The Exponent Set: The Third Fractal of the Mandelbrot Family

The Mandelbrot and Julia sets, defined by the iterative map zn+1=zn^2+c, are foundational objects in fractal geometry. We generalize this system to a three-parameter Mandelbrot family zn+1=zn^x+c, which generates three unique fractal sets by varying one of the complex parameters c, initial value z0, or complex exponent x with the rest fixed. When x=2, our family naturally reduces to the classic Mandelbrot-Julia system, ensuring consistency with existing theory. We focus on the exponent-varying set, named the Exponent Set (E-Set), which lacks systematic parameterization analysis in existing research. We conduct the first analysis of its two core properties: phase discontinuity of the divergence boundary and absence of a universal escape radius.