# Global topology of hyperbolic components: Cantor circle case

@article{Wang2016GlobalTO, title={Global topology of hyperbolic components: Cantor circle case}, author={Xiaoguang Wang and Yongcheng Yin}, journal={Proceedings of the London Mathematical Society}, year={2016}, volume={115} }

We prove that in the moduli space Md of degree d⩾2 rational maps, any hyperbolic component in the disconnectedness locus and of Cantor circle type is a finite quotient of R4d−4−n×Tn , where n is determined by dynamics. The proof uses some ideas from Riemann surface theory (Abel's Theorem), dynamical system and algebraic topology.

## 6 Citations

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We calculate the Hausdorff dimension 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…

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In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post‐critically finite hyperbolic tree mapping schemes for such maps, generalizing…

Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets

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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…

## References

SHOWING 1-10 OF 50 REFERENCES

Components of degree two hyperbolic rational maps

- Mathematics
- 1990

SummaryWe examine the structure of hyperbolic components and some boundary points of these for degree two rational maps. Rather than using quasi-conformal deformation theory, we use a technique from…

Bounded hyperbolic components of quadratic rational maps

- MathematicsErgodic Theory and Dynamical Systems
- 2000

Let $\mathcal{H}$ be a hyperbolic component of quadratic rational maps possessing two distinct attracting cycles. We show that $\mathcal{H}$ has compact closure in moduli space if and only if neither…

A characterization of hyperbolic rational maps

- Mathematics
- 2011

We give a topological characterization of rational maps with disconnected Julia sets. Our results extend Thurston’s characterization of postcritically finite rational maps. In place of iteration on…

Unbounded components in parameter space of rational maps

- Mathematics
- 2000

We apply the pinching construction to the study of boundaries of space of quasi<"onforni;il deformations of rational ma.ps.

The iteration of cubic polynomials Part I: The global topology of parameter space

- Mathematics
- 1988

Fonctions univalentes en dynamique analytique complexe. Torsion de la structure complexe. Topologie globale de l'espace parametre

The Riemann Surface

- Mathematics
- 1972

The purpose of this chapter is to show how to give a structure of analytic manifold to the set of points on a curve in the complex numbers, but our treatment also applies to more general fields like…

Dynamics of rational maps: a current on the bifurcation locus

- Mathematics
- 2001

Let fλ : P 1 → P1 be a family of rational maps of degree d > 1, parametrized holomorphically by λ in a complex manifold X. We show that there exists a canonical closed, positive (1,1)-current T on X…

Algebraic Topology

- Mathematics

The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.

A Course in Complex Analysis and Riemann Surfaces

- Mathematics
- 2014

From i to z: the basics of complex analysis From z to the Riemann mapping theorem: some finer points of basic complex analysis Harmonic functions Riemann surfaces: definitions, examples, basic…

The classification of polynomial basins of infinity

- Mathematics
- 2011

We consider the problem of classifying the dynamics of complex polynomials $f: \mathbb{C} \to \mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools ---…