Chaos: Generating Complexity from Simplicity

The most commonly used mapping to illustrate the phenomenon of chaos is the map x → 2x mod (1). This map is known as the 'unilateral shift' because, in the binary number system this map shifts all digits to the left by one decimal place, and truncates the integer. The second most commonly used paradigm of chaos is the Smale horseshoe whose complexity is essentially the bilateral shift obtained when we simply shift without truncation in some symbol system. Neither of these paradigms fully explains chaos since shifts cannot generate complex orbits from simple (rational) initial conditions. How chaos generates complexity from simplicity is an essential part that needs explanation. Providing this explanation is the objective of this paper.