture, Ergodic Theory Dynam. Systems 37 (2017), no. 8, 2487–2555. MR 3719268, DOI 10.1017/etds.2016.17
- Adrien Douady and John Hubbard, Exploring the Mandelbrot set, The Orsay notes, 1984.
- A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 (French). MR 762431
- Adrien Douady, Topological entropy of unimodal maps: monotonicity for quadratic polynomials, Real and complex dynamical systems (Hillerød, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 464, Kluwer Acad. Publ., Dordrecht, 1995, pp. 65–87. MR 1351519
- Dzmitry Dudko and Dierk Schleicher, Core entropy of quadratic polynomials, Arnold Math. J. 6 (2020), no. 3-4, 333–385. With an appendix by Wolf Jung. MR 4181716, DOI 10.1007/s40598-020-00134-y
- Yan Gao, On Thurston’s core entropy algorithm, Trans. Amer. Math. Soc. 373 (2020), no. 2, 747–776. MR 4068248, DOI 10.1090/tran/7122
- Yan Gao and Giulio Tiozzo, The core entropy for polynomials of higher degree, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 7, 2555–2603. MR 4546475, DOI 10.4171/JEMS/1154
- Wolf Jung, Core entropy and biaccessibility of quadratic polynomials, Preprint online at arXiv:1401.4792, 2014.
- Tao Li, A monotonicity conjecture for the entropy of Hubbard trees, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–State University of New York at Stony Brook. MR 2712186
- Kathryn Lindsey and Chenxi Wu, A characterization of Thurston’s Master Teapot, Preprint online at arXiv:1909.10675, 2019.
- Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, DOI 10.1007/BF02392694
- Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
- John Milnor and William Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465–563. MR 970571, DOI 10.1007/BFb0082847
- Johannes Riedl, Arcs in multibrot sets, locally connected Julia sets and their construction by quasiconformal surgery, PhD thesis, TU München, 2001.
- Dierk Schleicher, Rational parameter rays of the Mandelbrot set, Astérisque 261 (2000), xiv–xv, 405–443 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755449
- Stefano Silvestri and Rodrigo A. Pérez, Accessibility of the boundary of the Thurston set, Exp. Math. 32 (2023), no. 2, 405–422. MR 4592955, DOI 10.1080/10586458.2021.1974984
- William P. Thurston, Hyungryul Baik, Gao Yan, John H. Hubbard, Kathryn A. Lindsey, Lei Tan, and Dylan P. Thurston, Degree-$d$-invariant laminations, What’s next?—the mathematical legacy of William P. Thurston, Ann. of Math. Stud., vol. 205, Princeton Univ. Press, Princeton, NJ, 2020, pp. 259–325. MR 4205644, DOI 10.2307/j.ctvthhdvv.15
- Daniel J. Thompson, Generalized $\beta$-transformations and the entropy of unimodal maps, Comment. Math. Helv. 92 (2017), no. 4, 777–800. MR 3718487, DOI 10.4171/CMH/424
- William P. Thurston, Entropy in dimension one, Frontiers in complex dynamics, Princeton Math. Ser., vol. 51, Princeton Univ. Press, Princeton, NJ, 2014, pp. 339–384. MR 3289916
- Giulio Tiozzo, Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Harvard University. MR 3167282
- Giulio Tiozzo, Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set, Adv. Math. 273 (2015), 651–715. MR 3311773, DOI 10.1016/j.aim.2014.12.033
- Giulio Tiozzo, Continuity of core entropy of quadratic polynomials, Invent. Math. 203 (2016), no. 3, 891–921. MR 3461368, DOI 10.1007/s00222-015-0605-9
- Giulio Tiozzo, Galois conjugates of entropies of real unimodal maps, Int. Math. Res. Not. IMRN 2 (2020), 607–640. MR 4073928, DOI 10.1093/imrn/rny046
- Jinsong Zeng, Criterion for rays landing together, Trans. Amer. Math. Soc. 373 (2020), no. 9, 6479–6502. MR 4155183, DOI 10.1090/tran/8088
Bibliographic Information
- Kathryn Lindsey
- Affiliation: Department of Mathematics, Boston College, Maloney Hall, Chestnut Hill, Massachusetts 02467
- MR Author ID: 842785
- ORCID: 0000-0001-8164-6791
- Email: lindseka@bc.edu
- Giulio Tiozzo
- Affiliation: Department of Mathematics, University of Toronto, 40 St George St, Toronto, Ontario, Canada
- MR Author ID: 907703
- ORCID: 0000-0003-1075-2890
- Email: tiozzo@math.utoronto.ca
- Chenxi Wu
- Affiliation: Department of Mathematics, University of Wisconsin at Madison, Madison, Wisconsin 53706
- MR Author ID: 1125473
- ORCID: 0000-0001-5856-6435
- Email: wuchenxi2013@gmail.com
- Received by editor(s): February 1, 2023
- Received by editor(s) in revised form: October 3, 2024
- Published electronically: March 5, 2025
- Additional Notes: The first author was partially supported by NSF grant #1901247
The second author was partially supported by NSERC grant RGPIN-2017-06521 and an Ontario Early Researcher Award “Entropy in dynamics, geometry, and probability” - © Copyright 2025 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 378 (2025), 3297-3348
- MSC (2020): Primary 37F20, 37F46, 37B40
- DOI: https://doi.org/10.1090/tran/9346