Complex Polynomials Of Degree Research Articles

Bifurcations of dynamic rays in complex polynomials of degree two

AbstractIn 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.

On the minimum moduli of normalized polynomials with two prescribed values

WithPn denoting the set of complex polynomials of degree at mostn (n≥1), define, for any complex numberμ, the subset $$P_n (\mu ): = \{ p_n (z) \in P_n :p_n (0) = 1 and p_n (1) = \mu \} .$$ In this paper, we determine exactly the nonnegative quantity $$S_n (\mu ): = \mathop {\sup \{ \min |p_n (z)|\} }\limits_{p_n \in P_n (\mu )|z| \leqslant 1} ,$$ as a function ofn andμ. For fixedn≥2, the three-dimensional surface, generated by the points (Reμ, Imμ,Sn(μ)) for all complex numbersμ, has the interesting shape of a volcano.

Matrices, Eigenvalues, and Complex Projective Space

(1978). Matrices, Eigenvalues, and Complex Projective Space. The American Mathematical Monthly: Vol. 85, No. 9, pp. 727-733.