Periodic Patterns in the Binary Difference Field

A bst r a ct . T he di fference sequence of a binary sequence is th e bina ry sequence representing the presen ce of a difference in value at two neighboring sites in the or iginal seq uence . The difference field is the or dered ensemble of all difference seq uences aligne d one und er th e other. I t is equi valent to t he space-time pattern of a one-dimensional cellular au tom a ton un der a simple asymmet ric rule . Periodi c boundary conditions imposed at t he bou nd a.ries of the propagation net of changes, wh ich is ind uced by a fini te change of values in th e init ial state, give rise to pe riodic ban ds of t ilings along t hese boun da ry lines . Width and per iod of th ese bands evolve in a well-defined way, ex hibiting period and bandwidth doubling. A specia l kind of self-similarity is apparent, a nd t he pattern has a fractal skeleton. Periodic bou nd ary con ditions may resu lt from a conservation law imposed on t he states in the propagat ion net .