Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
Abstract
In this paper, we study holomorphic motion of Julia sets of polynomial-like maps. In particular, we prove that in the stable family of polynomial-like maps if all the continuos functions move continuously by parameter then the Julia sets move holomorphically. Moreover, we also study the relation between continuity of regular compact sets and their Green functions.
First Page
194
Last Page
201
References
1. Fabrizio Bianchi. Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree. Mathematische Annalen. Vol. 373, No. 3-4, pp. 901–928 (2019).
2. Fabrizio Bianchi, Tien-Cuong Dinh and Karim Rakhimov. Monotonicity of dynamical degrees for Hénon-like and polynomial-like maps (preprint) (2023). arXiv:2307.10665.
3. Fabrizio Bianchi and K.Rakhimov. Strong probabilistic stability in holomorphic families of endomorphisms of ℙk and polynomial-like maps. arXiv (2022).
4. Briend J.-Y. and J. Duval. Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de ℂℙk. Acta Math. Vol. 182, No. 2, pp. 143–157 (1999).
5. Tien-Cuong Dinh and Nessim Sibony. Dynamique des applications d’allure polynomiale. Journal de mathématiques pures et appliquées. Vol. 82, No. 4, pp. 367-423 (2003).
6. François Berteloot, Giovanni Bassanelli, Fabrizio Bianchi, and Christophe Dupont. Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of ℙk. No. 3, pp. 1-43 (2016).
7. Tien-Cuong Dinh and Nessim Sibony. Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings. Holomorphic dynamical systems. Eds. G. Gentili, J. Guenot, G. Patrizio, Lect. Notes in Math. Springer, Berlin, 165–294, (2010).
8. John-Erik Fornaess, Nessim Sibony. Dynamics of ℙ2 (examples). Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998). Contemporary Mathematics. Vol. 269, pp. 47-85 (2001).
9. Alexandre Freire, Artur Lopes, Ricardo Mañé. An invariant mesure for rational maps. Bulletin of the Brazilian Mathematical Society. Vol. 14, No. 1, pp. 45-62 (1983).
10. John H. Hubbard and Peter Papadopol. Superattractive fixed points in ℂn. Indiana University Mathematics Journal. Vol. 43, No. 1, pp. 321-365 (1994).
11. Mikhail Lyubich. Some typical properties of the dynamics of rational mappings. Russian Mathematical Surveys. Vol. 38, No. 5, pp. 154–155 (1983).
12. Mikhail Lyubich. Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory and Dynamical Systems. Vol. 3, pp. 351-385 (1983).
13. A.Sadullaev. Pluripotensial theory. Palmarium Academic Publishing (2012).
14. Ricardo Mañé, Paulo Sad, and Dennis Sullivan. On the dynamics of rational maps. Annales Scientifiques de l’École Normale Superiéure (4). Vol. 16, No. 2, pp. 193–217 (1983).
Recommended Citation
Sardor, Bazarbaev and Boymurodov, Sobir
(2023)
"Holomorphic motion of Julia sets of polynomial-like maps, and continuity of compact sets and their Green functions,"
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 6:
Iss.
4, Article 3.
DOI: https://doi.org/10.56017/2181-1318.1260