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Faculty Advisor

Joanna Furno

Abstract

We explore the complex dynamics of a family of polynomials defined on the complex plane by f(z) = azm(1+z/d)d where a is a complex number not equal to zero, and m and d are at least 2. These functions have three finite critical points, one of which has behavior that differs as we change our parameter values. We analyze the dynamical behavior at this critical point, with a particular interest in the structures that appear in the filled Julia set K(f) and the basin of infinity A_{\infty}(f). The behavior of the family is extremely sensitive to our inputs for a, m and d. We examine the connectedness of the filled Julia set, determine a region that is contained in the basin of infinity and a region contained in the component of the filled Julia set containing zero. Then we use these regions to find regions in the parameter space where the filled Julia set is disconnected or connected, respectively.

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