Combinatorics of distance doubling maps

Karsten Keller¹ ; Steffen Winter²

¹ Mathematical Institute University of Lübeck Wallstr. 40 D-23560 Lübeck, Germany
² Mathematical Institute University of Jena D-07740 Jena, Germany

Fundamenta Mathematicae, Tome 187 (2005) no. 1, pp. 1-35.

Résumé

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”.

Détail

DOI : 10.4064/fm187-1-1

Keywords: study combinatorics distance doubling maps circle mathbb mathbb prototypes beta beta bmod overline beta beta bmod representing orientation preserving orientation reversing respectively particular identify parts circle where iterates circ distance doubling map exhibit distance doubling behavior results include known statements related structure mandelbrot set overline suggest analogies structure tricorn antiholomorphic mandelbrot set