Ishikawa Iterates for Logarithmic Function

Abstract

ABSTRACT In this paper the dynamics of the complex logarithmic function is investigated using the Ishikawa iterates. The fractal images generated from the generalized transformation function z z c log( ) n , n 2 are analyzed. Keywords : . : Complex dynamics, Relative Superior Mandelbrot Set, Relative Superior Julia set, Ishikawa Iteration and Midgets. Equivalently, the Julia set is also closure of the set of the repelling periodic points. These two definitions clearly illustrates the chaotic 1. INTRODUCTION The fractals generated from the self-squared function, z z c 2 where z and c are complex quantities, have been studied extensively in the literature[2, 8, 9 & 10 ]. A multitude of interesting, intriguing and rich families of fractals are generated by changing the complex function Fz () . This paper explores the dynamics of a complex logarithmic function. In 1918, French mathematician Gaston Julia[12] investigated the iteration process of a complex function intensively and attained a Julia set, which is a landmark in the field of fractal theory. The object Mandelbrot set on the other hand was given by Benoit B. Mandelbrot [13 ] in 1979. Recently, R. L. Devaney [ 5], [6 ] and [7] studied widely the behavior of the exponential function and analyzed the Julia sets under different conditions. We briefly recall the well known result for the family of the quadratic polynomial