## On Thurston’s core entropy algorithm

HTML articles powered by AMS MathViewer

- by Yan Gao PDF
- Trans. Amer. Math. Soc.
**373**(2020), 747-776 Request permission

## Abstract:

The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant extending topological entropy for real maps to complex polynomials, whence providing a new tool to study the parameter space of polynomials. The base is a combinatorial algorithm allowing for the computation of the core entropy given by Thurston but without supplying a proof. In this paper, we will describe his algorithm and prove its validity.## References

- R. L. Adler, A. G. Konheim, and M. H. McAndrew,
*Topological entropy*, Trans. Amer. Math. Soc.**114**(1965), 309–319. MR**175106**, DOI 10.1090/S0002-9947-1965-0175106-9 - Lluís Alsedà and MichałMisiurewicz,
*Semiconjugacy to a map of a constant slope*, Discrete Contin. Dyn. Syst. Ser. B**20**(2015), no. 10, 3403–3413. MR**3411531**, DOI 10.3934/dcdsb.2015.20.3403 - Ben Bielefeld, Yuval Fisher, and John Hubbard,
*The classification of critically preperiodic polynomials as dynamical systems*, J. Amer. Math. Soc.**5**(1992), no. 4, 721–762. MR**1149891**, DOI 10.1090/S0894-0347-1992-1149891-3 - Mario Bonk and Daniel Meyer,
*Expanding Thurston maps*, Mathematical Surveys and Monographs, vol. 225, American Mathematical Society, Providence, RI, 2017. MR**3727134**, DOI 10.1090/surv/225 - 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** - A. Douady and J. H. Hubbard,
*Exploring the Mandelbrot set: The Orsay Notes*, www.math.cornell.edu/~hubbard/OrsayEnglish.pdf. - Harry Furstenberg,
*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1–49. MR**213508**, DOI 10.1007/BF01692494 - D. Dudko and D. Schleicher,
*Core entropy of quadratic polynomials*, arXiv:1412.9760v1, 2014. - Gao Yan,
*Dynatomic periodic curve and core entropy for polynomials*, thesis, Angers University, 2013. - W. Jung,
*Core entropy and biaccessibility of quadratic polynomials*, 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** - 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 - M. Misiurewicz and W. Szlenk,
*Entropy of piecewise monotone mappings*, Studia Math.**67**(1980), no. 1, 45–63. MR**579440**, DOI 10.4064/sm-67-1-45-63 - Edwin E. Moise,
*Geometric topology in dimensions $2$ and $3$*, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR**0488059**, DOI 10.1007/978-1-4612-9906-6 - R. L. Moore,
*Concerning upper semi-continuous collections of continua*, Trans. Amer. Math. Soc.**27**(1925), no. 4, 416–428. MR**1501320**, DOI 10.1090/S0002-9947-1925-1501320-8 - Alfredo Poirier,
*Critical portraits for postcritically finite polynomials*, Fund. Math.**203**(2009), no. 2, 107–163. MR**2496235**, DOI 10.4064/fm203-2-2 - Alfredo Poirier,
*Hubbard trees*, Fund. Math.**208**(2010), no. 3, 193–248. MR**2650982**, DOI 10.4064/fm208-3-1 - 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 - William P. Thurston,
*On the geometry and dynamics of iterated rational maps*, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher. MR**2508255**, DOI 10.1201/b10617-3 - W. Thurston, H. Baik, Y. Gao, J. Hubbard, K. Lindsey, L. Tan, and D. Thurston,
*degree-$d$ invariant lamination*, arXiv:1906.05324. - J. S. Zeng,
*Criterion for rays landing together*, arXiv:1503.05931, 2015.

## Additional Information

**Yan Gao**- Affiliation: Mathematical School of Sichuan University, Chengdu 610064, People’s Republic of China
- MR Author ID: 1062873
- Email: gyan@scu.edu.cn
- Received by editor(s): March 26, 2016
- Received by editor(s) in revised form: August 17, 2016, and November 4, 2016
- Published electronically: October 17, 2019
- Additional Notes: The author was partially supported by NSFC grant No. 11501383.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 747-776 - MSC (2010): Primary 37B40, 37F10, 37F20
- DOI: https://doi.org/10.1090/tran/7122
- MathSciNet review: 4068248