Epicycloids and Blaschke products

Abstract

It is well known that the bounding curve of the central hyperbolic component of the Multibrot set in the parameter space of unicritical degree d polynomials is an epicycloid with $d-1$ cusps. The interior of the epicycloid gives the polynomials of the form $z^d+c$ which have an attracting fixed point. We prove an analogous result for unicritical Blaschke products: in the parameter space of degree d unicritical Blaschke products, the parabolic functions are parameterized by an epicycloid with $d-1$ cusps and inside this epicycloid are the parameters which give rise to maps having an attracting fixed point in the unit disk. We further study in more detail the case when $d=2$ in which every Blaschke product is unicritical in the unit disk.

Keywords:

• Blaschke product
• parameter space
• epicycloid

AMS Subject Classifications:

• Primary: 30D05
• Secondary: 37F10
• 37F45

Acknowledgements

The first author would like to thank Department of Mathematical Sciences, Northern Illinois University for its hospitality during his visit. All authors would like to thank the referee for a particularly careful report, improving the readability of the paper and suggesting the normalization of degree two Blaschke products employed in Section 3.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported by China Scholarship Council and Beijing Institute of Technology. The second author was supported by the Simons Foundation [grant number #352034].