Abstract

This work is concerned with the question whether the Mandelbrot set is computable. The computability notions that we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components satisfy certain natural computability conditions. We conclude that the two-sided distance function of the Mandelbrot set is computable if the famous hyperbolicity conjecture is true. We also formulate the question whether the distance function of the Mandelbrot set is computable in terms of the escape time