Dyadic Transformation - Periodicity and Non-periodicity

Periodicity and Non-periodicity

Because of the simple nature of the dynamics when the iterates are viewed in binary notation, it is easy to categorize the dynamics based on the initial condition:

If the initial condition is irrational (as almost all points in the unit interval are), then the dynamics are non-periodic—this follows directly from the definition of an irrational 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 sub-interval 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.