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Abstract: The Binet formula provides a mechanism for the Fibonacci numbers to be viewed as a function of a complex variable. The Binet formula may be generalized by using other bases and multiplicative parameters that also give functions of a complex variable. Thus, filled-in Julia sets that exhibit escape time may be constructed. Moreover, these functions have computable critical points and hence we can create escape time images of the critical point based upon the underlying multiplicative parameter. Like the classic Mandelbrot set, these parameter space images provide an atlas of Julia sets. [Copyright &y& Elsevier]

Newton''s transformation fw(z)=z−1/(wzw−1) containing only one complex parameter w (w≠0 or 1) is constructed from the transcendental mapping F(z)=ezw+c. Although the number of critical points of fw(z) is countably infinite, a method based on the Valid Critical Point Set, vcps={zk∈C−π, is discussed for the generation of the generalized Mandelbrot set M of fw(z). The petal fragments, the multi-period fragments, and the classical “Mandelbrot set” fragments are found in M. The dynamical characteristics of fw(z) for different values of w are analyzed. The relationship between the parameter w in a classical “Mandelbrot set” fragment and the structure of the corresponding filled-in Julia set is investigated. A large number of novel images of filled-in Julia sets are generated based on the rate of convergence of orbits. [Copyright &y& Elsevier]

We study a certain class of equations involving the exponential function. First, we check the behavior of the roots of the equations and show that Newton''s method for solving the equations has infinitely many attractors including the attractor at infinity. We also find some regions contained in the basins of attraction of the roots of the equations for Newton''s method and then visualize the dynamics of Newton''s method for finding the roots. [Copyright &y& Elsevier]

Complex numbers are redefined from the plane to two non-Euclidean geometries, namely the surfaces of the sphere and the torus. The properties of the iterated equation zn+1=znp+c are investigated in these geometries, where z,p and c are complex numbers. Plots of the number of iterations required by this equation for convergence to a fixed point appear fractal and resemble multiple copies (at different scales) of the Julia set obtained when the same equation is iterated in the complex plane. [Copyright &y& Elsevier]