Self-Similar Cascades of Band Splittings of Linear Mod 1 Maps

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Abstract
A self-similar hierarchy of a new type is shown to exist for the two-parameter family of maps f(x) = βx + r, (0 ≤x ≤1, mod 1) with 2 ≥β> 1, 0 ≤r ≤2 − β. If 2 ≥ βQ > 1, (Q = 2, 3, 4, ⋯), then the interval [0, 1] is split into Q hands in a region of parameter space (β, r). Then the Q-th iterate fQ(x) for each band can be rescaled to take the form f̃Q(x) = βQx + R, (0 ≤x ≤1, mod 1) with 0 ≤R ≤2 − βQ. This is equivalent to the original family of maps. Therefore, each band is also split into q subbands if 2 ≥βqQ > 1, (q = 2, 3, 4, ⋯). Then the q-th iterate of f̃Q(x) for each subband can again be rescaled to take the form equivalent to the original family of maps. This is repeated ad infinitum as the critical line β= 1 is aproached, leading to a self-similar hierarchical structure of parameter space. Hence a self-similar cascade of Qn-band splittings is possible for any sequences of periods {Q1, Q2, ⋯} if we take an appropriate path to the critical line. If Qn = Q for each n, then the convergence ratio is δ= Q and the rescaling ratio at each band splitting point is α= 1/(21/Q − 1), where Q = 2, 3, 4, ⋯.

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