The Fractal Boundary of the Power Tower Function

We consider the function called the power tower function, defined by iterated exponentiation (or tetration) of the complex variable z. For real values x, it converges on the interval exp(−e)<x <exp(1/e). The function may be expressed as the inverse of the function x=y1/y, allowing an extension of the domain to 0< x <exp(1/e). It may also be expressed in terms of the Lambert W-function, enabling an analytical continuation to the complex plane.The boundary of the region of the complex plane for which the power tower converges to a finite value is fractal in nature. We show this by repeatedly zooming to higher magnifications, illustrating the deliciously intricate nature of the boundary.