**Preprint series:** 04-11, Reports on Algebra and Geometry

**MSC:**- 37F20 Combinatorics and topology
- 37E10 Maps of the circle
- 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations

**Abstract:** We study the combinatorics of distance doubling maps on

the circle ${\mathbb R}/{\mathbb Z}$ with prototypes

$h(\beta)=2\beta\bmod 1$ and $\bar{h}(\beta)=-2\beta\bmod 1$,

representing the orientation preserving and orientation reversing

case, respectively. In particular, we identify parts of the circle

where iterates $f^{\circ n}$ of a distance doubling map $f$

provide `distance doubling behavior'. The results include

well-known statements for $h$ related to the structure of the

Mandelbrot set $M$ and suggest some analogies to the structure of

the Tricorn - the `anti-holomorphic Mandelbrot set' - possibly

being related to the combinatorics of $\bar{h}$.

**Upload:** 2004-09-02

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