COURSES Alejandro B. Engel Department of Mathematics and Statistics Rochester Institute of Technology email: abesma@rit.edu Phone: (716) 475 2123 NICE MATH. RESEARCH OTHER

 Classic Sets Iterations of a complex function: Mandelbrot Sets and Julia Sets.By now you have an idea of what a Mandelbrot Set is. To create the color images of Mandelbrot Sets, some times -wrongly- called fractals, one starts with a complex one parameter map H(z;k). You can see how such a map looks like by going to either one of the three maps illustrated on the left. For instance, the Mandelbrot Map is F(z;c) = z2 + c; with z the variable and c the parameter. For short one just writes H(z) insted of H(z;k); this is the way you will see the maps in the pages hyperlinked on the left. The Mandelbrot Set: First we define two numbers: the maximum numbers of iterations maxit and the radius of the circle of divergence rdiv. In the complex plane of parameters k, each point gives a parameter value to map H(z). Starting at one critical point of H(z), one proceeds to iterate this point. The iterations are terminated either at maxit or when the iterates first leave the disc centered at the origin with radius rdiv -record this number of iterations as numit-. Now, associate with every number from 0 to maxit a color; preferably different color for different numbers. To 0 associate white and to maxit associate black. With this pallette you paint each pixel of your screen with color numit. The Julia Sets: As before, let us define two numbers: maxit and rdiv. Now, for a fixed parameter value let us look at the complex plane of variable values z. For each z iterate H(z) until either maxit is reached or until the iterates leave a circle of radius rdiv centered at the origin -as before, record this number as numit-. Paint the pixel in your screen, as before, with color numit. In this way, each parameter value generates its own Julia Set. There are "famous" Julia Sets. See if you discover one that will make you famous! Mandelbrot Sets: Mandelbrot Map Logistic Map Cubic Map Mandelbrot & Julia Sets: Mandelbrot Map Logistic Map Cubic Map For a given complex one parameter function, there is just one Mandelbrot Set. But, there are infinitely many Julia Sets. Sometimes, this is referred to as the dichotomy Mandelbrot Set-Julia Sets. These two type of sets exist in different spaces. The Mandelbrot Set lives in the complex plane of parameters, whereas the Julia Sets live in the complex plane of variables.