Correspondence and translation principles for the Mandelbrot set

Abstract

New insights into the combinatorial structure of the Mandelbrot set are given by `Correspondence' and `Translation' Principles both conjectured and partially proved by E. Lau and D. Schleicher. We provide complete proofs of these principles and discuss results related to them. Note: The `Translation' and `Correspondence' Principles given earlier turned out to be false in the general case. In April 1999, an errata was added to discuss which parts of the two statements are incorrect and which parts remain true.