Existence of the Mandelbrot set in the Parameter Planes of Certain Rational Functions:
Thursday, March 24 at 2pm
EMS Building, Room E408
In complex dynamics we compose a complex valued function with itself repeatedly and observe the orbits of values of that function. Particular interest is in the orbit of critical points of that function (critical orbits). One famous, studied example is the quadratic polynomial Pc(z) = z^2 + c and how changing the value of c makes a difference to the orbit of the critical point z = 0. The set of c values for which the critical orbit is bounded is called the Mandelbrot set.
Here we observe rational functions of the form Rn,a,c(z) = z^n + az^-n + c and their critical orbits. For the complex parameters a and c we consider fixing one while allowing the other to vary. It turns out that the parameter planes of these rational functions contain homeomorphic copies of the Mandelbrot set. We will discuss why and observe further characteristics of the parameter planes of these functions.