Dynamics of Complex Polynomials

The topic of my bachelor dissertation – which I wrote with Morten Bakkedal, a fellow student – was dynamics of Complex Polynomials, in particular quadratic polynomials. The quadratic polynomials can be written as Pc(z) = z2+c  through affine transformation. The dissertation explores the dynamic plane, where c is constant (Chapter 4).  And the parameter plane, where c is a variable (Chapter 6).

The parameter plane can be divided into values of c for which the sequence 

c, c2 +c, (c2 + c)2+c, . . .

goes to infinity and values for which it does not.

In this way the Mandelbrot set appears and the analysis of the parameter plane becomes an analysis of the famous set.

The dissertation shows some characteristic features of the Mandelbrot set e.g. that it is compact and its representation on the complex plane.

Further, the Mandelbrot set can be divided by the period of the c-values attracting cycles. In fact, the dissertation shows that the period of the attracting cycle is constant on each of the characteristic bulbs of the Mandelbrot set.

Chapter 5 is devoted to technical theorems that are required for the analysis of quadratic polynomials. It is shown that the dynamics of a general polynomial Pc is closely related to the dynamics of simple polynomials z2 near infinity. The development of the theory of complex functions has the past decades been closely connected to the increasing computer processing power. The dissertation takes advantage of this by illustrating concepts through computer graphics.

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