Bifurcations of dynamic rays in complex polynomials of degree two

Abstract

Abstract In the study of bifurcations of the family of degree-two complex polynomials, attention has been given mainly to parameter values within the Mandelbrot set M (e.g., connectedness of the Julia set and period doubling). The reason for this is that outside M , the Julia set is at all times a hyperbolic Cantor set. In this paper weconsider precisely this, values of the parameter in the complement of M . We find bifurcations occurring not on the Julia set itself but on the dynamic rays landing on itfrom infinity. As the parameter crosses the external rays of M , in the dynamic plane the points of the Julia set gain and lose dynamic rays. We describe these bifurcations with the aid of a family of circle maps and we study in detail the case of the fixed points.