Surgery sequences and self-similarity of the Mandelbrot set

Abstract

We introduce an analog in the context of rational maps of the idea of hyperbolic Dehn surgery from the theory of Kleinian groups. A surgery sequence is a sequence of postcritically finite maps limiting (in a precise manner) to a postcritically finite map with at least one strictly preperiodic critical orbit. As an application of this idea we give a new and elementary proof of Tan Lei's theorem on the asymptotic self-similarity of Julia Sets and the Mandelbrot Set at Misiurewicz points.

9 pages; version 4: figures of Mandelbrot and Julia sets drawn in vector format (postscript) using a numerical ODE to trace an approximate level set of the distance function from the sets. These replace bitmap figures in the previous version