Self-Similar Cascades of Band Splittings of Linear Mod 1 Maps : Condensed Matter and Statistical Physics

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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≧β^Q>1, (Q=2, 3, 4,ゥ, then the interval [0, 1] is split into Q bands in a region of parameter space (β, r ). Then the Q-th iterate f^Q(x) for each band can be rescaled to take the form f^^~^Q(x)=β^Qx+R, (0≦x≦1, mod 1) with 0≦R≦2-β^Q. This is equivalent to the original family of maps. Therefore, each band is also split into q subbands if 2≧β^<qQ>>1, (q=2, 3, 4, ゥ). Then the q-th iterate of f^^~^Q(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 approached, leading to a self-similar hierarchical structure of parameter space. Hence a self-similar cascade of Q_n-band splittings is possible for any sequence of periods { Q_1, Q_2,ゥ} if we take an appropriate path to the critical line. If Q_n= Q for each n, then the convergence ratio is δ=Q and the rescaling ratio at each band-splitting point is a=1/ (2^<1/Q>-1), where Q=2, 3. 4, ゥ
理論物理学刊行会の論文
1984-03-25