rational which generates immediately and periodically this parity cycle is For example, the parity cycle has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction as the latter leads to the rational cycle Any cyclic permutation of is associated to one of the above fractions. For instance, the cycle is produced by the fraction For a one-to-one correspondence, a parity cycle should be irreducible, that is, not partitionable into identical sub-cycles. As an illustration of this, the parity cycle and its sub-cycle are associated to the same fraction when reduced to lowest terms. In this context, assuming the validity of the Collatz conjecture implies that and are the only parity cycles generated by positive whole numbers (1 and 2, respectively). If the odd denominator of a rational is not a multiple of 3, then all the iterates have the same denominator and the sequence of numerators can be obtained by applying the " " generalization of the Collatz function 2-adic extension The function is well-defined on the ring of 2-adic integers, where it is continuous and measure-preserving with respect to the 2-adic measure. Moreover, its dynamics is known to be ergodic. Define the parity vector function acting on as The function is a 2-adic isometry. Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that almost all trajectories are acyclic in . An equivalent formulation of the Collatz conjecture is that Iterating on real or complex numbers The Collatz map (with shortcut) can be viewed as the restriction to the integers of the smooth map The iterations of this map on the real line lead to a dynamical system, further investigated by Chamberland. He showed that the conjecture does not hold for positive real numbers since there are infinitely many fixed points, as well as orbits escaping monotonically to infinity. The function has two attracting cycles of period 2, and . Moreover, the set of unbounded orbits is conjectured to be of measure 0. Letherman, Schleicher, and Wood extended the study to the complex plane, where most of the points have orbits that diverge to infinity (colored region on the illustration). The boundary between the colored region and the black components, namely the Julia set of , is a fractal pattern, sometimes called the "Collatz fractal". Optimizations Time–space tradeoff The section As a parity sequence above