×

Symmetries of Julia sets of nondegenerate polynomial skew products on \(\mathbb{C}^{2}\). (English) Zbl 1205.37059

The paper originates from the idea that the Julia sets of any kind of functions and maps can have symmetries. For the Julia set of a polynomial the symmetries are rotations. This problem being solved i.e. determination of the group of symmetries and of the polynomials having the same Julia set, the paper deals with the same problem in two dimensions. There are considered non-degenerate polynomials skew products in \(\mathbb{C}^2\), existence of vertical Green functions and Böttcher functions of the map, symmetries of the Julia set. It is then shown that the suitable transformations preserving the Julia set are conjugate to the rotational product map. A necessary and sufficient condition is given for the group of symmetries to be infinite.

MSC:

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37C80 Symmetries, equivariant dynamical systems (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. Atela and J. Hu, Commuting polynomials and polynomials with same Julia set, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996), 2427–2432. · Zbl 1298.37034 · doi:10.1142/S0218127496001570
[2] A. F. Beardon, Symmetries of Julia sets, Bull. London Math. Soc. 22 (1990), 576–582. · Zbl 0694.30023 · doi:10.1112/blms/22.6.576
[3] C. Favre and V. Guedj, Dynamique des applications rationnelles des espaces multiprojectifs, Indiana Univ. Math. J. 50 (2001), 881–934. · Zbl 1046.37026 · doi:10.1512/iumj.2001.50.1880
[4] S.-M. Heinemann, Julia sets of holomorphic endomorphisms of \(\mathbf C^n,\) Ergodic Theory Dynam. Systems 16 (1996), 1275–1296. · Zbl 0874.32008 · doi:10.1016/j.adolescence.2007.02.007
[5] M. Jonsson, Dynamics of polynomial skew products on \(\mathbf C^2,\) Math. Ann. 314 (1999), 403–447. · Zbl 0940.37018 · doi:10.1007/s002080050301
[6] W. Schmidt and N. Steinmetz, The polynomials associated with a Julia set, Bull. London Math. Soc. 27 (1995), 239–241. · Zbl 0826.30020 · doi:10.1112/blms/27.3.239
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.