We study the combinatorics of distance doubling maps on the circle ℝ/ℤ with prototypes h(β) = 2β mod 1 and h̅(β) = -2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates 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 h̅ they suggest some analogies to the structure of the tricorn, the “antiholomorphic Mandelbrot set”.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-1-1,
author = {Karsten Keller and Steffen Winter},
title = {Combinatorics of distance doubling maps},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {1-35},
zbl = {1086.37024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-1-1}
}
Karsten Keller; Steffen Winter. Combinatorics of distance doubling maps. Fundamenta Mathematicae, Tome 185 (2005) pp. 1-35. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-1-1/