Critical Points of the Multiplier Map for the Quadratic Family

Abstract

The multiplier λ_n of a periodic orbit of period n can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials $p_c(z)=z^2+c.$ We provide a numerical algorithm for computing critical points of this function (i.e., points where the derivative of the multiplier with respect to the complex parameter c vanishes). We use this algorithm to compute critical points of λ_n up to period n = 10.

Keywords:

• quadratic family
• multipliers of periodic orbits
• Mandelbrot set
• critical points of the multipliers

Acknowledgments

The authors would like to thank the Department of Mathematics at Uppsala University, where the main part of this work has been done. The authors would also like to thank Tanya Firsova for some valuable remarks and suggestions.

Notes

1 A Riemann mapping of a simply connected domain is a conformal diffeomorphism of the unit disk onto that domain.