number as one with a non-repeating binary expansion. This is the chaotic case. If x0 is rational the image of x0 contains a finite number of distinct values within [0, 1) and the forward orbit of x0 is eventually periodic, with period equal to the period of the binary expansion of x0. Specifically, if the initial condition is a rational number with a finite binary expansion of k bits, then after k iterations the iterates reach the fixed point 0; if the initial condition is a rational number with a k-bit transient (k ≥ 0) followed by a q-bit sequence (q > 1) that repeats itself infinitely, then after k iterations the iterates reach a cycle of length q. Thus cycles of all lengths are possible. For example, the forward orbit of 11/24 is: which has reached a cycle of period 2. Within any subinterval of [0, 1), no matter how small, there are therefore an infinite number of points whose orbits are eventually periodic, and an infinite number of points whose orbits are never periodic. This sensitive dependence on initial conditions is a characteristic of chaotic maps. Periodicity via bit shifts The periodic and non-periodic orbits can be more easily understood not by working with the map directly, but rather with the bit shift map defined on the Cantor space . That is, the homomorphism is basically a statement that the Cantor set can be mapped into the reals. It is a surjection: every dyadic rational has not one, but two distinct representations in the Cantor set. For example, This is just the binary-string version of the famous 0.999... = 1 problem. The doubled representations hold in general: for any given finite-length initial sequence of length , one has The initial sequence corresponds to the non-periodic part of the orbit, after which iteration settles down to all zeros (equivalently, all-ones). Expressed as bit strings, the periodic orbits of the map can be seen to the rationals. That is, after an initial "chaotic" sequence of , a periodic orbit settles down into a repeating string of length . It is not hard to see that such repeating sequences correspond to rational numbers. Writing one then clearly has Tacking on the initial non-repeating sequence, one clearly has a rational number. In fact, every rational number can be expressed in this way: an initial "random" sequence, followed by a cycling repeat. That