Combinatorics of distance doubling maps

Abstract

We study the combinatorics of distance doubling maps on the circle R/Z with prototypes h(β) = 2β mod 1 and h(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 •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 h they suggest some analogies to the structure of the tricorn, the "antiholomorphic Mandelbrot set". c (resp. p c (R β c) = R h(β) c). This property indicates that the structure of J c is strongly related to h, and similarly the structure of J c to h. The "antidoubling" map h plays a similar role for antiquadratic maps in the complex plane as h for quadratic maps. A dynamic ray having exactly one accumulation point z in the Julia set is said to land at z. In case the Julia set is locally connected, each dynamic ray lands and the Julia set can be considered as the quotient of T with respect to the equivalence relation that identifies angles β 1 , β 2 iff the corresponding dynamic rays land at the same point. (For a general discussion of the structure of polynomial Julia sets see [8].) Here we are only interested in the landing behavior of rays corresponding to periodic angles β ∈ T (periodic with respect to h or h). The crucial point is that all dynamic rays for periodic angles land at a periodic point of the Julia set different from 0 and that only finitely many such rays land at the same periodic point. We come back to this fact when defining bifurcation chords. Kneading sequences. In order to describe the dynamics of a distance doubling map f on T we look at the orbits of angles and how they differ. For this we use kneading sequences, first introduced by Milnor and Thurston [12] for interval maps. The idea is to divide T appropriately into two half-circles. The kneading sequence of an angle indicates in which half-circle each of its iterates lies. Definition 1.1. Let f be a distance doubling map on T and for α ∈ T let β 1 , β 2 be the two angles with f (β 1) = f (β 2) = α. From the two open halfcircles ]β 1 , β 2 [ and ]β 2 , β 1 [, taken in counterclockwise direction, let T 0 be the one containing α and T 1 the other one. (In case α = β 1 set T 0 = ]β 1 , β 2 [.) The kneading sequence of α with respect to f is the sequence s 1 s 2 s 3. .. ∈ {0, 1, * } N defined by s i =      * if f •i−1 (α) ∈ {β 1 , β 2 }, 0 if f •i−1 (α) ∈ T 0 , 1 if f •i−1 (α) ∈ T 1 ,