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Full-Text Articles in Dynamical Systems
Mapping Polynomial Dynamics, Devin Becker
Mapping Polynomial Dynamics, Devin Becker
DePaul Discoveries
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 …
Dynamics Of The Family Lambda Tan Z^2, Santanu Nandi
Dynamics Of The Family Lambda Tan Z^2, Santanu Nandi
Dissertations, Theses, and Capstone Projects
We prove some topological properties of the dynamical plane ($z$-plane) and a combinatorial structure of the parameter plane of a holomorphic family of meromorphic maps $\lambda \tan z^2$. In the dynamical plane, we prove that there is no Herman ring and the Julia set is a Cantor set for the map when the parameter is in the central capture component. Julia set is connected for the maps when the parameters are in other hyperbolic components. In the parameter plane, I prove that the capture components are simply connected and there are always four hyperbolic shell components attached to a virtual …
Geometric Limits Of Julia Sets Of Maps Z^N + Exp(2Πiθ) As N → ∞, Scott R. Kaschner, Reaper Romero, David Simmons
Geometric Limits Of Julia Sets Of Maps Z^N + Exp(2Πiθ) As N → ∞, Scott R. Kaschner, Reaper Romero, David Simmons
Scholarship and Professional Work - LAS
We show that the geometric limit as n → ∞ of the Julia sets J(Pn,c) for the maps Pn,c(z) = zn + c does not exist for almost every c on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle.