## Institute for Mathematical Sciences

## Preprint ims97-13

** D. Schleicher**
* Rational Parameter Rays of the Mandelbrot Set*

Abstract: We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the
external angle of such a ray and the dynamics at the landing
point. Our proof is different from the original one, given by
Douady and Hubbard and refined by Lavaurs, in several ways: it
replaces analytic arguments by combinatorial ones; it does not
use complex analytic dependence of the polynomials with respect
to parameters and can thus be made to apply for non-complex
analytic parameter spaces; this proof is also technically
simpler. Finally, we derive several corollaries about
hyperbolic components of the Mandelbrot set.
Along the way, we introduce partitions of dynamical and
parameter planes which are of independent interest, and we
interpret the Mandelbrot set as a symbolic parameter space of
kneading sequences and internal addresses.

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