Exploring parameter spaces in complex dynamics
DOI:
https://doi.org/10.17268/sel.mat.2023.01.06Keywords:
Complex dynamics, Blaschke products, Mandelbrot set, Julia setAbstract
We show the structure of the parameter space for a family of rational maps containing Blaschke products. Through numerical simulations using the orbit of a single critical point, we reveal the existence of infinitely many Mandelbrot-like sets along the unit circle, as well as eight-like structures in other regions of parameter space. We pose some open questions related to the parameter space of these functions.
References
Horgan J. Mandelbrot set-to. Scientific American. 1990 Apr 1; 262(4):30-5.
Douady A, Hubbard JH. Exploring the mandelbrot set. the orsay notes. Publ. Math. Orsay. 1984.
Roesch P, Wang X, Yin Y. Moduli space of cubic Newton maps. Advances in Mathematics. 2017 Dec 15; 322:1-59.
Devaney R. Structure of the McMullen domain in the parameter planes for rational maps. Fundamenta Mathematicae. 2005; 3(185):267-85.
Devaney R. The McMullen domain: satellite Mandelbrot sets and Sierpinski holes. Conformal Geometry and Dynamics of the American Mathematical Society. 2007; 11(12):164-90.
Fagella N, Keen L. Stable components in the parameter plane of transcendental functions of finite type. The Journal of Geometric Analysis. 2021; 31(5):4816-55.
Garcia SR, Mashreghi J, Ross WT. Finite Blaschke products and their connections. Cham: Springer; 2018 May 24.
Herman MR. Sur la conjugaison diff´erentiable des diff´eomorphismes du cercle `a des rotations. Publications Math´ematiques de l'IHÉS. 1979; 49:5-233.
Milnor J. Hyperbolic components. Conformal dynamics and hyperbolic geometry. 2012; 573:183-232.
Boyd B, Boyd S. Dynamics explorer, Computer software[Internet]. 2012 Mar. 24 Available on https://sourceforge.net/projects/detool/
Milnor J. Dynamics in One Complex Variable.(AM-160):(AM-160)-. Princeton University Press; 2011 Feb 11.
Branner B, Fagella N, Buff X. Quasiconformal surgery in holomorphic dynamics. Cambridge University Press; 2014 Jan 23.
Canela J, Fagella N, Garijo A. Tongues in degree 4 Blaschke products. Nonlinearity. 2016 Sep 21; 29(11):3464.
Navarro PI. Dinámica de um produto de Blaschke[Tese de Douorado], Universidade de Sao Paulo; 2021.
Steinmetz N. Rational iteration: complex analytical dynamical systems.De gruter. InRational Iteration 2011 Jul 22. de Gruyter studies in mathematics; 1993.
Schleicher D, editor. Complex dynamics: families and friends. CRC Press; 2009 Nov 3.
Gallas JA. Structure of the parameter space of the Hénon map. Physical Review Letters. 1993 May 3; 70(18):2714.
Douady A, Hubbard JH. On the dynamics of polynomial-like mappings. InAnnales scientifiques de l'École normale supérieure 1985; Vol. 18(2):287-343).
McMullen C. The Mandelbrot set is universal, in “The Mandelbrot set, theme and variations”, LMS Lecture Note Series 274. Cambridge University Press, 2000.
Alves AM, Silva e Silva BP, Salarinoghabi M. Geometric limit of Julia set of a family of rational functions with odd degree. Dynamical Systems. 2021 Oct 2; 36(4):699-713.
Roesch P. On local connectivity for the Julia set of rational maps: Newton's famous example. Annals of mathematics. 2008; 168(1):127-74.
Qiu W, Wang X, Yin Y. Dynamics of McMullen maps. Advances in Mathematics. 2012 Mar 1; 229(4):2525-77.
Suarez PI. Julia-like sets in the parameter space of a family of quartic rational maps. Poster presented at Congress Complex Dynamics in the Tropics, IMPA; 2022 Nov 8- Dic 2; Rio de Janeiro. Brazil. Available from: https://impa.br/en_US/eventos-do-impa/2022-2/complex-dynamics-in-the-tropics/.
Buff X, Henriksen C. Julia sets in parameter spaces. Communications in Mathematical Physics. 2001 Jul; 220(2):333-75.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.
