Combinatorics of distance doubling maps

We study the combinatorics of distance doubling maps on the circle ℝ/ℤ with prototypes h(β) = 2β mod 1 and h̄(β) = -2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates f^on 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 h̄ they suggest some analogies to the structure of the tricorn, the " antiholomorphic Mandelbrot set".