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Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets
- Source :
- Transactions of the American Mathematical Society. 374:5191-5223
- Publication Year :
- 2021
- Publisher :
- American Mathematical Society (AMS), 2021.
-
Abstract
- For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which are Cantor circles, and prove that for any Cantor circle hyperbolic component $\mathcal{H}$ in the space of rational maps, the infimum of the Hausdorff dimensions of the Julia sets of the maps in $\mathcal{H}$ is equal to the conformal dimension of the Julia set of any representative $f_0\in\mathcal{H}$, and that the supremum of the Hausdorff dimensions is equal to $2$.
- Subjects :
- Mathematics::Dynamical Systems
Degree (graph theory)
Mathematics::Complex Variables
Applied Mathematics
General Mathematics
010102 general mathematics
Hausdorff space
Mathematics::General Topology
01 natural sciences
Julia set
Infimum and supremum
Moduli space
Conformal dimension
Combinatorics
Hausdorff dimension
0101 mathematics
Uniformization (set theory)
Mathematics
Subjects
Details
- ISSN :
- 10886850 and 00029947
- Volume :
- 374
- Database :
- OpenAIRE
- Journal :
- Transactions of the American Mathematical Society
- Accession number :
- edsair.doi...........b09b8e0f511b1cca1a8566f925c7b3df