Complex Behavior in Digit Sequences Produced with Simple Iterated Maps

Christopher Maes
Massachusetts Institute of Technology

Iterated maps operating on x and the representation of x in base b are studied. Particular attention is paid to the reversal addition map given by x_ (n + 1) = x_n + R(x_n, b) (where R(y, b) is the reversal of y in base b). Qualitiative classes of behavior similar to those in cellular automata are found in successive terms in the sequence. An algorithm for classifying unique sequences and determining the behavior of a sequence is presented. All simple maps are enumerated and those maps which produce complex behavior are found. Carry propagation and dependence on initial conditions are analyzed. A conjecture on maps in the form f(x) + R((g(x), b) producing complex and thus universal behavior is presented.

Created by Mathematica  (April 20, 2004)

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