Combinatorics of distance doubling maps
Volume 187 / 2005
Fundamenta Mathematicae 187 (2005), 1-35
MSC: Primary 37F20, 37E10; Secondary 37F45.
DOI: 10.4064/fm187-1-1
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 $\overline{h}(\beta)=-2\beta\bmod 1$, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates $f^{\circ n}$ of a distance doubling map $f$ exhibit “distance doubling behavior”. The results include well known statements for $h$ related to the structure of the Mandelbrot set $M$. For $\overline{h}$ they suggest some analogies to the structure of the tricorn, the “antiholomorphic Mandelbrot set”.
