Combinatorics of distance doubling maps

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 $\bar{h}$.