Some investigations on the dynamics of the family λz exp(z)

Abstract

The paper examines some properties of the dynamics of a family of transcendental entire functions, We prove that for any integer p there exist functions in the family with each having attractive p;-cycle. Moreover, the components of the bifurcation diagram are simply-connected. We characterize the Fatou sets of functions of the family having attractive fixed noints end show that these functions are structural stable

For the Julia set we prove that it is a null set for certain parameters and that the Hausdonf dimension of the Julia set is 2 for the entire family.

Keywords:

• Complex dynamical systems
• Iteration
• Mandelbrot set

AMS Subject Classifications:

• 39B12
• 30D05
• 26A18