Dissonance of multiple Devil's staircases

摘要

The multiple Devil's staircase reported by Qu, Wu and He in 1997 was composed of many tower-like structures. Each of which include two branches of the conventional complete Devil's staircases that connect the "single-exit" top and bottom steps. Their analytic conclusion show that all the conventional Devil's staircases are confined by two smooth curves with similar function forms W proportional to 1/ln epsilon. These may be addressed as a kind of consonance. Our recent study found that it happens only in a few cases. Actually, in their system, 16 different kinds of tower branches exist in most parts of the parameter space. A lot of the steps lose the consonant property which are the so-called dissonant structures. The number of types of the corresponding dissonant branches is employed to describe the dissonance of the staircase. When the number of the discontinuous regions n, in the system develops, the dissonance of the staircase increases with 2n(3) - n rule. The numerical result shows that the conclusion is valid for a general discontinuous circle snap.