On Some One-Complex Dimensional Slices of the Boundedness Locus of a Multi-Parameter Rational Family - (:unkn) unknown (My University, 2017)
Complex dynamics involves the study of the behavior of complex-valued functions when they are composed with themselves repeatedly. We observe the orbits of a function by passing starting values through the function iteratively. Of particular interest are the orbits of any critical points of the function, called critical orbits. The behavior of a family of functions can be determined by examining the change in the critical orbit(s) of the functions as the values of the associated parameters vary. These behaviors are often separated into two categories: parameter values where one or more critical orbits remain bounded, and parameter values where all critical orbits are unbounded. A famous example of this is the Mandelbrot set, which consists of all c-values at which the sole critical orbit of the polynomial P_c(z) = z^2 + c is bounded. In this paper we discuss some dynamics of the family of complex rational functions Rnca (z) = z^n + a/z^n + c. If we fix the variables n and c while allowing a to vary, we see what look like small copies of the Mandelbrot set within the a-parameter plane. It turns out that for particular values of n, c, and a the function Rnca(z) behaves locally like a quadratic polynomial. We prove that at these parameter values the 'baby' Mandelbrot sets which appear are in fact homeomorphic copies of the original Mandelbrot set. We then examine other interesting parameter slices by fixing parameters or letting them vary in a predictable way. We once again observe the appearance of what look like Mandelbrot sets within these slices, and prove some properties regarding their locations.
Keywords: Complex Dynamics