Chip-Firing And A Devil’s Staircase
نویسنده
چکیده
The devil’s staircase – a continuous function on the unit interval [0,1] which is not constant, yet is locally constant on an open dense set – is the sort of exotic creature a combinatorialist might never expect to encounter in “real life.” We show how a devil’s staircase arises from the combinatorial problem of parallel chip-firing on the complete graph. This staircase helps explain a previously observed “mode locking” phenomenon, as well as the surprising tendency of parallel chip-firing to find periodic states of small period.
منابع مشابه
Parallel Chip-firing on the Complete Graph: Devil’s Staircase and Poincaré Rotation Number
We study how parallel chip-firing on the complete graph Kn changes behavior as we vary the total number of chips. Surprisingly, the activity of the system, defined as the average number of firings per time step, does not increase smoothly in the number of chips; instead it remains constant over long intervals, punctuated by sudden jumps. In the large n limit we find a “devil’s staircase” depend...
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